Financial Markets Atlas: Instrument Catalog

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One card per financial instrument of the flows pages: what it is, its conventions in each jurisdiction, its cash flows, how it is valued, the risks to monitor and the rules that apply, with the diagrams where it appears. The formulas open in a separate panel. The worked example of each card is the one of its diagram on the sell-side page; all numbers illustrative. The jurisdiction buttons switch the local conventions, rules and links.

FX›FX Spot

Formulas

Quotes and amounts

Pip value
pip value=N×0.0001\displaystyle \text{pip value} = N \times 0.0001

One pip is the fourth decimal of the rate (the second for USD/JPY); its value is in the terms currency.

NN
base currency notional

Worked example EUR 10,000,000 × 0.0001 = USD 1,000

Mid and spread
m=12 (b+a)s=a−b\displaystyle m = \tfrac{1}{2}\,(b + a) \qquad s = a - b

The market mid is the average of the best bid and offer; the spread is their difference.

b, ab,\ a
bid and offer

Worked example 1.0850 / 1.0851: mid 1.08505, spread 1 pip

Counter-currency amount
AUSD=NEUR×S\displaystyle A_{\text{USD}} = N_{\text{EUR}} \times S

The terms currency paid or received for the base currency notional at the deal rate.

SS
deal rate, USD per EUR

Worked example 10,000,000 × 1.0853 = USD 10,853,000

Dealer margin
margin=N (Sclient−Scover)\displaystyle \text{margin} = N\,\left(S_{\text{client}} - S_{\text{cover}}\right)

What the dealer keeps when it buys back in the market the currency it sold to the client.

Worked example 10,000,000 × (1.0853 − 1.0851) = USD 2,000; electronic, against the mid: USD 1,000

Inverse quote
(1S)bid=1Soffer(1S)offer=1Sbid\displaystyle \left(\tfrac{1}{S}\right)^{\text{bid}} = \frac{1}{S^{\text{offer}}} \qquad \left(\tfrac{1}{S}\right)^{\text{offer}} = \frac{1}{S^{\text{bid}}}

Turning a pair around swaps the bid and the offer.

Worked example USD/EUR 0.92157 / 0.92166

Cross rates

Cross through USD, USD in the middle
SEUR/JPY=SEUR/USD×SUSD/JPYbEUR/JPY=bEUR/USD bUSD/JPY\displaystyle S_{\text{EUR/JPY}} = S_{\text{EUR/USD}} \times S_{\text{USD/JPY}} \qquad b_{\text{EUR/JPY}} = b_{\text{EUR/USD}}\,b_{\text{USD/JPY}}

A cross is priced from the two USD pairs; when USD is the terms of one and the base of the other, the bids multiply and the offers multiply.

Worked example USD/JPY 152.40 / 152.41: EUR/JPY 165.35 / 165.38

Cross through USD, USD the terms of both
SEUR/GBP=SEUR/USDSGBP/USDbEUR/GBP=bEUR/USDaGBP/USD\displaystyle S_{\text{EUR/GBP}} = \frac{S_{\text{EUR/USD}}}{S_{\text{GBP/USD}}} \qquad b_{\text{EUR/GBP}} = \frac{b_{\text{EUR/USD}}}{a_{\text{GBP/USD}}}

When USD is the terms currency of both pairs the cross divides them, and the bid divides a bid by an offer.

Worked example GBP/USD 1.2950 / 1.2951: EUR/GBP 0.83777 / 0.83792

No triangular arbitrage
bEUR/USD bUSD/JPY≤SEUR/JPY≤aEUR/USD aUSD/JPY\displaystyle b_{\text{EUR/USD}}\,b_{\text{USD/JPY}} \le S_{\text{EUR/JPY}} \le a_{\text{EUR/USD}}\,a_{\text{USD/JPY}}

A cross quoted outside the bounds of its two legs can be traded against them at a profit, so the pricing engines keep crosses inside them.

Value dates and rolls

Spot date
Tspot=T+2 business days(T+1 for USD/CAD)\displaystyle T_{\text{spot}} = T + 2\ \text{business days} \qquad (T + 1\ \text{for USD/CAD})

Business days in both currencies’ centers, and a New York business day for USD.

TT
trade date

Worked example Oct 28, 2026 + 2 = Oct 30, 2026

Today and tomorrow
Ftom=S−pTNFtoday=S−pTN−pON\displaystyle F_{\text{tom}} = S - p_{\text{TN}} \qquad F_{\text{today}} = S - p_{\text{TN}} - p_{\text{ON}}

Value before the spot date is priced off spot less the points of the days in between.

pTN, pONp_{\text{TN}},\ p_{\text{ON}}
tom-next and overnight points

Worked example tom-next about 0.66 pips a day at the rates of the example

Tom-next roll
Croll=N pTN≈N S (rd−rf) 1360\displaystyle C_{\text{roll}} = N\,p_{\text{TN}} \approx N\,S\,(r_d - r_f)\,\frac{1}{360}

A position carried past its value date is rolled with a tom-next swap: the holder of the lower-yielding currency pays the points for each day.

rd, rfr_d,\ r_f
USD and EUR overnight rates

Worked example a long EUR 10,000,000 position pays about USD 660 a day

Settlement

Gross settlement exposure
E=Npaidfrom the payment of one leg to the receipt of the other\displaystyle E = N_{\text{paid}} \quad \text{from the payment of one leg to the receipt of the other}

Settled gross, the whole principal of the leg paid first is at risk until the counter-currency arrives (Herstatt risk).

Worked example EUR 10,000,000, about USD 10,850,500, from the European morning to the New York afternoon

Net pay-in in CLS
Pc=max⁡ ⁣(0, ∑kpayk,c−∑krcvk,c)\displaystyle P_c = \max\!\Big(0,\ \sum_k \text{pay}_{k,c} - \sum_k \text{rcv}_{k,c}\Big)

CLS settles each trade gross across its members’ accounts but funds them with one net pay-in per currency; a member long a currency receives a pay-out.

cc
currency
kk
the member’s trades

Worked example the bank: EUR 10,000,000 − 10,000,000 = 0; USD 10,853,000 − 10,851,000 = 2,000 to receive

Liquidity saving
1−∑c∣netc∣∑cgrossc\displaystyle 1 - \frac{\sum_c \left|\text{net}_c\right|}{\sum_c \text{gross}_c}

The share of the gross payments that the netting removes, all currencies in USD at spot.

Worked example the bank’s two trades: 99.995%

Valuation and risk

Revaluation of a balance
V=B×Sclose\displaystyle V = B \times S_{\text{close}}

A currency balance is translated at the closing rate.

BB
balance in the foreign currency
Unsettled trade
V=N (Fclose(Tv)−Sdeal) Pd(t,Tv)\displaystyle V = N\,\left(F_{\text{close}}(T_v) - S_{\text{deal}}\right)\,P_d(t, T_v)

A trade not yet settled is a short-dated forward: the closing outright to its value date less the deal rate, discounted to today.

TvT_v
value date
PdP_d
USD discount factor

Worked example the client’s trade at the closing mid 1.08505: −USD 2,499, the margin and half the spread

Position P&L
P&L=N (St−S0)\displaystyle \text{P\&L} = N\,(S_t - S_0)

An open position gains or loses one pip value per pip.

Worked example unhedged, the bank short EUR 10,000,000 at 1.0853 loses USD 1,000 for each pip the EUR rises

Net open position
NOP=max⁡ ⁣(∑cLc, ∑c∣Sc∣)\displaystyle \text{NOP} = \max\!\Big(\sum_c L_c,\ \sum_c \left|S_c\right|\Big)

The larger of the sum of the net long positions and the sum of the net short positions, in the reporting currency: the measure of the FX limits and of the standardized capital rules.

Lc, ScL_c,\ S_c
net long and net short position in currency c

Worked example zero for the bank once the cover is done

Market data of the worked example, illustrative: EUR/USD 1.0850 / 1.0851, USD rate 4.09% and EUR rate 1.90% (Act/360); USD/JPY 152.40 / 152.41 and GBP/USD 1.2950 / 1.2951 for the crosses.
FX›FX Forward

Formulas

Forward price (covered interest parity)

Outright
F=S+p\displaystyle F = S + p

The forward rate is spot plus the forward points of the tenor.

pp
forward points, in units of the rate

Worked example 1.0851 + 0.0120 = 1.0971 offer; the client 1.0974

Covered interest parity
F=S 1+rd τ1+rf τ\displaystyle F = S\,\frac{1 + r_d\,\tau}{1 + r_f\,\tau}

Buying the base currency forward costs the same as borrowing the terms currency, buying spot and depositing the base currency to the date.

rd, rfr_d,\ r_f
terms (USD) and base (EUR) money-market rates
τ\tau
days/360 for USD and EUR, days/365 for GBP and CAD

Worked example 1.08505 × (1 + 4.09% × 182/360) / (1 + 1.90% × 182/360) = 1.09695

Forward points
p=F−S=S (rd−rf) τ1+rf τ\displaystyle p = F - S = S\,\frac{(r_d - r_f)\,\tau}{1 + r_f\,\tau}

The points carry the interest differential: positive when the terms currency pays the higher rate.

Worked example +119.0 pips at mid, quoted +118 / +120

Implied rate
rd=[FS (1+rf τ)−1]1τ\displaystyle r_d = \left[\frac{F}{S}\,(1 + r_f\,\tau) - 1\right]\frac{1}{\tau}

The terms currency rate that the points and the base currency rate imply.

Worked example 4.09% from the mid points and EUR at 1.90%

Annualized premium
F−SS⋅360d\displaystyle \frac{F - S}{S}\cdot\frac{360}{d}

The forward premium of the base currency as a yearly rate, close to the rate differential.

dd
days to the value date

Worked example 2.17% a year

Dates, curve and basis

Points for other dates
p(d)=p1+(p2−p1) d−d1d2−d1\displaystyle p(d) = p_1 + (p_2 - p_1)\,\frac{d - d_1}{d_2 - d_1}

Points for a date between two tenors are interpolated, linearly in days or on the two interest curves.

d1, d2d_1,\ d_2
days to the surrounding tenors

Worked example Feb 26, 2027, 119 days: 59 + (119 − 91) / (182 − 91) × (119 − 59) = +77.5 pips at mid

Discount factor form
F=S Pf(0,T)Pd(0,T)\displaystyle F = S\,\frac{P_f(0,T)}{P_d(0,T)}

The same parity with the two currencies’ discount factors to the value date, as the pricing engines compute it.

Pd, PfP_d,\ P_f
USD and EUR discount factors
Cross-currency basis
F=S 1+rd τ1+(rf+b) τ\displaystyle F = S\,\frac{1 + r_d\,\tau}{1 + (r_f + b)\,\tau}

Parity with the cash rates holds only up to a basis b: the premium paid to borrow USD through the FX market, negative for EUR/USD when USD funding is scarce.

bb
cross-currency basis on the base currency rate

Worked example with USD cash at 3.95% (illustrative), the +119 points imply b = −0.14%

Valuation

Mark to market
Vt=N (Ft(T)−K) Pd(t,T)\displaystyle V_t = N\,\left(F_t(T) - K\right)\,P_d(t, T)

The forward is worth the difference between today’s outright to its value date and the contract rate, on the notional, discounted.

KK
contract rate
Ft(T)F_t(T)
outright today to the value date

Worked example Jan 27, 2027, spot 1.1050: 10,000,000 × (1.1111 − 1.0974) × 0.98954 = +USD 135,447

Value at the trade
V0=N (F0mid−K) Pd(0,T)\displaystyle V_0 = N\,\left(F_0^{\text{mid}} - K\right)\,P_d(0, T)

The client’s mark at mid on the trade date: the dealer’s margin and half the spread.

Worked example 10,000,000 × (1.09695 − 1.0974) × 0.97952 = −USD 4,418

Delta
Δ=N Pf(0,T)\displaystyle \Delta = N\,P_f(0, T)

The spot-equivalent position in the base currency: the notional discounted at its own rate.

Worked example EUR 9,903,823

Points sensitivity
∂V∂(rd−rf)≈N S τ1+rf τ Pd(0,T)\displaystyle \frac{\partial V}{\partial (r_d - r_f)} \approx N\,S\,\frac{\tau}{1 + r_f\,\tau}\,P_d(0,T)

The change in value for a change in the rate differential, quoted per basis point.

Worked example about USD 532 per bp

Credit and capital

Replacement cost
RC=max⁡(Vt, 0)\displaystyle RC = \max(V_t,\ 0)

What the bank loses if the client defaults today: the positive mark of the forward for the bank.

Expected exposure
EE(t)≈φ(0) N S σt≈0.4 N S σt\displaystyle EE(t) \approx \varphi(0)\,N\,S\,\sigma\sqrt{t} \approx 0.4\,N\,S\,\sigma\sqrt{t}

For a forward struck at the market, the expected positive mark grows with the square root of time.

σ\sigma
FX volatility
φ\varphi
standard normal density

Worked example at the value date: 0.399 × 10,850,500 × 6.5% × √(184/365) = USD 199,772

SA-CCR exposure
EAD=1.4 (RC+PFE)AddOnFX=4%×N S×min⁡(M, 1)\displaystyle EAD = 1.4\,\left(RC + PFE\right) \qquad \text{AddOn}_{FX} = 4\% \times N\,S \times \sqrt{\min(M,\,1)}

The standardized exposure of the capital rules: a 4% supervisory factor for FX, scaled by the maturity factor of an unmargined trade.

MM
remaining maturity, years
PFEPFE
multiplier × add-on

Worked example add-on USD 308,157; EAD 1.4 × (4,418 + 308,157) = USD 437,604

CVA
CVA≈(1−R)∑kEE(tk) PD(tk−1,tk) Pd(0,tk)\displaystyle \text{CVA} \approx (1 - R)\sum_k EE(t_k)\,PD(t_{k-1}, t_k)\,P_d(0, t_k)

The expected loss from the client’s default over the life of the forward, charged in the price.

RR
recovery rate
PDPD
default probability between two dates

Settlement and cover

Value date flows
receives N EURpays N K USD\displaystyle \text{receives } N \text{ EUR} \qquad \text{pays } N\,K \text{ USD}

Both currencies move in full on the value date.

Worked example EUR 10,000,000 against USD 10,974,000

The bank’s cover
Kcover=Soffer+poffer\displaystyle K_{\text{cover}} = S^{\text{offer}} + p^{\text{offer}}

An outright from the market, or spot plus a swap for the points.

Worked example 1.0851 + 0.0120 = 1.0971

Dealer margin
N (K−Kcover)\displaystyle N\,\left(K - K_{\text{cover}}\right)

The pips added to the market outright, including the credit charge.

Worked example 10,000,000 × (1.0974 − 1.0971) = USD 3,000

Result on the value date

Result against spot
N (ST−K)\displaystyle N\,\left(S_T - K\right)

What the bought forward gains or loses against buying at spot on the value date.

STS_T
spot on the value date

Worked example 1.1200: +USD 226,000; 1.0800: −USD 174,000

Hedged cost
Chedged=N KCunhedged=N ST\displaystyle C_{\text{hedged}} = N\,K \qquad C_{\text{unhedged}} = N\,S_T

The importer’s cost is fixed by the forward; unhedged, it follows spot.

Worked example USD 10,974,000 whatever the spot rate

Market data of the worked example, illustrative: EUR/USD 1.0850 / 1.0851 on Oct 28, 2026, USD rate 4.09%, EUR rate 1.90% (Act/360), 182 days from the spot date to the value date, EUR/USD volatility 6.5% for the exposure.
FX›Window Forward

Formulas

Pricing the window

Forward to a date in the window
F(t)=S+p(t)t1≤t≤t2\displaystyle F(t) = S + p(t) \qquad t_1 \le t \le t_2

Each possible drawdown date has its own outright.

t1, t2t_1,\ t_2
window start and end

Worked example offers 1.0872 on Nov 30, 2026, 1.0911 on Jan 29, 2027

Client buying the base currency
K=max⁡t1≤t≤t2Foffer(t)+m\displaystyle K = \max_{t_1 \le t \le t_2} F^{\text{offer}}(t) + m

The client buys at the highest offer in the window: the window end when the base currency is at a premium.

mm
margin

Worked example 1.0911 + 0.0005 = 1.0916

Client selling the base currency
K=min⁡t1≤t≤t2Fbid(t)−m\displaystyle K = \min_{t_1 \le t \le t_2} F^{\text{bid}}(t) - m

An exporter selling at a premium gets the lowest bid: the window start.

Cost of the flexibility
C=∑iNi (K−Fmid(ti))\displaystyle C = \sum_i N_i\,\left(K - F^{\text{mid}}(t_i)\right)

What the client pays over dated forwards on the dates it actually draws.

Ni, tiN_i,\ t_i
amount and date of drawdown i

Worked example 8,488 + 6,669 + 1,095 + 286 = USD 16,538

Drawdowns

Drawdown payment
Ai=Ni×K\displaystyle A_i = N_i \times K

Each drawdown is paid at the contract rate.

Worked example 2,000,000 × 1.0916 = USD 2,183,200; 2,500,000 × 1.0916 = USD 2,729,000; 1,000,000 × 1.0916 = USD 1,091,600

Undrawn balance
B(t)=N−∑ti≤tNi\displaystyle B(t) = N - \sum_{t_i \le t} N_i

The parent deal keeps what is left to draw.

Worked example EUR 500,000 on Jan 29, 2027

Close-out of the balance
X=B (Sclose−K)\displaystyle X = B\,\left(S_{\text{close}} - K\right)

Instead of delivery, the balance can be closed out at the market rate, the difference paid in USD.

Worked example at 1.0950 (illustrative): 500,000 × (1.0950 − 1.0916) = +USD 1,700 to the client

The bank’s cover

Cover to the window end
Kcover=Soffer+poffer(t2)\displaystyle K_{\text{cover}} = S^{\text{offer}} + p^{\text{offer}}(t_2)

The bank buys the whole amount spot and swaps it to the window end.

Worked example 1.0851 + 0.0060 = 1.0911

Swap back on a drawdown
Gi=Ni (p(t2)−p(ti))\displaystyle G_i = N_i\,\left(p(t_2) - p(t_i)\right)

The bank buys the drawn amount for the drawdown date and sells it for the window end, earning the points in between.

Worked example drawdown 1: 2,000,000 × (59.8 − 23.1) pips = USD 7,344

The bank’s result
Π=N m+∑iGi\displaystyle \Pi = N\,m + \sum_i G_i

The margin plus the points of the dates the client did not use.

Worked example USD 3,000 + USD 13,106 = USD 16,106 at the trade-date points

Valuation

Most favorable date for the client
t∗=arg max⁡t1≤t≤t2 (F(t)−K)Pd(0,t)\displaystyle t^{*} = \operatorname*{arg\,max}_{t_1 \le t \le t_2}\ \left(F(t) - K\right)P_d(0, t)

The bank values the balance as if the client drew on the date worth most to it.

Worked example Jan 29, 2027 while the EUR is at a forward premium

Mark of the balance
V=B (F(t∗)−K) Pd(0,t∗)\displaystyle V = B\,\left(F(t^{*}) - K\right)\,P_d(0, t^{*})

The undrawn balance as a forward to that date.

Delta
Δ=B Pf(0,t∗)\displaystyle \Delta = B\,P_f(0, t^{*})

The spot-equivalent position of the balance in the base currency.

Credit and timing

Expected exposure to the window end
EE(t)≈0.4 N S σt\displaystyle EE(t) \approx 0.4\,N\,S\,\sigma\sqrt{t}

The exposure on the client’s line grows to the window end.

Worked example 0.399 × 6,510,300 × 6.5% × √(93/365) = USD 85,216

Points risk of the swaps back
δΠ=∑iNi δ ⁣(p(t2)−p(ti))\displaystyle \delta\Pi = \sum_i N_i\,\delta\!\left(p(t_2) - p(t_i)\right)

The points the bank earns are those of the drawdown day, so a change in the differential before the drawdowns changes its result.

Market data of the worked example, illustrative: EUR/USD 1.0850 / 1.0851 on Oct 28, 2026, USD rate 4.09%, EUR rate 1.90% (Act/360); the points of the drawdown dates from parity at mid, as on the trade date.
FX›FX Swap

Formulas

Pricing the points

Far rate
F=S0+p\displaystyle F = S_0 + p

The far leg’s rate is the near rate plus the points agreed.

S0S_0
near rate
pp
swap points

Worked example 1.08505 + 0.0062 = 1.09125

Points from parity
p=S(1+rd τ1+rf τ−1)\displaystyle p = S\left(\frac{1 + r_d\,\tau}{1 + r_f\,\tau} - 1\right)

The points that make borrowing USD through the swap cost the same as in the money market.

τ\tau
days/360

Worked example 91 days: +59.8 pips at mid, quoted +58 / +60

Implied rate
rdimpl=[FS0 (1+rf τ)−1]1τ\displaystyle r_d^{\text{impl}} = \left[\frac{F}{S_0}\,(1 + r_f\,\tau) - 1\right]\frac{1}{\tau}

The USD rate the client pays through the swap, given the EUR rate its EUR would have earned.

Worked example (1.09125 / 1.08505 × (1 + 1.90% × 91/360) − 1) × 360/91 = 4.17%

Cross-currency basis
b=1τ[SF (1+rdcash τ)−1]−rf\displaystyle b = \frac{1}{\tau}\left[\frac{S}{F}\,(1 + r_d^{\text{cash}}\,\tau) - 1\right] - r_f

The EUR rate implied by the swap less the EUR cash rate: negative when USD costs more through FX swaps than in the cash market.

Worked example with USD cash at 3.95% (illustrative): b = −0.14% at the 91-day mid

The two legs

Near leg
pays N EURreceives N S0 USD\displaystyle \text{pays } N \text{ EUR} \qquad \text{receives } N\,S_0 \text{ USD}

The near leg settles on the spot date like a spot.

NN
EUR amount

Worked example 10,000,000 × 1.08505 = USD 10,850,500

Far leg
receives N EURpays N F USD\displaystyle \text{receives } N \text{ EUR} \qquad \text{pays } N\,F \text{ USD}

The far leg reverses the exchange at the far rate.

Worked example 10,000,000 × 1.09125 = USD 10,912,500

Net cost
N (F−S0)=N p\displaystyle N\,(F - S_0) = N\,p

The USD repaid less the USD received: the points, the net interest of the period.

Worked example 10,000,000 × 0.0062 = USD 62,000

Dealer margin
N (pclient−pcover)\displaystyle N\,\left(p_{\text{client}} - p_{\text{cover}}\right)

The pips added to the market points.

Worked example 10,000,000 × 0.0002 = USD 2,000

Valuation

Value of the swap
V=NS0Pd(T1)−NSPf(T1)+NSPf(T2)−NFPd(T2)\displaystyle V = N S_0 P_d(T_1) - N S P_f(T_1) + N S P_f(T_2) - N F P_d(T_2)

The four amounts discounted on their own currency’s curve, the EUR converted at spot; for the client of the example (sell/buy).

T1, T2T_1,\ T_2
near and far dates
Pd, PfP_d,\ P_f
USD and EUR discount factors

Worked example at the trade, at mid: −USD 2,197, the margin and half the spread

After the near leg
Vt=N (Ft(T2)−F) Pd(t,T2)\displaystyle V_t = N\,\left(F_t(T_2) - F\right)\,P_d(t, T_2)

Once the near leg has settled, the far leg is a forward like any other.

Points sensitivity
∂V∂(rd−rf)≈N S τ1+rf τ Pd(T2)\displaystyle \frac{\partial V}{\partial (r_d - r_f)} \approx N\,S\,\frac{\tau}{1 + r_f\,\tau}\,P_d(T_2)

The client that borrowed USD gains when the differential widens: it has locked its cost.

Worked example about USD 270 per bp of differential

Credit and exposure

Exposure on the far leg
EE(t)≈0.4 N S σt\displaystyle EE(t) \approx 0.4\,N\,S\,\sigma\sqrt{t}

The far leg’s replacement cost grows with spot moves until it settles.

Worked example 0.399 × 10,850,500 × 6.5% × √(93/365) = USD 142,026

Variation margin
VMt=Vt−Vt−1\displaystyle VM_t = V_t - V_{t-1}

Under a CSA the change in the far leg’s mark moves as collateral each day.

Settlement exposure
Enear=N S0Efar=N F\displaystyle E_{\text{near}} = N\,S_0 \qquad E_{\text{far}} = N\,F

Each leg puts its full principal at risk on its value date when settled gross.

Worked example USD 10,850,500 and USD 10,912,500

Rolls and short dates

Tom-next points
pTN≈S (rd−rf) 1360\displaystyle p_{\text{TN}} \approx S\,(r_d - r_f)\,\frac{1}{360}

One day of points: the cost of carrying a spot position overnight.

Worked example about 0.66 pips, USD 660 a day on EUR 10,000,000

Rolling a hedge
CFroll=N (Kold−Sroll)Knew=Sroll+p\displaystyle CF_{\text{roll}} = N\,\left(K_{\text{old}} - S_{\text{roll}}\right) \qquad K_{\text{new}} = S_{\text{roll}} + p

A maturing forward is rolled with a swap: the gain or loss of the old forward is paid in cash and a new forward starts from the current spot.

Worked example EUR 10,000,000 sold forward at 1.0900, rolled at spot 1.1000 (illustrative): USD 100,000 paid by the hedger, the new forward from 1.1000

Value today
Ftoday=S−pTN−pON\displaystyle F_{\text{today}} = S - p_{\text{TN}} - p_{\text{ON}}

A swap from today to spot uses the overnight and tom-next points.

Market data of the worked example, illustrative: EUR/USD mid 1.08505 on Oct 28, 2026, USD rate 4.09%, EUR rate 1.90% (Act/360), 91 days between the legs, EUR/USD volatility 6.5% for the exposure.
FX›Non-Deliverable Forward

Formulas

Pricing

NDF outright
F=S+p\displaystyle F = S + p

The offshore forward: spot plus the NDF points of the tenor.

Worked example 5.1500 + 0.0740 = 5.2240 offer; the client 5.2260

Implied offshore yield
rBRLoffshore=[FS (1+rUSD τ)−1]1τ\displaystyle r_{\text{BRL}}^{\text{offshore}} = \left[\frac{F}{S}\,(1 + r_{\text{USD}}\,\tau) - 1\right]\frac{1}{\tau}

The BRL rate that the offshore points imply; it can differ from the onshore rate (the DI curve) by the premium of the offshore market.

τ\tau
days/360

Worked example mid 5.2230: 9.76% (Act/360)

Dealer margin
NBRL(1Kcover−1K)\displaystyle N_{\text{BRL}}\left(\frac{1}{K_{\text{cover}}} - \frac{1}{K}\right)

The 20 points over the market, in USD.

Worked example 50,000,000 × (1/5.2240 − 1/5.2260) = USD 3,663

Fixing and settlement

USD notional
NUSD=NBRLK\displaystyle N_{\text{USD}} = \frac{N_{\text{BRL}}}{K}

The USD the client buys at the contract rate, on paper.

KK
contract rate, BRL per USD

Worked example 50,000,000 / 5.2260 = USD 9,567,547

Settlement amount
X=NUSD−NBRLR=NBRL(1K−1R)\displaystyle X = N_{\text{USD}} - \frac{N_{\text{BRL}}}{R} = N_{\text{BRL}}\left(\frac{1}{K} - \frac{1}{R}\right)

The difference between the USD bought and the USD value of the BRL at the fixing, paid to the client when positive.

RR
fixing, BRL per USD

Worked example 9,567,547 − 9,433,962 = USD 133,585

Same, from the USD notional
X=NUSD R−KR\displaystyle X = N_{\text{USD}}\,\frac{R - K}{R}

The rate difference divided by the fixing, because the payment is in USD.

Worked example 9,567,547 × 0.0740 / 5.3000 = USD 133,585

The bank’s hedge
Xhedge=NBRL(1Kcover−1R)\displaystyle X_{\text{hedge}} = N_{\text{BRL}}\left(\frac{1}{K_{\text{cover}}} - \frac{1}{R}\right)

The offsetting NDF fixes on the same rate.

Worked example 50,000,000 × (1/5.2240 − 1/5.3000) = USD 137,248

Fixing date
Tfix=Tsettle−2 business days\displaystyle T_{\text{fix}} = T_{\text{settle}} - 2\ \text{business days}

For most currencies, in the calendar of the reference currency.

Worked example Jan 29, 2027 − 2 = Jan 27, 2027

Valuation

Mark to market
Vt=NBRL(1K−1Ft)PUSD(t,T)\displaystyle V_t = N_{\text{BRL}}\left(\frac{1}{K} - \frac{1}{F_t}\right)P_{\text{USD}}(t, T)

The settlement amount at today’s NDF forward, discounted from the settlement date.

FtF_t
NDF outright to the fixing date

Worked example at the trade, at the mid 5.2230: −USD 5,438, the margin and half the spread

Delta
∂X∂R=NBRLR2\displaystyle \frac{\partial X}{\partial R} = \frac{N_{\text{BRL}}}{R^2}

The USD gained per unit rise of the BRL-per-USD rate.

Worked example about USD 95,730 for a 1% move at the forward

Convexity
∂2X∂R2=−2 NBRLR3\displaystyle \frac{\partial^2 X}{\partial R^2} = -\frac{2\,N_{\text{BRL}}}{R^3}

Paid in USD on a BRL notional, the settlement is concave in the rate: each further move is worth a little less.

Payoff and the hedged asset

Hedged value
NBRLR+X=NUSD\displaystyle \frac{N_{\text{BRL}}}{R} + X = N_{\text{USD}}

BRL assets plus the NDF are worth the USD notional whatever the fixing.

Worked example USD 9,567,547

Breakeven
X=0  ⟺  R=K\displaystyle X = 0 \iff R = K

The client gains when the fixing is above the contract rate (BRL weaker).

Worked example 5.3000: +USD 133,585; 5.1500: −USD 141,191

Credit and margin

Expected exposure
EE(t)≈0.4 NUSD σt\displaystyle EE(t) \approx 0.4\,N_{\text{USD}}\,\sigma\sqrt{t}

The exposure on the net difference grows with the currency’s volatility.

Worked example 0.399 × 9,567,547 × 14% × √(93/365) = USD 269,733

Initial margin, standard schedule
IM=6%×NUSD\displaystyle IM = 6\% \times N_{\text{USD}}

The schedule of the uncleared margin rules for FX, before netting and the thresholds; dealers use ISDA SIMM instead.

Worked example 6% × 9,567,547 = USD 574,053

Variation margin
VMt=Vt−Vt−1\displaystyle VM_t = V_t - V_{t-1}

The daily change in the mark settled as collateral.

Market data of the worked example, illustrative: USD/BRL spot 5.1500 on Oct 28, 2026, USD rate 4.09% (Act/360), 91 days to settlement, USD/BRL volatility 14% for the exposure.
FX›FX Option

Formulas

Pricing (Garman-Kohlhagen)

Forward
F=S e(rd−rf) T\displaystyle F = S\,e^{(r_d - r_f)\,T}

The forward rate the option is priced off: spot carried by the interest rate differential.

SS
spot, USD per EUR
rd, rfr_d,\ r_f
USD and EUR interest rates
TT
time to expiry, years

Worked example 1.0855 × e^(2.00% × 92/365) = 1.0910

Call and put
C=e−rdT[F N(d1)−K N(d2)]P=e−rdT[K N(−d2)−F N(−d1)]\displaystyle C = e^{-r_d T}\left[F\,N(d_1) - K\,N(d_2)\right] \qquad P = e^{-r_d T}\left[K\,N(-d_2) - F\,N(-d_1)\right]

The Black-Scholes formula applied to a currency pair: the foreign currency’s rate plays the part of a dividend yield.

NN
standard normal distribution
KK
strike

Worked example C = 0.010316 USD per EUR, × EUR 10,000,000 = USD 103,160

d₁ and d₂
d1,2=ln⁡(F/K)±12 σ2TσT\displaystyle d_{1,2} = \frac{\ln(F/K) \pm \tfrac{1}{2}\,\sigma^2 T}{\sigma\sqrt{T}}

The distance between the forward and the strike, in standard deviations of the exchange rate at expiry.

σ\sigma
implied volatility of the strike and expiry

Worked example σ = 6.6%: d₁ = −0.232, d₂ = −0.265

Put-call parity
C−P=e−rdT (F−K)\displaystyle C - P = e^{-r_d T}\,(F - K)

A bought call and a sold put at the same strike make a forward: the link that keeps call and put volatilities consistent.

Premium conventions
USD pips=C% EUR=CSEUR pips=CS K\displaystyle \text{USD pips} = C \qquad \%\,\text{EUR} = \frac{C}{S} \qquad \text{EUR pips} = \frac{C}{S\,K}

The same premium quoted in the terms or the base currency, per unit of notional.

Worked example 103.2 USD pips = 0.95% of the EUR notional

Greeks

Delta
Δ=e−rfT N(d1)\displaystyle \Delta = e^{-r_f T}\,N(d_1)

The amount of base currency that hedges the option against small spot moves (spot delta, premium excluded).

Worked example 0.406: the bank buys EUR 4,060,000

Premium-included delta
Δpi=Δ−CS\displaystyle \Delta_{\text{pi}} = \Delta - \frac{C}{S}

The convention when the premium is paid in the base currency, as for USD/JPY with the premium in USD.

Gamma
Γ=e−rfT φ(d1)S σT\displaystyle \Gamma = \frac{e^{-r_f T}\,\varphi(d_1)}{S\,\sigma\sqrt{T}}

How fast the delta changes with spot: largest at the money and close to expiry.

φ\varphi
standard normal density

Worked example a 1% spot move changes the hedge by about EUR 1,170,000

Vega
ν=S e−rfT φ(d1)T\displaystyle \nu = S\,e^{-r_f T}\,\varphi(d_1)\sqrt{T}

The change in value for a change in volatility, quoted per volatility point.

Worked example about USD 21,100 per vol point on EUR 10,000,000

Hedged P&L
dΠ≈12 Γ S2(σrealized2−σimplied2)dt\displaystyle d\Pi \approx \tfrac{1}{2}\,\Gamma\,S^2\left(\sigma_{\text{realized}}^2 - \sigma_{\text{implied}}^2\right)dt

A delta-hedged long option gains when the volatility actually realized exceeds the implied volatility paid, and loses its time value otherwise.

The volatility smile

At-the-money strike
KATM=F e12σATM2T\displaystyle K_{\text{ATM}} = F\,e^{\frac{1}{2}\sigma_{\text{ATM}}^2 T}

The delta-neutral straddle strike that the at-the-money volatility refers to.

Worked example 1.0915 at 6.3 vol

25-delta volatilities
σ25C=σATM+BF25+12RR25σ25P=σATM+BF25−12RR25\displaystyle \sigma_{25C} = \sigma_{\text{ATM}} + BF_{25} + \tfrac{1}{2}RR_{25} \qquad \sigma_{25P} = \sigma_{\text{ATM}} + BF_{25} - \tfrac{1}{2}RR_{25}

The market quotes the smile as a risk reversal (call volatility less put volatility) and a butterfly (the wings over the at-the-money volatility).

RR25RR_{25}
25-delta risk reversal
BF25BF_{25}
25-delta butterfly
Strike from delta
K=F exp⁡ ⁣(−σT  N−1 ⁣(erfTΔ)+12σ2T)\displaystyle K = F\,\exp\!\left(-\sigma\sqrt{T}\;N^{-1}\!\left(e^{r_f T}\Delta\right) + \tfrac{1}{2}\sigma^2 T\right)

Turns a quoted delta into the strike it stands for.

Payoffs at expiry

Call and put
max⁡(ST−K, 0)max⁡(K−ST, 0)\displaystyle \max(S_T - K,\,0) \qquad \max(K - S_T,\,0)

What the option pays at expiry per unit of notional, before the premium.

STS_T
spot at the expiry cut

Worked example (1.1150 − 1.1000) × 10,000,000 = USD 150,000, less USD 103,160 = +USD 46,840

Collar
max⁡(ST−Kc, 0)−max⁡(Kp−ST, 0)\displaystyle \max(S_T - K_c,\,0) - \max(K_p - S_T,\,0)

A bought call and a sold put: the put’s premium pays for the call.

Kc, KpK_c,\ K_p
call and put strikes

Worked example 1.1200 / 1.0632, zero cost

Call spread
max⁡(ST−K1, 0)−max⁡(ST−K2, 0)\displaystyle \max(S_T - K_1,\,0) - \max(S_T - K_2,\,0)

A bought call and a sold call at a higher strike: a cheaper option with a capped gain.

Worked example 1.1000 / 1.1300: USD 76,140

Up-and-out call
1{max⁡t≤TSt<H} max⁡(ST−K, 0)\displaystyle \mathbf{1}\{\max_{t \le T} S_t < H\}\,\max(S_T - K,\,0)

The call pays only if spot never touched the barrier during the life.

HH
barrier

Worked example H = 1.1400: USD 32,220

Barrier parity
Cup-out+Cup-in=C\displaystyle C_{\text{up-out}} + C_{\text{up-in}} = C

A knock-out and the matching knock-in make the vanilla.

Chance of touching
P(touch)≈2 N ⁣(−ln⁡(H/S)σT)\displaystyle P(\text{touch}) \approx 2\,N\!\left(-\frac{\ln(H/S)}{\sigma\sqrt{T}}\right)

The probability that spot touches the barrier before expiry, without drift.

Worked example 2 × N(−1.479) ≈ 14%

Settlement and credit

Physical delivery
pays N K USDreceives N EUR\displaystyle \text{pays } N\,K \text{ USD} \qquad \text{receives } N \text{ EUR}

On exercise the option becomes a spot trade at the strike.

NN
EUR notional

Worked example USD 11,000,000 against EUR 10,000,000

Cash settlement
N max⁡(Sfix−K, 0)\displaystyle N\,\max(S_{\text{fix}} - K,\,0)

Only the intrinsic value moves, on a fixing named in the confirmation.

Worked example USD 150,000

Exposure of the buyer
EE(t)=E[max⁡(Vt, 0)]=E[Vt]\displaystyle EE(t) = \mathbb{E}\left[\max(V_t,\,0)\right] = \mathbb{E}\left[V_t\right]

A bought option is worth something to its holder at all times, so the exposure to the seller is its whole value.

Market data of the worked example, illustrative: spot 1.0855, USD rate 3.90%, EUR rate 1.90%, 92 days to expiry, volatility 6.6% for the client.
FX›Average Rate Option

Formulas

Pricing the average

Average of the fixings
A=1n∑i=1nSti\displaystyle A = \frac{1}{n}\sum_{i=1}^{n} S_{t_i}

The rate the option settles against: the arithmetic mean of the fixings over the period.

StiS_{t_i}
fixing on date i
nn
number of fixings

Worked example 1.0790 on the 21 fixings of the illustrative month

Average forward
M1=1n∑i=1nF(ti)\displaystyle M_1 = \frac{1}{n}\sum_{i=1}^{n} F(t_i)

The expected average: the mean of the forwards to the fixing dates, also the fair rate of the average rate forward.

F(ti)F(t_i)
forward to the value date of fixing i

Worked example 1.0861, against 1.0871 for the 1-month outright

Second moment
M2=1n2∑i=1n∑j=1nF(ti) F(tj) eσ2min⁡(ti, tj)\displaystyle M_2 = \frac{1}{n^2}\sum_{i=1}^{n}\sum_{j=1}^{n} F(t_i)\,F(t_j)\,e^{\sigma^2 \min(t_i,\,t_j)}

The spread of the average: two fixings share the randomness up to the earlier of their dates.

σ\sigma
volatility of the exchange rate
Effective volatility
σA2 T=ln⁡M2M12σA≈σ3\displaystyle \sigma_A^2\,T = \ln\frac{M_2}{M_1^2} \qquad \sigma_A \approx \frac{\sigma}{\sqrt{3}}

The volatility of a lognormal with the same two moments; for an average running from today to T it is close to the volatility over the square root of three.

TT
time to the last fixing

Worked example 3.91% from 6.5% (σ/√3 = 3.75%)

Price by moment matching
P=e−rdTp[K N(−d2)−M1 N(−d1)]d1,2=ln⁡(M1/K)±12σA2TσAT\displaystyle P = e^{-r_d T_p}\left[K\,N(-d_2) - M_1\,N(-d_1)\right] \qquad d_{1,2} = \frac{\ln(M_1/K) \pm \tfrac{1}{2}\sigma_A^2 T}{\sigma_A\sqrt{T}}

Black’s formula on the average forward with the effective volatility (Levy’s approximation).

TpT_p
time from the premium date to the settlement date
KK
strike

Worked example 0.002310 USD per EUR, × EUR 5,000,000 = USD 11,550

Against the vanilla
PavgPvanilla\displaystyle \frac{P_{\text{avg}}}{P_{\text{vanilla}}}

The ratio of the two premiums at the same strike and volatility.

Worked example USD 11,550 / USD 24,275 = 0.48

Payoffs at the end

Average rate put
N max⁡(K−A, 0)\displaystyle N\,\max(K - A,\,0)

What the option pays after the last fixing, before the premium.

NN
EUR notional

Worked example (1.0800 − 1.0790) × 5,000,000 = USD 5,000

Average rate call
N max⁡(A−K, 0)\displaystyle N\,\max(A - K,\,0)

The importer’s version: protection against a rising average.

Average rate forward
N (Kf−A)\displaystyle N\,(K_f - A)

For the seller of EUR: the difference between the agreed rate and the average, either way.

KfK_f
agreed rate

Worked example (1.0856 − 1.0790) × 5,000,000 = USD 33,000

Average strike call
N max⁡(ST−A, 0)\displaystyle N\,\max(S_T - A,\,0)

The average as the strike: pays if the final rate ends above the period’s average.

STS_T
rate at the end

Risk and hedging

Delta
Δ=−e−rdTp N(−d1) M1S\displaystyle \Delta = -e^{-r_d T_p}\,N(-d_1)\,\frac{M_1}{S}

Every forward moves with spot, so the average forward does too.

SS
spot

Worked example -0.302: the bank, short the put, sells about EUR 1,510,000 forward (a vanilla: -0.356)

Strip hedge of the forward
hi=NnΔk≈N n−kn\displaystyle h_i = \frac{N}{n} \qquad \Delta_k \approx N\,\frac{n-k}{n}

The average rate forward hedges as one forward per fixing date; after k fixings only the rest is at risk.

hih_i
EUR hedged to fixing date i
kk
fixings published

Worked example EUR 238,095 per fixing date

Vega
ν=∂P∂σ\displaystyle \nu = \frac{\partial P}{\partial \sigma}

The change in premium for one volatility point, smaller than a vanilla’s because the average moves less.

Worked example about USD 3,260 per vol point, against USD 5,700 for the vanilla

Strike left after k fixings
K∗=n K−∑i=1kStin−k\displaystyle K^{*} = \frac{n\,K - \sum_{i=1}^{k} S_{t_i}}{n-k}

Part way through, the option is an option on the remaining fixings with an adjusted strike.

Worked example after 11 fixings: (21 × 1.0800 − 11.9249) / 10 = 1.0755

Settlement

Cash settlement
X=N max⁡(K−A, 0) paid on tn+2\displaystyle X = N\,\max(K - A,\,0) \text{ paid on } t_n + 2

One USD payment two business days after the last fixing; no EUR is delivered.

tnt_n
last fixing date

Worked example USD 5,000 on Nov 30, 2026

The exporter’s floor
R≥K−Pe−rdTp\displaystyle R \ge K - \frac{P}{e^{-r_d T_p}}

Receipts sold at spot evenly over the period average A; with the put the rate achieved is never below the strike less the premium.

RR
rate achieved on the EUR sold

Worked example 1.0800 − 0.0023 = 1.0777

Forward at inception
V0=e−rdTp N (Kf−M1)\displaystyle V_0 = e^{-r_d T_p}\,N\,(K_f - M_1)

The average rate forward’s value to the client on the trade date: the bank’s margin.

Worked example 0.99669 × 5,000,000 × (1.0856 − 1.0861) = −USD 2,492

Market data of the worked example, illustrative: spot 1.0850, 1-month points +21 to Nov 30, 2026, USD rate 3.90%, 1-month volatility 6.5%, 21 fixings from Oct 29, 2026 to Nov 26, 2026, settlement Nov 30, 2026.
FX›Structured Forward

Formulas

Zero-cost structures

Zero-cost collar
P(Kp, σp)=C(Kc∗, σc)\displaystyle P(K_p,\,\sigma_p) = C(K_c^{*},\,\sigma_c)

The call strike that the bought put pays for, each option priced at the volatility of its own strike.

Kp, Kc∗K_p,\ K_c^{*}
put strike, fair call strike
σp, σc\sigma_p,\ \sigma_c
volatilities of the two strikes on the smile

Worked example put 1.0700 at 6.39% = call 1.1330 at 7.15% = 87.2 USD pips, USD 87,214

Smile quotes
σ25C=σATM+BF25+12RR25σ25P=σATM+BF25−12RR25\displaystyle \sigma_{25C} = \sigma_{\text{ATM}} + BF_{25} + \tfrac{1}{2}RR_{25} \qquad \sigma_{25P} = \sigma_{\text{ATM}} + BF_{25} - \tfrac{1}{2}RR_{25}

A positive risk reversal makes EUR calls dearer than puts, so a sold call pays for a put further from the forward.

RR25, BF25RR_{25},\ BF_{25}
25-delta risk reversal and butterfly

Worked example RR +0.86 puts the fair call at 1.1330; a flat 6.6% would give 1.1255

Margin in the strike
M=N[C(Kc)−C(Kc∗)]\displaystyle M = N\left[C(K_c) - C(K_c^{*})\right]

What the client gives up by selling the call at a lower strike than the fair one.

NN
EUR notional

Worked example 10,000,000 × (93.4 − 87.2) pips = USD 6,171

Participating forward
P(K)=α C(K)\displaystyle P(K) = \alpha\,C(K)

The bought put at K paid by a call sold on a fraction α of the notional at the same strike.

α\alpha
share of the upside given up

Worked example α = 50%: K = 1.0830 at a flat 6.6%

Knock-in forward
P(K)=CUI(K,H)=C(K)−CUO(K,H)\displaystyle P(K) = C_{\text{UI}}(K, H) = C(K) - C_{\text{UO}}(K, H)

The sold call exists only if spot touches the barrier; barrier parity prices it from the up-and-out call.

HH
barrier

Worked example H = 1.1400: K = 1.0940, 185.4 USD pips each side

Payoffs at expiry

Collar
max⁡(Kp−ST, 0)−max⁡(ST−Kc, 0)\displaystyle \max(K_p - S_T,\,0) - \max(S_T - K_c,\,0)

A bought put and a sold call, per EUR of notional.

STS_T
spot at the expiry cut

Worked example (1.0700 − 1.0600) × 10,000,000 = USD 100,000

Rate achieved with a collar
R=min⁡ ⁣(max⁡(ST, Kp), Kc)\displaystyle R = \min\!\left(\max(S_T,\,K_p),\,K_c\right)

The exporter’s rate on the EUR it sells, between the two strikes.

Worked example between 1.0700 and 1.1300

Participating forward
R=K+(1−α)max⁡(ST−K, 0)\displaystyle R = K + (1-\alpha)\max(S_T - K,\,0)

The worst rate K, and the share 1 − α of any rise above it.

Worked example at 1.1300: 1.0830 + 50% × 470 pips = 1.1065

Knock-in forward
max⁡(K−ST, 0)−1{max⁡t≤TSt≥H} max⁡(ST−K, 0)\displaystyle \max(K - S_T,\,0) - \mathbf{1}\{\max_{t \le T} S_t \ge H\}\,\max(S_T - K,\,0)

The put always; the sold call only if the barrier traded during the life.

Chance of touching
P(touch)≈2 N ⁣(−ln⁡(H/S)σT)\displaystyle P(\text{touch}) \approx 2\,N\!\left(-\frac{\ln(H/S)}{\sigma\sqrt{T}}\right)

The probability that spot touches the barrier before expiry, without drift.

Worked example 2 × N(−1.061) ≈ 29%

Hedging the package

Net delta
Δbank=−Δput+Δcall\displaystyle \Delta_{\text{bank}} = -\Delta_{\text{put}} + \Delta_{\text{call}}

The bank is short the put and long the call: both make it long EUR.

Worked example 0.280 + 0.283 = 0.563: the bank sells about EUR 5,630,000 spot

Net vega
νbank=νcall−νput\displaystyle \nu_{\text{bank}} = \nu_{\text{call}} - \nu_{\text{put}}

The two legs have similar vegas, so a collar has little exposure to the level of volatility.

Worked example about USD 120 per vol point

Risk reversal exposure
δV≈νcall δσc−νput δσp\displaystyle \delta V \approx \nu_{\text{call}}\,\delta\sigma_c - \nu_{\text{put}}\,\delta\sigma_p

A collar is a risk reversal: its value moves with the skew between calls and puts.

Worked example +0.1 vol of risk reversal: USD 2,060 for the bank

Exposure to the client
EE(t)=E[max⁡(Vt, 0)]\displaystyle EE(t) = \mathbb{E}\left[\max(V_t,\,0)\right]

The bank’s exposure is the value of the package to the bank, positive when the sold call is in the money.

VtV_t
value of the package to the bank

Target redemption forward

Each fixing
Xi=N[(K−Si)+−ℓ (Si−K)+]\displaystyle X_i = N\left[(K - S_i)^{+} - \ell\,(S_i - K)^{+}\right]

Against selling at the market: a gain on N below the strike, a loss on ℓ N above it.

ℓ\ell
leverage, 2
SiS_i
fixing i
NN
EUR a month

Worked example K = 1.1320, N = EUR 1,000,000

Accrued gain and target
Gi=∑j≤i(K−Sj)+τ=min⁡{i:Gi≥G}\displaystyle G_i = \sum_{j \le i}(K - S_j)^{+} \qquad \tau = \min\{i : G_i \ge G\}

The TARF ends at the first fixing where the gains reach the target; that fixing pays only what is left of it.

GG
target

Worked example G = 0.1500, USD 150,000

Value by Monte Carlo
V=1M∑m=1M∑i≤τmP(0,ti) Xi(m)\displaystyle V = \frac{1}{M}\sum_{m=1}^{M}\sum_{i \le \tau_m} P(0,t_i)\,X_i^{(m)}

The average over simulated paths of the discounted settlements up to the target.

MM
number of paths

Worked example fair strike 1.1362 (16,000 paths); at 1.1320 the client gives up about USD 28,900

Expected life
E[τ]P(τ≤12)\displaystyle \mathbb{E}[\tau] \qquad P(\tau \le 12)

How many fixings the TARF lasts on average, and how often the target is reached.

Worked example 5.8 fixings of 12; the target is reached on 79% of the paths

Market data of the worked example, illustrative: spot 1.0850, 6-month forward 1.0970, USD rate 3.90%, EUR rate 1.69% (from the forward), 182 days to expiry; volatility 6.6% at the money, 25-delta butterfly 0.20, risk reversal +0.86. The participating forward, the knock-in forward and the TARF at a flat 6.6%.
FX›FX Future

Formulas

Price

Fair futures price
F=S 1+rUSD d/3601+rEUR d/360\displaystyle F = S\,\frac{1 + r_{\text{USD}}\,d/360}{1 + r_{\text{EUR}}\,d/360}

Spot carried to the delivery date by the two money-market rates: the forward rate for that date.

SS
spot, USD per EUR
dd
days from the spot date to delivery

Worked example 1.0867 × (1 + 3.90% × 47/360) / (1 + 1.90% × 47/360) = 1.0895

Futures-forward basis
Ffut−Ffwd≈convexity+funding\displaystyle F^{\text{fut}} - F^{\text{fwd}} \approx \text{convexity} + \text{funding}

Daily margining makes the future differ slightly from the forward: the gains are reinvested and the losses financed as rates move; dealers arbitrage the rest through EFPs.

Calendar spread
FMar−FDec=FDec[1+rUSD d′/3601+rEUR d′/360−1]\displaystyle F_{\text{Mar}} - F_{\text{Dec}} = F_{\text{Dec}}\left[\frac{1 + r_{\text{USD}}\,d^{\prime}/360}{1 + r_{\text{EUR}}\,d^{\prime}/360} - 1\right]

The price of the roll: the forward points between the two delivery dates.

d′d^{\prime}
days from December to March delivery

Worked example 91 days: +54.8 pips

Implied differential
rUSD−rEUR≈(FS−1)360d\displaystyle r_{\text{USD}} - r_{\text{EUR}} \approx \left(\frac{F}{S} - 1\right)\frac{360}{d}

The interest rate gap the futures price implies, the check against the money markets.

Worked example (1.0895 / 1.0867 − 1) × 360 / 47 = 1.97%

Margin

Variation margin
VMt=n Q (Pt−Pt−1)\displaystyle VM_t = n\,Q\,(P_t - P_{t-1})

The day’s gain or loss on the position, settled in cash the next morning.

nn
contracts
QQ
contract size, EUR 125,000
PtP_t
settlement price of day t

Worked example 500 × 125,000 × (1.0911 − 1.0895) = USD 100,000

Tick and pip value
tick=0.00005 Qpip=0.0001 Q\displaystyle \text{tick} = 0.00005\,Q \qquad \text{pip} = 0.0001\,Q

The cash value of one price step for one contract.

Worked example USD 6.25 and USD 12.50; USD 6,250 a pip on 500 contracts

Cumulative margin
∑s≤tVMs=n Q (Pt−P0)\displaystyle \sum_{s \le t} VM_s = n\,Q\,(P_t - P_0)

The margin paid and received adds up to the change from the trade price.

P0P_0
trade price

Worked example 43 pips after five days: USD 268,750

Initial margin
IM=n×m\displaystyle IM = n \times m

A deposit per contract set by the clearing house, called through the FCM.

mm
margin per contract

Worked example 500 × USD 2,500 = USD 1,250,000

Hedging

Number of contracts
n=EQ\displaystyle n = \frac{E}{Q}

The contracts that hedge a currency exposure, before the rounding to whole contracts.

EE
EUR exposure

Worked example EUR 62,500,000 / 125,000 = 500

Value of a 1% move
δV=n Q P×1%\displaystyle \delta V = n\,Q\,P\times 1\%

The position’s change in USD for a 1% move in EUR/USD.

Worked example USD 680,938

Stressed variation margin
VMstress=n Q δP\displaystyle VM^{\text{stress}} = n\,Q\,\delta P

The cash a hedger must find in one day if the rate moves against it.

δP\delta P
stressed price move

Worked example a 2% fall, 218 pips: USD 1,362,500

Basis of the hedge
Π=n Q (δF−δS)\displaystyle \Pi = n\,Q\,(\delta F - \delta S)

What is left of a spot exposure hedged with futures: the change in the basis.

Expiry and delivery

Last trading day
TLTD=W3−2 business days\displaystyle T_{\text{LTD}} = W_3 - 2 \text{ business days}

Trading ends two business days before the third Wednesday of the contract month, the spot date of the last trading day.

W3W_3
third Wednesday

Worked example Dec 14, 2026, for delivery on Dec 16, 2026

Physical delivery
long receives n Q EURpays n Q Pfinal USD\displaystyle \text{long receives } n\,Q \text{ EUR} \qquad \text{pays } n\,Q\,P_{\text{final}} \text{ USD}

Open positions turn into a spot exchange settled through CLS.

PfinalP_{\text{final}}
final settlement price

Worked example EUR 62,500,000 against USD for 500 contracts

Exchange for physical
PEFPfut=Pfwd+points to delivery\displaystyle P_{\text{EFP}}^{\text{fut}} = P^{\text{fwd}} + \text{points to delivery}

A future and an OTC forward or spot traded together at linked prices, to move a position between the two markets.

Market data of the worked example, illustrative: USD rate 3.90%, EUR rate 1.90% (money market, Act/360), spot date Oct 30, 2026, delivery Dec 16, 2026; the settlement prices of the first five days are illustrative.
Money Markets›Treasury Bill

Formulas

Price and yields

Price from the discount rate (US)
P=100(1−d t360)\displaystyle P = 100\left(1 - d\,\frac{t}{360}\right)

The US quotes a bill by its discount rate on the face value; the price follows on Act/360, rounded to six decimals by Treasury.

dd
discount rate
tt
days to maturity

Worked example d = 5.200%, t = 91: P = 98.685556; at the 5.205% high rate 98.684292

Discount in currency
D=F d t360\displaystyle D = F\,d\,\frac{t}{360}

The amount the buyer earns if it holds the bill to maturity.

FF
face value

Worked example USD 500,000,000 × 5.200% × 91/360 = USD 6,572,220

Money market yield
yMM=360 d360−d t=100−PP⋅360t\displaystyle y_{\text{MM}} = \frac{360\,d}{360 - d\,t} = \frac{100 - P}{P}\cdot\frac{360}{t}

The return on the price paid, Act/360: the rate to compare with deposits, CP, repo and SOFR.

Worked example 5.269%

Bond-equivalent yield (investment rate)
yBE=365 d360−d t=100−PP⋅365t\displaystyle y_{\text{BE}} = \frac{365\,d}{360 - d\,t} = \frac{100 - P}{P}\cdot\frac{365}{t}

Treasury’s investment rate for bills of up to half a year: the return on the price paid on Act/365, to compare with note and bond yields.

Worked example 5.342%

Bond-equivalent yield beyond half a year
yBE=−b+b2−4ac2aa=t2Y−14,b=tY,c=P−100P\displaystyle y_{\text{BE}} = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \qquad a = \frac{t}{2Y} - \frac{1}{4},\quad b = \frac{t}{Y},\quad c = \frac{P - 100}{P}

For a 52-week bill the investment rate allows for the coupon a note would have paid at six months.

YY
365, or 366 when the year after the issue date contains February 29

Worked example 364 days at 5.200% discount: 5.490%

Price from a yield (UK, euro area, Canada)
P=1001+y tB\displaystyle P = \frac{100}{1 + y\,\frac{t}{B}}

Sterling, euro and Canadian bills are quoted as a yield on the price paid; the base B is 365 for sterling and Canadian dollars, 360 for euro.

yy
money market yield
BB
day count base

Worked example UK 3-month at 3.945%, 91 days: P = 99.0260; Bubill at 1.950%: 99.5095

Discount rate and yield
d=y1+y tB\displaystyle d = \frac{y}{1 + y\,\frac{t}{B}}

For the same price the discount rate sits below the yield, by more as the rate and the maturity grow.

Worked example 5.269% yield on Act/360 gives back 5.200% discount; the UK bill at 3.945% is a 3.907% discount on Act/365

Cash flows and accretion

Settlement amount
S=F P100\displaystyle S = F\,\frac{P}{100}

What the buyer pays on the issue or settlement date; there is no accrued interest on a discount instrument.

Worked example USD 500,000,000 × 98.685556% = USD 493,427,780

Return earned
F−S=S yMM t360\displaystyle F - S = S\,y_{\text{MM}}\,\frac{t}{360}

The discount earned is the money market yield applied to the amount paid.

Worked example USD 493,427,780 × 5.269% × 91/360 = USD 6,572,220

Amortized cost
ACτ=S(1+yMM τ360)\displaystyle AC_{\tau} = S\left(1 + y_{\text{MM}}\,\frac{\tau}{360}\right)

Between purchase and maturity the carrying value accretes toward par; money funds and held-to-collect portfolios carry bills this way.

τ\tau
days since settlement

Worked example day 14: USD 494,438,891; the 77-day bill at the same 5.200% costs USD 494,438,890

Holding period return
rHP=Psell−PbuyPbuy⋅360τ\displaystyle r_{\text{HP}} = \frac{P_{\text{sell}} - P_{\text{buy}}}{P_{\text{buy}}}\cdot\frac{360}{\tau}

A bill sold before maturity earns the price change, including the effect of any rate move since purchase.

Worked example bought at 5.200% for 91 days, sold after 14 days at 5.150%: 5.548% annualized

Risk and carry

DV01 of a US bill
DV01=F t360×0.0001\displaystyle \text{DV01} = F\,\frac{t}{360}\times 0.0001

The price change for one basis point of discount rate: a fixed amount per day to maturity, whatever the rate.

Worked example USD 500,000,000 × 91/360 × 0.0001 = USD 12,639

DV01 of a yield-quoted bill
DV01=F t/B(1+y t/B)2×0.0001\displaystyle \text{DV01} = F\,\frac{t/B}{\left(1 + y\,t/B\right)^2}\times 0.0001

The same measure when the quote is a yield on the price.

Worked example UK bill: GBP 12,224 per bp

Modified duration
Dmod=t/Y1+yBE t/Y\displaystyle D_{\text{mod}} = \frac{t/Y}{1 + y_{\text{BE}}\,t/Y}

The price sensitivity to the yield, in years: close to the time to maturity for a bill.

Worked example 0.246 years for the 91-day bill

Carry in repo
carry≈S (yMM−rrepo)τ360\displaystyle \text{carry} \approx S\,\left(y_{\text{MM}} - r_{\text{repo}}\right)\frac{\tau}{360}

A dealer that finances a bill in repo earns the bill yield less the repo rate; on an upward-sloping curve the bill also rolls down as it shortens.

rrepor_{\text{repo}}
repo rate

Worked example 5.269% against the 5.31% GC rate: −USD 558 a day on USD 493,427,780

Dealer margin on an allocation
M=F (dstop−dclient)t360\displaystyle M = F\,\left(d_{\text{stop}} - d_{\text{client}}\right)\frac{t}{360}

The bank’s margin on the bills passed from its award: the rate gap times the DV01.

Worked example 0.5 bp × USD 12,639 = USD 6,320

Auctions

Single-price auction (US)
d∗=min⁡{d:∑di≤dQi≥Q−QNC}\displaystyle d^{*} = \min\Big\{d : \sum_{d_i \le d} Q_i \ge Q - Q_{\text{NC}}\Big\}

Competitive bids are accepted from the lowest rate up until the offering, less the noncompetitive bids, is covered; every accepted bid pays the high rate.

di, Qid_i,\ Q_i
rate and amount of bid i
QQ
offering amount
QNCQ_{\text{NC}}
noncompetitive bids

Worked example high rate 5.205%

Proration at the high rate
qi=Qi Q−QNC−∑dj<d∗Qj∑dj=d∗Qj\displaystyle q_i = Q_i\,\frac{Q - Q_{\text{NC}} - \sum_{d_j < d^{*}} Q_j}{\sum_{d_j = d^{*}} Q_j}

Bids exactly at the high rate share what is left; Treasury publishes the percentage allotted at the high rate.

Multiple-price auction (UK, euro area, Canada)
yˉ=∑iqi yi∑iqi\displaystyle \bar{y} = \frac{\sum_i q_i\,y_i}{\sum_i q_i}

Each successful bidder pays its own yield; the debt manager publishes the average and the highest accepted yields.

qi, yiq_i,\ y_i
amount allotted and yield of bid i
Bid-to-cover
BTC=amount bidamount accepted\displaystyle \text{BTC} = \frac{\text{amount bid}}{\text{amount accepted}}

The demand at the auction for each unit sold.

Worked example 2.9 at the 13-week auction

Worked example: the US 13-week bill, 91 days, USD 500,000,000 face, bought by the fund at 5.200% discount from the bank’s award at 5.205% (illustrative). The UK, euro area and Canadian bills use the yield formula with their day count base, as in the card.
Money Markets›Commercial Paper

Formulas

Price and yield

Price from the discount rate (US)
P=100(1−d t360)\displaystyle P = 100\left(1 - d\,\frac{t}{360}\right)

US paper is quoted as a discount rate on the face value, like a bill.

dd
discount rate
tt
days to maturity

Worked example d = 5.28%, t = 30: P = 99.5600

Price from a yield (ECP, NEU CP, sterling and Canadian paper)
P=1001+y tB\displaystyle P = \frac{100}{1 + y\,\frac{t}{B}}

Elsewhere the quote is a yield on the price paid; B is 360 for euro and dollar ECP, 365 for sterling and Canadian dollars.

BB
day count base

Worked example NEU CP at 1.97%: 99.8361; sterling CP at 3.99%: 99.6731

Money market yield
y=d1−d t360\displaystyle y = \frac{d}{1 - d\,\frac{t}{360}}

The return on the price paid, to compare with OIS and with the yield-quoted markets.

Worked example 5.303%

Spread to OIS
s=y−OIS(t)\displaystyle s = y - \text{OIS}(t)

What the investor earns over the risk-free rate for the term: the price of the issuer’s credit.

Worked example 5.303% − 5.28% = 2.3 bp

Cash flows

Investor pays
S=F P100\displaystyle S = F\,\frac{P}{100}

The settlement amount on the issue date.

FF
face value

Worked example USD 150,000,000 × 99.5600% = USD 149,340,000

Discount earned
F−S\displaystyle F - S

The investor’s return over the term, received with the par repayment.

Worked example USD 660,000

Dealer fee
fee=Fissue f t360\displaystyle \text{fee} = F_{\text{issue}}\,f\,\frac{t}{360}

The issuer pays the dealer a fee quoted in basis points a year on the face placed.

ff
fee rate, bp a year

Worked example USD 300,000,000 × 0.05% × 30/360 = USD 12,500

Issuer proceeds
Snet=Fissue P100−fee\displaystyle S_{\text{net}} = F_{\text{issue}}\,\frac{P}{100} - \text{fee}

What reaches the issuer’s account on the issue date.

Worked example USD 298,680,000 − USD 12,500 = USD 298,667,500

All-in cost
c=(FissueSnet−1)360t\displaystyle c = \left(\frac{F_{\text{issue}}}{S_{\text{net}}} - 1\right)\frac{360}{t}

The issuer’s funding cost including the fee, as a money market yield.

Worked example 5.354%

The roll and the backstop

Outstanding
O(t)=∑iFi 1{tiissue≤t<timat}\displaystyle O(t) = \sum_i F_i\,\mathbf{1}\{t_i^{\text{issue}} \le t < t_i^{\text{mat}}\}

The program’s outstanding paper on a date: what the backup line must cover.

Average cost of a rolled program
cˉ=∑iFi ci ti∑iFi ti\displaystyle \bar{c} = \frac{\sum_i F_i\,c_i\,t_i}{\sum_i F_i\,t_i}

The cost of the program weighted by amount and days, compared with the revolver’s drawn margin.

Backstop coverage
coverage=undrawn committed linesO(t)≥100%\displaystyle \text{coverage} = \frac{\text{undrawn committed lines}}{O(t)} \ge 100\%

Rating agencies expect top-tier issuers to cover all the paper outstanding with committed lines.

Risk and value

DV01
DV01=F t360×0.0001\displaystyle \text{DV01} = F\,\frac{t}{360}\times 0.0001

The price change for one basis point on the discount rate.

Worked example USD 150,000,000 × 30/360 × 0.0001 = USD 1,250

Value against OIS
V=F1+OIS t360−S\displaystyle V = \frac{F}{1 + \text{OIS}\,\frac{t}{360}} - S

The paper discounted at the risk-free rate less its price: the credit spread earned over the term, in currency.

Worked example USD 2,891

Mark to market
Vτ=F(1−dτ t−τ360)\displaystyle V_{\tau} = F\left(1 - d_{\tau}\,\frac{t - \tau}{360}\right)

Paper held by a fund or a dealer marked at today’s rate for the remaining days.

dτd_{\tau}
discount rate on day τ
Spread and default intensity
s≈λ (1−R)\displaystyle s \approx \lambda\,(1 - R)

The credit triangle: the spread pays for the expected loss rate.

λ\lambda
default intensity, a year
RR
recovery rate

Worked example 2.3 bp with R = 40%: λ ≈ 0.039% a year

Worked example: USD 300,000,000 of 30-day paper at 5.28% discount, USD 150,000,000 bought by the fund, a dealer fee of 5 bp a year and a 30-day OIS of 5.28% (illustrative).
Money Markets›CD and Deposit

Formulas

Interest and cash flows

Interest at maturity
I=N r tB\displaystyle I = N\,r\,\frac{t}{B}

Simple interest on the money market basis of the currency.

NN
principal
rr
deposit rate
tt
days
BB
360 for USD and EUR, 365 for GBP and CAD

Worked example USD 100,000,000 × 5.35% × 182/360 = USD 2,704,722

Amount at maturity
M=N(1+r tB)\displaystyle M = N\left(1 + r\,\frac{t}{B}\right)

Principal and interest paid together at maturity.

Worked example USD 102,704,722

Day count by currency
I365=I360 360365\displaystyle I_{365} = I_{360}\,\frac{360}{365}

The same quoted rate earns less on Act/365 than on Act/360.

Worked example GBP 100,000,000 × 4.05% × 182/365 = GBP 2,019,452

Floating-rate CD coupon
C=N[∏d(1+rd ndB)−1]+N s tB\displaystyle C = N\left[\prod_{d}\left(1 + \frac{r_d\,n_d}{B}\right) - 1\right] + N\,s\,\frac{t}{B}

A floating CD compounds the overnight rate in arrears over the period and adds its spread.

rd, ndr_d,\ n_d
overnight fixing and its days
ss
spread

Valuation

Value at a market yield
V=N(1+c TB)1+y trB\displaystyle V = \frac{N\left(1 + c\,\frac{T}{B}\right)}{1 + y\,\frac{t_r}{B}}

A negotiable CD sold before maturity is worth its maturity amount discounted at the current yield for the days left.

cc
CD rate
TT
original days
yy
market yield
trt_r
days remaining

Worked example day 91 at 5.30%: USD 101,346,955

Accrued interest
AI=N c τB\displaystyle AI = N\,c\,\frac{\tau}{B}

The interest earned since issue, part of the value above.

τ\tau
days since issue

Worked example day 91: USD 1,352,361

Seller’s return
rHP=(VN−1)Bτ\displaystyle r_{\text{HP}} = \left(\frac{V}{N} - 1\right)\frac{B}{\tau}

Rates fell after the purchase, so the seller earns a little more than the CD rate.

Worked example 5.329%

Value against OIS
PV=M1+OIS tB−N\displaystyle PV = \frac{M}{1 + \text{OIS}\,\frac{t}{B}} - N

The deposit discounted at the risk-free rate: the spread the bank pays, in currency.

Worked example USD 73,891

Funds transfer pricing and liquidity

Funds transfer price
rFTP(t)=OIS(t)+sfund(t)−ℓ(t)\displaystyle r_{\text{FTP}}(t) = \text{OIS}(t) + s_{\text{fund}}(t) - \ell(t)

Treasury’s rate for money of term t: the risk-free rate, the bank’s funding spread, less the liquidity value of the deposit.

sfunds_{\text{fund}}
marginal funding spread
ℓ\ell
liquidity value

Worked example 5.20% + 15 bp = 5.35%

Break cost
BC≈N max⁡(rnow(tr)−r, 0)trB+fee\displaystyle BC \approx N\,\max\left(r_{\text{now}}(t_r) - r,\ 0\right)\frac{t_r}{B} + \text{fee}

A deposit broken early leaves the bank to replace the funding at today’s rate for the days left.

rnow(tr)r_{\text{now}}(t_r)
funding rate today for the remaining term
Liquidity coverage ratio
LCR=HQLAnet cash outflows over 30 days≥100%\displaystyle \text{LCR} = \frac{\text{HQLA}}{\text{net cash outflows over 30 days}} \ge 100\%

A term deposit enters the outflows when its maturity is 30 days away.

LCR outflow of a wholesale deposit
O=ρ N 1{tr≤30}\displaystyle O = \rho\,N\,\mathbf{1}\{t_r \le 30\}

Run-off rates under the Basel standard: 40% for non-operational deposits of non-financial companies, 100% for financial institutions.

ρ\rho
run-off rate

Worked example the client’s USD 100,000,000 adds USD 40,000,000 of outflow in its last 30 days

Net stable funding ratio
NSFR=available stable fundingrequired stable funding≥100%\displaystyle \text{NSFR} = \frac{\text{available stable funding}}{\text{required stable funding}} \ge 100\%

Funding from non-financial companies with less than a year to run counts at 50%, more than a year at 100%.

Risk

DV01 of a CD
DV01≈V tr/B1+y tr/B×0.0001\displaystyle \text{DV01} \approx V\,\frac{t_r/B}{1 + y\,t_r/B}\times 0.0001

The holder’s sensitivity to one basis point of yield.

Worked example day 91: USD 2,528 per bp

Spread to OIS
s=r−OIS(t)\displaystyle s = r - \text{OIS}(t)

The bank’s credit and term funding spread.

Worked example 5.35% − 5.20% = 15 bp

Worked example: USD 100,000,000 for 182 days at 5.35%, Act/360, against a 6-month OIS of 5.20% (illustrative); the CD sold on day 91 at 5.30%.
Money Markets›Overnight Loan

Formulas

Interest

Overnight interest
I=N r nB\displaystyle I = N\,r\,\frac{n}{B}

Simple interest for the calendar days to the next business day.

NN
amount lent
rr
overnight rate
nn
calendar days: 1, or 3 over a weekend
BB
360 for USD and EUR, 365 for GBP and CAD

Worked example USD 2,000,000,000 × 5.33% × 1/360 = USD 296,111

Repayment
N+I\displaystyle N + I

Principal and interest repaid the next morning.

Worked example USD 2,000,296,111

Over a weekend
IFri=N r 3B\displaystyle I_{\text{Fri}} = N\,r\,\frac{3}{B}

A Friday loan earns three days of interest at Friday’s rate.

Worked example USD 888,333

Converting the day count
r365=r360 365360\displaystyle r_{365} = r_{360}\,\frac{365}{360}

The same daily interest expressed on the other basis.

Worked example 5.33% on Act/360 = 5.404% on Act/365

The benchmarks

Volume-weighted median (EFFR, SOFR)
r∗=min⁡{rk:∑ri≤rkVi≥12∑iVi}\displaystyle r^{*} = \min\Big\{r_k : \sum_{r_i \le r_k} V_i \ge \tfrac{1}{2}\sum_i V_i\Big\}

The rate at which half of the day’s volume traded at or below.

ri, Vir_i,\ V_i
rate and volume of trade i
Volume-weighted trimmed mean (SONIA, €STR)
r∗=∑i∈CVi ri∑i∈CVi\displaystyle r^{*} = \frac{\sum_{i \in C} V_i\,r_i}{\sum_{i \in C} V_i}

The mean of the central 50% of the volume, after trimming the top and bottom quarters; CORRA trims only the bottom quarter.

CC
the trades kept
Compounded in arrears
R=[∏i(1+ri niB)−1]BD\displaystyle R = \left[\prod_{i}\left(1 + \frac{r_i\,n_i}{B}\right) - 1\right]\frac{B}{D}

The overnight rates compounded over a period, as in the floating leg of an OIS or a rolled loan.

DD
days in the period

Worked example Wednesday to Monday at 5.33% (n = 1, 1, 3): 5.331%

Compounded index
Xt=Xt−1(1+rt−1 nt−1B)R=(XendXstart−1)BD\displaystyle X_t = X_{t-1}\left(1 + \frac{r_{t-1}\,n_{t-1}}{B}\right) \qquad R = \left(\frac{X_{\text{end}}}{X_{\text{start}}} - 1\right)\frac{B}{D}

The administrators publish an index of the compounded rate (the SOFR Averages and Index, the SONIA Compounded Index), so a period rate is the ratio of two values.

The policy framework

The corridor
rfloor≤r≤rceiling\displaystyle r_{\text{floor}} \le r \le r_{\text{ceiling}}

The central bank’s administered rates frame the overnight rate.

Worked example 5.30% (ON RRP) ≤ 5.33% ≤ 5.50% (standing repo facility)

Floor arbitrage
π=N (IORB−r)n360\displaystyle \pi = N\,\left(\text{IORB} - r\right)\frac{n}{360}

A bank that borrows below IORB and holds the cash as reserves earns the gap, before its balance sheet costs.

Worked example USD 2,000,000,000 × 7 bp / 360 = USD 3,889 a day

Spread to the floor
s=r−rfloor\displaystyle s = r - r_{\text{floor}}

Where the market trades against the rate paid on reserves: below it when non-banks without access to the floor are the lenders.

Worked example EFFR − IORB = −7 bp; SONIA − Bank Rate = −5 bp; €STR − deposit facility = −8 bp

End-of-day position
borrow=max⁡(T−Bclose, 0)Bclose=Bopen+∑in−∑out\displaystyle \text{borrow} = \max\left(T - B_{\text{close}},\ 0\right) \qquad B_{\text{close}} = B_{\text{open}} + \sum \text{in} - \sum \text{out}

Treasury borrows what the account will lack at the close against its target.

TT
target balance

Worked example USD 2,000,000,000 short at 15:00 ET

Credit and liquidity

Credit exposure
E=N+I\displaystyle E = N + I

The lender’s whole exposure is unsecured until repayment.

Worked example USD 2,000,296,111

LCR outflow
O=100%×N\displaystyle O = 100\%\times N

Overnight unsecured funding from a financial firm runs off in full in the LCR.

Worked example USD 2,000,000,000

Brokerage
fee=N b n360\displaystyle \text{fee} = N\,b\,\frac{n}{360}

The broker’s commission, quoted in basis points a year.

bb
brokerage rate

Worked example half a basis point: USD 278

Worked example: USD 2,000,000,000 borrowed overnight at 5.33%, Act/360, with IORB at 5.40% and the ON RRP at 5.30% (illustrative).
Money Markets›Repo

Formulas

The two legs

Purchase price with a margin ratio
PP=MVm\displaystyle PP = \frac{MV}{m}

The cash paid on the start leg for collateral worth MV; US triparty quotes the margin as 102%.

MVMV
collateral market value, accrued interest included
mm
margin ratio

Worked example USD 1,020,000,000 / 1.02 = USD 1,000,000,000

Haircut
PP=MV (1−h)h=1−1m\displaystyle PP = MV\,(1 - h) \qquad h = 1 - \frac{1}{m}

The same protection quoted as a cut to the collateral value (the GMRA’s haircut).

hh
haircut

Worked example a margin of 102% is a haircut of 1.96%; a 2% haircut on USD 1,020,000,000 would lend USD 999,600,000

Repurchase price
RP=PP(1+r tB)\displaystyle RP = PP\left(1 + r\,\frac{t}{B}\right)

What the cash borrower pays on the close leg.

rr
repo rate
BB
360 for USD and EUR, 365 for GBP and CAD

Worked example USD 1,000,000,000 × (1 + 5.32% × 1/360) = USD 1,000,147,778

Repo interest
I=PP r tB\displaystyle I = PP\,r\,\frac{t}{B}

The price differential between the two legs.

Worked example USD 147,778

Open repo
I=PP∑drd ndB\displaystyle I = PP\sum_{d} \frac{r_d\,n_d}{B}

An open repo is renewed daily at a rate reset each day until either side ends it.

rd, ndr_d,\ n_d
the day’s rate and its calendar days

Exposure and margin

Collateral value
MV=Q Pclean+AI100\displaystyle MV = Q\,\frac{P_{\text{clean}} + AI}{100}

Bonds count at their dirty price.

QQ
face amount
AIAI
accrued interest per 100
Exposure of the cash lender
Eτ=PP(1+r τB)−MVτ\displaystyle E_{\tau} = PP\left(1 + r\,\frac{\tau}{B}\right) - MV_{\tau}

What the lender would lose if the borrower defaulted today and the collateral were sold at its value; negative while the margin holds.

Margin call
call=max⁡(m PP(1+r τB)−MVτ, 0)\displaystyle \text{call} = \max\left(m\,PP\left(1 + r\,\frac{\tau}{B}\right) - MV_{\tau},\ 0\right)

The collateral the borrower must add to restore the margin ratio.

Worked example the collateral falls 0.5% to USD 1,014,900,000: a call of USD 5,250,733

Rates and specials

Specialness
s=rGC−rspecial\displaystyle s = r_{\text{GC}} - r_{\text{special}}

How far below general collateral a bond in demand finances.

Worked example 5.31% − 5.10% = 21 bp

Cost of borrowing a special bond
cost=N (rGC−rspecial)tB\displaystyle \text{cost} = N\,\left(r_{\text{GC}} - r_{\text{special}}\right)\frac{t}{B}

The interest the bond borrower gives up on its cash against a GC investment.

Worked example USD 100,000,000 at 5.08% against 5.31%: USD 639 a day

Matched book spread
π=N (rreverse−rrepo)tB\displaystyle \pi = N\,\left(r_{\text{reverse}} - r_{\text{repo}}\right)\frac{t}{B}

The dealer lends cash at one rate and borrows at another against the same bond.

Worked example 5.10% against 5.08% on USD 100,000,000: USD 56 a day

Forward price of a bond
F=(P+AI0)(1+r tB)−∑ici(1+r t−tiB)−AIt\displaystyle F = \left(P + AI_0\right)\left(1 + r\,\frac{t}{B}\right) - \sum_i c_i\left(1 + r\,\frac{t - t_i}{B}\right) - AI_t

Buying a bond and financing it in repo locks in a forward price; the repo rate is the carry.

cic_i
coupons received before t
Implied repo rate
IRR=Ffut CF+AIt−(P+AI0)P+AI0⋅Bt\displaystyle \text{IRR} = \frac{F_{\text{fut}}\,CF + AI_t - \left(P + AI_0\right)}{P + AI_0}\cdot\frac{B}{t}

The financing rate implied by a bond and its futures contract (no coupon before delivery): the basis trade compares it with repo.

CFCF
conversion factor

Balance sheet and clearing

Gross and net exposure
Egross=∑repo+∑reverseEnet=∣∑same cpty, date(repo−reverse)∣\displaystyle E_{\text{gross}} = \sum \text{repo} + \sum \text{reverse} \qquad E_{\text{net}} = \Big|\sum_{\text{same cpty, date}} (\text{repo} - \text{reverse})\Big|

Repo and reverse repo with the same counterparty and maturity can be netted on the balance sheet; a CCP makes them the same counterparty.

Worked example USD 1,000,000,000 of repo and of reverse repo through FICC net to zero

Leverage ratio
LR=Tier 1 capitalleverage exposure\displaystyle \text{LR} = \frac{\text{Tier 1 capital}}{\text{leverage exposure}}

Repo that cannot be netted consumes leverage exposure whatever its collateral.

Worked example: USD 1,000,000,000 overnight at 5.32%, Act/360, collateral at a margin of 102%; GC 5.31%, the special 10-year at 5.10% to the bank and 5.08% to the hedge fund on USD 100,000,000 (illustrative).
Rates›Interest Rate Swap

Formulas

Cash flows

Fixed coupon
Cifix=N K τi\displaystyle C^{\text{fix}}_i = N\,K\,\tau_i

Each fixed payment is the notional times the fixed rate times the accrual fraction of the period.

NN
notional
KK
fixed rate
τi\tau_i
accrual fraction: days/360 for SOFR and €STR, days/365 for SONIA and CORRA, 30/360 for the EURIBOR swap’s fixed leg

Worked example USD 100,000,000 × 3.848% × 365/360 = USD 3,901,444 for a 365-day period

Overnight rate compounded in arrears
Rj=1τj[∏d=1Dj(1+rd nd360)−1]\displaystyle R_j = \frac{1}{\tau_j}\left[\prod_{d=1}^{D_j}\left(1+\frac{r_d\,n_d}{360}\right)-1\right]

The floating rate of a period compounds the daily fixings published during it, so the coupon is known at the end of the period.

rdr_d
the overnight fixing for business day d
ndn_d
the calendar days it applies (3 over a weekend)
DjD_j
the business days in period j
Floating coupon
Cjflt=N Rj τj\displaystyle C^{\text{flt}}_j = N\,R_j\,\tau_j

The floating payment applies the compounded rate to the notional.

Net payment
Xi=Cifix−Ciflt\displaystyle X_i = C^{\text{fix}}_i - C^{\text{flt}}_i

When the legs pay on the same date one net amount moves, here seen from the fixed receiver.

Worked example year 1 (364 days): 3,890,756 − 4,004,000 = −USD 113,244 for the client

Valuation

Discount factor
P(0,T)=e−z(T) T\displaystyle P(0,T) = e^{-z(T)\,T}

The value today of one unit paid at T, from the zero rate of the curve.

z(T)z(T)
zero rate to T, continuously compounded
Forward rate
Fj=1τj(P(0,Tj−1)P(0,Tj)−1)\displaystyle F_j = \frac{1}{\tau_j}\left(\frac{P(0,T_{j-1})}{P(0,T_j)}-1\right)

The rate the curve implies for a future period, used to project an unknown floating coupon.

Annuity
A=∑i=1nτi P(0,Ti)\displaystyle A = \sum_{i=1}^{n}\tau_i\,P(0,T_i)

The present value of one unit of fixed rate paid on every fixed date: the scale of every swap risk number.

Worked example A ≈ 8.30 for the 10-year example

Leg values
PVfix=N K APVflt=N∑j=1mτj Fj P(0,Tj)=N[P(0,T0)−P(0,Tm)]\displaystyle PV_{\text{fix}} = N\,K\,A \qquad PV_{\text{flt}} = N\sum_{j=1}^{m}\tau_j\,F_j\,P(0,T_j) = N\left[P(0,T_0)-P(0,T_m)\right]

With forwards and discounting from the same overnight curve, the floating leg telescopes to two discount factors.

Swap value
Vrec=PVfix−PVfltVpay=−Vrec\displaystyle V_{\text{rec}} = PV_{\text{fix}} - PV_{\text{flt}} \qquad V_{\text{pay}} = -V_{\text{rec}}

The receiver of fixed holds the fixed leg and owes the floating leg; the payer is the mirror image.

Par swap rate
S=P(0,T0)−P(0,Tn)A\displaystyle S = \frac{P(0,T_0)-P(0,T_n)}{A}

The fixed rate that makes the swap worth zero: the rate quoted on the screens.

Worked example S = 3.850% on the example curve

Mark to market
Vrec(t)=N (K−St) At\displaystyle V_{\text{rec}}(t) = N\,(K - S_t)\,A_t

After the trade the swap is worth the difference between its fixed rate and today’s par rate, times today’s annuity.

StS_t
par rate today for the remaining maturity
AtA_t
annuity today

Worked example par falls to 3.800%: 100,000,000 × (3.848% − 3.800%) × 8.30 ≈ +USD 398,000 for the client

Two curves (EURIBOR swap)
PVflt=N∑jτj Fj 6M P €STR(0,Tj)\displaystyle PV_{\text{flt}} = N\sum_{j}\tau_j\,F^{\,6\text{M}}_j\,P^{\,\text{€STR}}(0,T_j)

The forwards come from the EURIBOR curve and the discounting from the €STR curve, so the floating leg no longer telescopes.

The curve

Bootstrapping
P(0,Tn)=1−Sn∑i=1n−1τi P(0,Ti)1+Sn τn\displaystyle P(0,T_n) = \frac{1 - S_n\sum_{i=1}^{n-1}\tau_i\,P(0,T_i)}{1+S_n\,\tau_n}

Each discount factor is solved from the par rate of the next tenor, starting from the shortest; futures set the short end and interpolation fills between quotes.

SnS_n
par rate of the n-year swap
P(0,T0)P(0,T_0)
1 for a spot-starting curve
Futures convexity adjustment
Ffwd≈Ffut−12 σ2 t1 t2\displaystyle F^{\text{fwd}} \approx F^{\text{fut}} - \tfrac{1}{2}\,\sigma^2\,t_1\,t_2

A futures rate sits above the forward rate of the same period because futures margin daily; the gap grows with maturity.

σ\sigma
normal volatility of the short rate
t1, t2t_1,\ t_2
start and end of the period, in years

Risk

DV01
DV01=−∂V∂y×0.0001≈N A×0.0001\displaystyle \text{DV01} = -\frac{\partial V}{\partial y}\times 0.0001 \approx N\,A\times 0.0001

The change in value for a one basis point parallel move of the curve; for a swap close to par it is the notional times the annuity.

Worked example 100,000,000 × 8.30 × 0.0001 ≈ USD 83,000 per bp

Bucketed delta
Δk=V(qk+1 bp)−V(qk)∑kΔk≈DV01\displaystyle \Delta_k = V(q_k + 1\,\text{bp}) - V(q_k) \qquad \sum_k \Delta_k \approx \text{DV01}

The sensitivity to each quote of the curve, so a hedge can match the risk tenor by tenor.

qkq_k
par quote of curve instrument k
P&L explain
ΔV≈∑kΔk δqk+12∑kΓk δqk2+Θ δt+new trades+unexplained\displaystyle \Delta V \approx \sum_k \Delta_k\,\delta q_k + \tfrac{1}{2}\sum_k \Gamma_k\,\delta q_k^2 + \Theta\,\delta t + \text{new trades} + \text{unexplained}

The daily P&L split into its sources; a large unexplained part points to a model, data or booking issue.

Γk\Gamma_k
gamma by bucket
Θ\Theta
carry and roll-down
Carry of a receiver
carry per day≈N (K−r) 1360\displaystyle \text{carry per day} \approx N\,(K - r)\,\frac{1}{360}

What the swap earns each day if rates do not move, before roll-down.

rr
today’s overnight rate

Counterparty and margin

Variation margin
VMt=Vt−Vt−1PAIt=Mt−1 rt−1 n360\displaystyle VM_t = V_t - V_{t-1} \qquad PAI_t = M_{t-1}\,r_{t-1}\,\frac{n}{360}

The change in value is settled in cash each day; the CCP charges or pays price alignment interest on the cumulative margin so that the swap’s value is unaffected.

Mt−1M_{t-1}
cumulative variation margin received
nn
days since the last payment
CVA
CVA≈(1−R)∑k=1KEE(tk)  PD(tk−1,tk)  P(0,tk)\displaystyle \text{CVA} \approx (1-R)\sum_{k=1}^{K} EE(t_k)\;PD(t_{k-1},t_k)\;P(0,t_k)

The expected loss from the counterparty’s default, from the simulated exposure profile and the default probabilities implied by its CDS.

RR
recovery rate
EE(tk)EE(t_k)
expected positive exposure at t_k
PDPD
probability of default between two dates
Initial margin, ISDA SIMM (rate delta, one currency)
WSk=RWk skIM=∑kWSk2+∑k∑l≠kρkl WSk WSl\displaystyle WS_k = RW_k\,s_k \qquad IM = \sqrt{\sum_k WS_k^2 + \sum_k\sum_{l\neq k}\rho_{kl}\,WS_k\,WS_l}

Uncleared swaps exchange initial margin computed from weighted sensitivities by tenor; simplified here (the concentration and basis terms left out).

sks_k
sensitivity to tenor k, per bp
RWkRW_k
risk weight set by ISDA
ρkl\rho_{kl}
correlation between tenors

Conventions of the worked example (USD, SOFR, Act/360); the other jurisdictions change the day count and the benchmark, as in the card.
Rates›Swaption

Formulas

Pricing (Bachelier on the swap rate)

Annuity of the forward swap
A=∑i=1nτi P(0,Ti)\displaystyle A = \sum_{i=1}^{n} \tau_i\,P(0,T_i)

The present value of 1 paid on each fixed date of the underlying swap: the value of one unit of swap rate, and the scale of every swaption number.

τi\tau_i
accrual fraction of period i, Act/360
P(0,Ti)P(0,T_i)
discount factor to the payment date of period i

Worked example A = 7.930 for the 10 annual periods from Nov 1, 2027

Forward swap rate
S0=∑jτj Fj P(0,Tj)A\displaystyle S_0 = \frac{\sum_{j} \tau_j\,F_j\,P(0,T_j)}{A}

The fixed rate that makes the forward-starting swap worth zero today: the at-the-money strike.

FjF_j
forward SOFR rate of period j from the curve

Worked example S₀ = 3.900%

Payer and receiver
Vpay=NA[(S0−K) Φ(d)+σNT φ(d)]Vrec=NA[(K−S0) Φ(−d)+σNT φ(d)]\displaystyle V_{\text{pay}} = N A\left[(S_0-K)\,\Phi(d) + \sigma_N\sqrt{T}\,\varphi(d)\right] \qquad V_{\text{rec}} = N A\left[(K-S_0)\,\Phi(-d) + \sigma_N\sqrt{T}\,\varphi(d)\right]

The normal model applied to the forward swap rate, with the annuity as the unit of account: the market standard since rates went close to zero.

NN
notional
KK
strike
σN\sigma_N
normal volatility, in rate units a year (98.0 bp = 0.0098)
TT
time to expiry, years
Φ, φ\Phi,\ \varphi
standard normal distribution and density

Worked example 100,000,000 × 7.930 × 0.0098 × √1 × 0.3989 = USD 3,100,425

d
d=S0−KσNT\displaystyle d = \frac{S_0 - K}{\sigma_N\sqrt{T}}

The distance between the forward swap rate and the strike in standard deviations of the swap rate at expiry.

Worked example d = 0.00 at the money

At the money
VATM=NA σNT2π≈0.4 NA σNT\displaystyle V_{\text{ATM}} = N A\,\sigma_N\sqrt{\frac{T}{2\pi}} \approx 0.4\,N A\,\sigma_N\sqrt{T}

The rule of thumb traders use for a straddle leg at the forward.

Worked example 0.3989 × 7.930 × 0.98% = 3.10% of the notional; at the 96.5 bp mid USD 3,052,970

Put-call parity
Vpay−Vrec=NA (S0−K)\displaystyle V_{\text{pay}} - V_{\text{rec}} = N A\,(S_0 - K)

A bought payer and a sold receiver at the same strike make a forward-starting swap paying the strike.

Worked example zero at the money: the payer and the receiver at 3.90% cost the same

Normal and lognormal volatility
σN≈σB S0K  ≈  σB S0 at the money\displaystyle \sigma_N \approx \sigma_B\,\sqrt{S_0 K} \;\approx\; \sigma_B\,S_0 \ \text{at the money}

The Black (lognormal) volatility quoted before the 2010s and its normal equivalent; shifted Black adds a constant to the rates to allow negative levels, as in euro.

σB\sigma_B
Black volatility, % of the rate

Worked example 98 bp at 3.90% is about 25.1% Black volatility

Greeks and the hedge

Delta
Δpay=Φ(d)Δrec=−Φ(−d)\displaystyle \Delta_{\text{pay}} = \Phi(d) \qquad \Delta_{\text{rec}} = -\Phi(-d)

The share of the notional of the forward swap that hedges the option against small moves of the swap rate; a payer behaves like part of a pay-fixed swap.

Worked example 0.50: the bank, short the payer, pays fixed on USD 50,000,000 of the 1y-forward 10-year swap

DV01
DV01=NA Φ(d)×0.0001\displaystyle \text{DV01} = N A\,\Phi(d)\times 0.0001

The change in value for a 1 bp rise of the forward swap rate.

Worked example USD 39,651 per bp

Gamma
Γ=φ(d)σNT\displaystyle \Gamma = \frac{\varphi(d)}{\sigma_N\sqrt{T}}

How fast the delta changes as the swap rate moves: largest at the money and close to expiry.

Worked example a 10 bp move changes the hedge by about USD 4,070,000 of forward swap

Vega
ν=NAT φ(d)×0.0001\displaystyle \nu = N A\sqrt{T}\,\varphi(d)\times 0.0001

The change in value for a 1 bp change of the normal volatility.

Worked example USD 31,637 per bp of volatility

Theta
Θ≈−NA σN φ(d)2T\displaystyle \Theta \approx -\frac{N A\,\sigma_N\,\varphi(d)}{2\sqrt{T}}

The time decay of the option, a year; divided by 365 for a day.

Worked example about −USD 4,247 a day today

Delta-hedged P&L
dΠ≈12 NA Γ(δS2−σN2 δt)\displaystyle d\Pi \approx \tfrac{1}{2}\,N A\,\Gamma\left(\delta S^2 - \sigma_N^2\,\delta t\right)

A delta-hedged long swaption gains when the swap rate moves more than the volatility paid implies and loses its time value otherwise.

δS\delta S
move of the forward swap rate over the day

Payoffs at expiry

Payer
N AT max⁡(ST−K, 0)\displaystyle N\,A_T\,\max(S_T - K,\,0)

What the payer swaption is worth at expiry: the rate difference times the annuity of the swap on the expiry curve.

STS_T
10-year swap rate at expiry
ATA_T
annuity at expiry

Worked example (4.40% − 3.90%) × 8.045 × 100,000,000 = USD 4,022,425, less the premium +USD 922,000

Receiver
N AT max⁡(K−ST, 0)\displaystyle N\,A_T\,\max(K - S_T,\,0)

The mirror image, for protection against lower rates.

Worked example 3.60% strike: USD 2,055,040 premium, breakeven 3.36%

Straddle
N AT ∣ST−K∣\displaystyle N\,A_T\,\left|S_T - K\right|

A payer and a receiver at the same strike: the value grows with the size of the move either way.

Worked example USD 6,200,850, breakevens 3.18% and 4.68%

Breakeven of the payer
S∗=K+VpayN AT\displaystyle S^{*} = K + \frac{V_{\text{pay}}}{N\,A_T}

The swap rate at expiry where the settlement repays the premium.

Worked example 4.28%

Exercise and settlement

Exercise decision
payer: exercise if ST>Kreceiver: exercise if ST<K\displaystyle \text{payer: exercise if } S_T > K \qquad \text{receiver: exercise if } S_T < K

European exercise at 11:00 New York on the expiry date, against the market swap rate.

Worked example 4.40% > 3.90%: exercised

Cash settlement, collateralized cash price
CS=N (ST−K)∑i=1nτi PT(Ti)\displaystyle CS = N\,(S_T - K)\sum_{i=1}^{n}\tau_i\,P_T(T_i)

The value at expiry of the swap at the strike, discounted on the overnight curve of the settlement date, as in the 2021 ISDA Definitions; here the curve is flat at the settlement rate.

PT(Ti)P_T(T_i)
discount factor at expiry to date T_i

Worked example annuity 8.045: USD 4,022,425 on Nov 1, 2027

Physical settlement
Vswap(T)=N (ST−K) AT\displaystyle V_{\text{swap}}(T) = N\,(S_T - K)\,A_T

The client receives a swap paying the strike, cleared at the CCP: the same value, kept as a position instead of cash.

Exposure of the buyer
EE(t)=E[max⁡(Vt, 0)]=E[Vt]\displaystyle EE(t) = \mathbb{E}\left[\max(V_t,\,0)\right] = \mathbb{E}\left[V_t\right]

A bought swaption is worth something to its holder at all times, so the client’s exposure to the bank is its whole value, covered by the variation margin the bank posts.

Market data of the worked example, illustrative: trade on Oct 28, 2026, expiry Oct 28, 2027, swap from Nov 1, 2027 to Nov 2, 2037, annual SOFR legs Act/360, a flat SOFR curve (zero rate 3.878%, continuous) giving a 3.900% forward swap rate, normal volatility 98.0 bp for the client.
Rates›Overnight Rate Future

Formulas

Price and final settlement

Price and rate
P=100−R\displaystyle P = 100 - R

The futures price is 100 minus the rate of the reference quarter, in percent, so the price rises when rates fall.

PP
futures price
RR
rate of the reference quarter, %

Worked example 96.1475: R = 3.8525%

Final settlement rate
R=[∏d=1D(1+rd nd360)−1]360NQ×100\displaystyle R = \left[\prod_{d=1}^{D}\left(1+\frac{r_d\,n_d}{360}\right)-1\right]\frac{360}{N_Q}\times 100

At expiry the rate is the overnight benchmark compounded over the reference quarter from the published daily fixings.

rdr_d
fixing of business day d, decimal
ndn_d
calendar days it applies (3 over a weekend)
NQN_Q
days in the quarter (91)

Worked example SOFR at 3.70% to May 5 and 3.45% after (illustrative): R = 3.6033%, final price 96.3967

Basis point value
BPV=N×0.0001×90360=25\displaystyle \text{BPV} = N\times 0.0001\times\frac{90}{360} = 25

One basis point on one million of notional for a standard quarter: the value of one contract’s move.

NN
notional, 1,000,000 (the contract unit is 2,500 times the price)

Worked example USD 25 per basis point; USD 12.50 for a tick of 0.005

Result to expiry
P&L=(Pfinal−P0)×100×BPV×n\displaystyle \text{P\&L} = (P_{\text{final}} - P_0)\times 100\times\text{BPV}\times n

Held to expiry, the daily margin adds up to the change from the trade price to the final settlement price.

nn
contracts, negative for a seller

Worked example (96.3967 − 96.1475) × 100 × 25 × 2,000 = +USD 1,246,000

Variation margin

Daily variation margin
VMt=(St−St−1)×100×BPV×n\displaystyle VM_t = (S_t - S_{t-1})\times 100\times\text{BPV}\times n

The change in the settlement price in basis points, times 25, times the contracts; on the trade day the trade price stands in for the previous settlement.

StS_t
settlement price of day t

Worked example Oct 30: (96.185 − 96.140) × 100 × 25 × 2,000 = +USD 225,000

Cumulative margin
∑t=1TVMt=(ST−P0)×100×BPV×n\displaystyle \sum_{t=1}^{T} VM_t = (S_T - P_0)\times 100\times\text{BPV}\times n

The daily amounts add up to the change from the trade price.

Worked example (96.215 − 96.1475) × 100 × 25 × 2,000 = +USD 337,500

Margin account
Mt=Mt−1+VMtcall to Minit when Mt<Mmaint\displaystyle M_t = M_{t-1} + VM_t \qquad \text{call to } M^{\text{init}} \text{ when } M_t < M^{\text{maint}}

The account at the broker absorbs the daily margin; below the maintenance level the broker calls it back to the initial level.

Minit, MmaintM^{\text{init}},\ M^{\text{maint}}
initial and maintenance margin

Worked example initial margin 2,000 × USD 500 = USD 1,000,000

Curve and convexity

Implied futures rate
Ffut=100−P\displaystyle F^{\text{fut}} = 100 - P

The rate the futures price implies for the reference quarter.

Worked example 96.155 at the first settlement: 3.845%

Convexity adjustment
Ffwd≈Ffut−12 σ2 t1 t2\displaystyle F^{\text{fwd}} \approx F^{\text{fut}} - \tfrac{1}{2}\,\sigma^2\,t_1\,t_2

A buyer’s margin gains arrive when rates are low and its losses when rates are high, so the futures rate sits above the forward rate; the gap grows with the square of the maturity.

σ\sigma
normal volatility of the short rate
t1, t2t_1,\ t_2
start and end of the reference quarter, years

Worked example 0.10 bp for March 2027; 11.2 bp for the December 2031 contract

Strip rate
Rstrip=[∏k=1K(1+Rk dk360)−1]360∑kdk\displaystyle R_{\text{strip}} = \left[\prod_{k=1}^{K}\left(1+R_k\,\frac{d_k}{360}\right)-1\right]\frac{360}{\sum_k d_k}

Consecutive contracts compound into the rate of a longer period: the price of a one-year deposit or of the first year of a swap.

RkR_k
rate of contract k (after convexity for back contracts)
dkd_k
days in quarter k

Worked example Dec 2026 to Sep 2027 at 96.050, 96.155, 96.300, 96.405: 3.827% for 364 days

Discount factor from the strip
P(0,TK)=∏k=1K(1+Rk dk360)−1\displaystyle P(0,T_K) = \prod_{k=1}^{K}\left(1+R_k\,\frac{d_k}{360}\right)^{-1}

The futures strip gives the discount factors of the short end of the curve, from the first IMM date.

Rate after a meeting inside the quarter
r2≈R NQ−r1 d1NQ−d1\displaystyle r_2 \approx \frac{R\,N_Q - r_1\,d_1}{N_Q - d_1}

When a policy meeting falls inside the reference quarter, the contract’s rate and the rate before the meeting give the rate the market expects after it, ignoring compounding.

r1r_1
overnight rate before the meeting
d1d_1
days of the quarter before the new rate applies

Worked example R = 3.6033%, 3.70% for 50 days: r₂ ≈ 3.49% for the 41 days after, against 3.45% fixed (the gap is the compounding)

Risk and hedging

DV01 of a position
DV01=n×BPV\displaystyle \text{DV01} = n\times\text{BPV}

The change in value for a one basis point move of the contract’s rate.

Worked example 2,000 × 25 = USD 50,000 per bp

Hedge ratio
n=DV01exposureBPV\displaystyle n = \frac{\text{DV01}_{\text{exposure}}}{\text{BPV}}

Contracts that offset an exposure, using its DV01 in present value: the futures margin is paid now, the hedged flows later.

Worked example a USD 50,000,000 3x6 FRA with a DV01 of USD 1,236: about 50 contracts

Initial margin
IM=n×m\displaystyle IM = n\times m

The clearing house’s margin per contract times the contracts, before the broker’s add-on.

mm
margin per contract

Worked example 2,000 × USD 500 = USD 1,000,000

Conventions of the US worked example (SR3, SOFR, Act/360, USD 25 per basis point); SONIA and CORRA compound on Act/365F and the EURIBOR future settles to a single fixing. Volatility for the convexity adjustment 0.90% normal (90 bp), illustrative.
Rates›Bond Future

Formulas

Price, invoice and delivery

Contract and tick
1 point=1,000132=31.25164=15.625\displaystyle 1 \text{ point} = 1{,}000 \qquad \tfrac{1}{32} = 31.25 \qquad \tfrac{1}{64} = 15.625

TY is USD 100,000 face, so a point of price is USD 1,000 and the tick, half a 32nd, USD 15.625.

Worked example 110-16 = 110.5; one 32nd on 500 contracts = USD 15,625

Invoice amount
Invoice=Face100 (F×CF+AIT)\displaystyle \text{Invoice} = \frac{\text{Face}}{100}\,\left(F\times CF + AI_T\right)

What the long pays for the bond delivered: the futures settlement price times the conversion factor, plus accrued interest.

FF
futures settlement price
CFCF
conversion factor of the bond delivered
AITAI_T
accrued interest on the delivery day, per 100

Worked example 500,000 × (110.5 × 0.8937 + 1.4917) = USD 50,122,781

Theoretical futures price
F∗=FPCTDCFCTD−delivery option valueCFCTD\displaystyle F^{*} = \frac{FP_{\text{CTD}}}{CF_{\text{CTD}}} - \frac{\text{delivery option value}}{CF_{\text{CTD}}}

Without the short’s options the future would trade at the CTD’s forward price over its factor; the options lower it.

Worked example 110.575 (110-18.4) without options, against 110.500 traded

Conversion factor and cheapest to deliver

Conversion factor (CBOT)
CF=a[c2+k+c0.06 (1−k)]−ba=1.03−v/6b=c2⋅6−v6\displaystyle CF = a\left[\frac{c}{2} + k + \frac{c}{0.06}\,(1-k)\right] - b \qquad a = 1.03^{-v/6} \quad b = \frac{c}{2}\cdot\frac{6-v}{6}

The price per 1 of face at a 6% yield, from the first day of the delivery month to the maturity rounded down to the quarter, less accrued interest.

cc
coupon, decimal
n, zn,\ z
whole years, and months beyond them rounded down to 0, 3, 6 or 9
vv
z if z < 7, otherwise 3
kk
1.03 to the power −2n, or −(2n + 1) when z is 7 or more

Worked example 4% Feb 2034, 6 years 6 months from Jun 1, 2027: 0.8937; 4.375% May 2034: 0.9108

Gross basis
B=P−F×CF\displaystyle B = P - F\times CF

The cash price less the futures price converted to that bond, quoted in 32nds.

PP
clean price of the bond

Worked example CTD: 98.892 − 110.5 × 0.8937 = 4.4/32

Cheapest to deliver
CTD=arg⁡max⁡i IRRi=arg⁡min⁡i NBi\displaystyle \text{CTD} = \arg\max_i\ \text{IRR}_i = \arg\min_i\ NB_i

The short delivers the bond with the highest implied repo rate: buying it in repo and delivering it into the future costs the least.

Worked example 4% Feb 2034 3.59%; 4.375% May 2034 3.26%; 4.25% Nov 2034 2.37%; 4.625% Feb 2035 2.10%

Which bond is cheapest
y<6%⇒lowest durationy>6%⇒highest duration\displaystyle y < 6\% \Rightarrow \text{lowest duration} \qquad y > 6\% \Rightarrow \text{highest duration}

With yields below the notional coupon the factors favor the short, low-duration bonds; above it the long ones.

Worked example yields near 4.2%: the shortest note, 4% Feb 2034, is cheapest

Basis and implied repo

Forward price
FP=(P+AI0)(1+r t360)−C(1+r tc360)−AIT\displaystyle FP = (P + AI_0)\left(1 + r\,\frac{t}{360}\right) - C\left(1 + r\,\frac{t_c}{360}\right) - AI_T

The price at which the bond can be bought for delivery: the dirty price financed in repo to the delivery date, less a coupon received, less the accrued then.

rr
term repo rate
tt
days from settlement to delivery
C, tcC,\ t_c
coupon received before delivery and the days from it to delivery

Worked example CTD: (98.892 + 0.177) × (1 + 3.80% × 119/360) − 1.492 = 98.821

Carry
Carry=P−FP\displaystyle \text{Carry} = P - FP

Coupon income less financing over the period, per 100: positive when the bond’s coupon yield exceeds the repo rate.

Worked example CTD: 2.3/32

Net basis
NB=B−Carry=FP−F×CF\displaystyle NB = B - \text{Carry} = FP - F\times CF

The gross basis less carry: the price of the short’s delivery options, and of any mispricing.

Worked example CTD: 4.4/32 − 2.3/32 = 2.2/32; 4.375% May 2034: 5.8/32

Implied repo rate
IRR=F×CF+AIT+C−(P+AI0)(P+AI0) t360−C tc360\displaystyle \text{IRR} = \frac{F\times CF + AI_T + C - (P + AI_0)}{(P + AI_0)\,\frac{t}{360} - C\,\frac{t_c}{360}}

The return on buying the bond, selling the future and delivering; below the repo rate the basis trade earns the options’ value only if they pay off.

Worked example CTD: 3.59% against 3.80% term repo

Variation margin

Daily variation margin
VMt=(St−St−1)×1,000×n\displaystyle VM_t = (S_t - S_{t-1})\times 1{,}000\times n

The change in the settlement price in points, times USD 1,000, times the contracts, negative for a short.

StS_t
settlement price of day t; the trade price on the first day
nn
contracts, −500 for the client

Worked example Mar 5: (111-02 − 110-19) = 15/32 × 1,000 × −500 = −USD 234,375

Cumulative margin
∑t=1TVMt=(ST−F0)×1,000×n\displaystyle \sum_{t=1}^{T} VM_t = (S_T - F_0)\times 1{,}000\times n

The five days add up to the price change since the trade.

Worked example (110-27 − 110-16) × 1,000 × −500 = −USD 171,875

Risk and hedging

DV01 of the future
DV01F≈DV01CTDCFCTD\displaystyle \text{DV01}_F \approx \frac{\text{DV01}_{\text{CTD}}}{CF_{\text{CTD}}}

The future moves with the cheapest bond scaled by its factor.

DV01CTD\text{DV01}_{\text{CTD}}
per USD 100,000 face

Worked example USD 59.4 / 0.8937 = USD 66.5 per contract

Hedge ratio
n=−β DV01portfolioDV01F\displaystyle n = -\beta\,\frac{\text{DV01}_{\text{portfolio}}}{\text{DV01}_F}

Contracts to sell so that the futures offset the portfolio’s rate risk; β scales for a portfolio whose yields move more or less than the CTD’s.

β\beta
yield beta, 1 here

Worked example USD 33,249 per bp / 66.5 = 500 contracts sold

Cost of the roll
roll=Fnear−Fnext≈carry3 monthsCF\displaystyle \text{roll} = F_{\text{near}} - F_{\text{next}} \approx \frac{\text{carry}_{\text{3 months}}}{CF}

Rolling a short into the next contract sells the deferred future below the near one when carry is positive.

Worked example on TY June 2027, trade Mar 2, 2027, cash settlement Mar 3, 2027, delivery on the last day (Jun 30, 2027), term repo 3.80%; the deliverable notes and their yields are illustrative. The gilt, Bund and CGB contracts use the same quantities with their own notional coupon and factor rules.
Rates›Option on Futures

Formulas

Pricing (Black-76)

Call and put on a future
C=e−rT[F Φ(d1)−K Φ(d2)]P=e−rT[K Φ(−d2)−F Φ(−d1)]\displaystyle C = e^{-rT}\left[F\,\Phi(d_1) - K\,\Phi(d_2)\right] \qquad P = e^{-rT}\left[K\,\Phi(-d_2) - F\,\Phi(-d_1)\right]

Black’s model: the futures price has no carry, so it plays the part of the forward; the premium paid up front is discounted from expiry.

FF
futures price
KK
strike
rr
discount rate
TT
time to expiry, years
Φ\Phi
standard normal distribution

Worked example P = 0.5000 points = 0’32 at σ = 7.55%

d₁ and d₂
d1,2=ln⁡(F/K)±12 σ2TσT\displaystyle d_{1,2} = \frac{\ln(F/K) \pm \tfrac{1}{2}\,\sigma^2 T}{\sigma\sqrt{T}}

The distance between the futures price and the strike in standard deviations of the price at expiry.

σ\sigma
implied volatility of the futures price

Worked example F = 110.78125, K = 110, T = 23/365: d₁ = 0.383, d₂ = 0.364

Put-call parity
C−P=e−rT (F−K)\displaystyle C - P = e^{-rT}\,(F - K)

A bought call and a sold put at the same strike make a long future at the strike.

Premium in 64ths
premium=m64×USD 1,000 per contract\displaystyle \text{premium} = \frac{m}{64}\times \text{USD } 1{,}000 \text{ per contract}

TY options are quoted in points and 64ths; one point is USD 1,000 on a USD 100,000 contract.

mm
premium in 64ths

Worked example 32/64 × 1,000 = USD 500 a contract, USD 500,000 for 1,000

Price and yield volatility
σyield≈σprice FDF\displaystyle \sigma_{\text{yield}} \approx \frac{\sigma_{\text{price}}\,F}{D_F}

Converts the price volatility of the option into basis points of yield, to compare it with swaption volatilities.

DFD_F
change in the futures price for a 1 percentage point change in yield
Early exercise
PAmerican=PEuropean+ε,ε≥0\displaystyle P_{\text{American}} = P_{\text{European}} + \varepsilon, \qquad \varepsilon \ge 0

The right to exercise early is worth a little for a deep in-the-money option: exercising frees the premium’s value as variation margin.

ε\varepsilon
early exercise premium

Greeks and the hedge

Delta
ΔC=e−rT Φ(d1)ΔP=−e−rT Φ(−d1)\displaystyle \Delta_C = e^{-rT}\,\Phi(d_1) \qquad \Delta_P = -e^{-rT}\,\Phi(-d_1)

The number of futures that hedges one option against small moves.

Worked example Δ = −0.350: the market maker short 1,000 puts sells 350 futures

Gamma
Γ=e−rT φ(d1)F σT\displaystyle \Gamma = \frac{e^{-rT}\,\varphi(d_1)}{F\,\sigma\sqrt{T}}

How fast the delta changes with the futures price: largest at the strike and close to expiry.

φ\varphi
standard normal density

Worked example a 1-point fall adds about 176 futures to the market maker’s hedge

Vega
ν=e−rT F φ(d1)T\displaystyle \nu = e^{-rT}\,F\,\varphi(d_1)\sqrt{T}

The change in value for a change in volatility, quoted per volatility point.

Worked example about USD 102,800 per point on 1,000 puts

Theta
Θ=∂V∂t\displaystyle \Theta = \frac{\partial V}{\partial t}

The value the option loses each day if nothing moves.

Worked example about −USD 17,000 a day on 1,000 puts today

Hedge ratio for a bond portfolio
n=DV01portfolio∣Δ∣ DV01future\displaystyle n = \frac{\text{DV01}_{\text{portfolio}}}{\left|\Delta\right|\,\text{DV01}_{\text{future}}}

The number of puts whose delta offsets the rate sensitivity of a portfolio; it changes as the delta moves.

DV01future\text{DV01}_{\text{future}}
the cheapest to deliver’s DV01 divided by its conversion factor

Payoffs at expiry

Put and call
max⁡(K−FT, 0)max⁡(FT−K, 0)\displaystyle \max(K - F_T,\,0) \qquad \max(F_T - K,\,0)

What the option is worth at expiry per point of the contract, before the premium; exercised into a futures position.

FTF_T
futures price at expiry

Worked example (110 − 108.50) × 1,000 × 1,000 = USD 1,500,000, less USD 500,000 = +USD 1,000,000

Breakeven of the put
F∗=K−P\displaystyle F^{*} = K - P

The futures price at expiry where the put repays its premium.

Worked example 110 − 0’32 = 109-16 (109.50)

Put spread
max⁡(K1−FT, 0)−max⁡(K2−FT, 0)\displaystyle \max(K_1 - F_T,\,0) - \max(K_2 - F_T,\,0)

A bought put and a sold put at a lower strike: cheaper protection, capped at the difference of the strikes.

K1>K2K_1 > K_2
strikes

Worked example 110 / 108: 0’26 net, USD 406,250

Overwrite
(FT−F0)−max⁡(FT−K, 0)\displaystyle (F_T - F_0) - \max(F_T - K,\,0)

A long futures position and a sold call: income from the premium for a capped gain.

Worked example 112 calls at 0’24: USD 375,000 received

Exercise and margin

Exercise into futures
VM1=(K−Fsettle)×1,000×n\displaystyle \text{VM}_1 = (K - F_{\text{settle}}) \times 1{,}000 \times n

The exercised put opens a short future at the strike; its first mark to the settlement price pays the intrinsic value as variation margin.

nn
number of contracts

Worked example (110 − 108.50) × 1,000 × 1,000 = USD 1,500,000

Premium-paid style (CME, MX)
cashT+1=−P×1,000×nIMlong=0\displaystyle \text{cash}_{T+1} = -P\times 1{,}000\times n \qquad \text{IM}_{\text{long}} = 0

The buyer pays the premium in full the next day and has no further margin; the writer is margined.

Worked example USD 500,000 on T+1

Futures-style (ICE, Eurex)
VMt=Vt−Vt−1∑tVMt+V0=VT\displaystyle \text{VM}_t = V_t - V_{t-1} \qquad \sum_t \text{VM}_t + V_0 = V_T

No premium at purchase: the option is marked to market every day like a future, and the original premium is paid when the option is exercised or expires.

Scenario margin (simplified)
IM=max⁡s[−ΔV(F+δFs, σ+δσs)]\displaystyle \text{IM} = \max_{s}\left[-\Delta V\left(F + \delta F_s,\ \sigma + \delta\sigma_s\right)\right]

The clearing house revalues the writer’s portfolio under price and volatility scenarios and calls the worst loss; SPAN 2 at CME and Prisma at Eurex add correlation and liquidity terms.

Market data of the worked example, illustrative: trade on Oct 28, 2026, expiry Nov 20, 2026 (23 days), TY December future 110-25, discount rate 3.90%, implied volatility 7.55% solved so that the 110 put is worth 0’32; the other strikes at the same volatility.
Rates›Cap and Floor

Formulas

Valuation

Caplet (Bachelier)
ci=N τi P(0,Tip)[(Fi−K) Φ(di)+σiTie φ(di)],di=Fi−KσiTie\displaystyle c_i = N\,\tau_i\,P(0,T^{p}_i)\left[(F_i - K)\,\Phi(d_i) + \sigma_i\sqrt{T^{e}_i}\,\varphi(d_i)\right], \quad d_i = \frac{F_i - K}{\sigma_i\sqrt{T^{e}_i}}

Each caplet is a call on the rate of its period, priced in the normal model and discounted from its payment date.

FiF_i
forward rate of period i
τi\tau_i
accrual fraction
TipT^{p}_i
payment date
TieT^{e}_i
effective expiry
σi\sigma_i
normal volatility of the caplet

Worked example period 5: forward 4.950%, 94 days

Effective expiry, rate compounded in arrears
Tie=ts,i+te,i−ts,i3\displaystyle T^{e}_i = t_{s,i} + \frac{t_{e,i} - t_{s,i}}{3}

A compounded rate keeps moving until the last day of its period, but less and less: its variance is that of a rate fixed one third of the way through.

ts,i, te,it_{s,i},\ t_{e,i}
start and end of period i, in years from today

Worked example period 5: 1.089 years

Cap and floor
Cap=∑iciFloor=∑ifi\displaystyle \text{Cap} = \sum_{i} c_i \qquad \text{Floor} = \sum_{i} f_i

A cap is the sum of its caplets, a floor the sum of its floorlets.

Worked example 1.38% = USD 1,380,000 at 111.2 bp; mid 1.30% at 104.8 bp

Flat and caplet volatilities
Cap(σflat)=∑ici(σi)\displaystyle \text{Cap}(\sigma_{\text{flat}}) = \sum_{i} c_i(\sigma_i)

The market quotes one flat volatility per cap maturity; stripping finds caplet volatilities consistent with every quoted maturity.

Cap-floor parity
Cap(K)−Floor(K)=N∑iτi P(0,Tip) (Fi−K)\displaystyle \text{Cap}(K) - \text{Floor}(K) = N\sum_{i}\tau_i\,P(0,T^{p}_i)\,(F_i - K)

A bought cap and a sold floor at the same strike make a swap paying the strike.

Premium as a running cost
π=CapN∑iτi P(0,Tip)\displaystyle \pi = \frac{\text{Cap}}{N\sum_{i}\tau_i\,P(0,T^{p}_i)}

The premium spread over the life of the cap, the way a borrower compares it with its loan margin.

Worked example 1.38% / 2.808 = 0.49% a year

Cash flows

Caplet payment
Xi=N τi max⁡(Ri−K, 0)\displaystyle X_i = N\,\tau_i\,\max(R_i - K,\,0)

What the cap pays for period i, two business days after the period ends for SOFR.

RiR_i
compounded SOFR of period i
KK
strike

Worked example (5.40% − 5.00%) × 100,000,000 × 94/360 = USD 104,444

Rate compounded in arrears
Ri=1τi[∏d=1Di(1+rd nd360)−1]\displaystyle R_i = \frac{1}{\tau_i}\left[\prod_{d=1}^{D_i}\left(1+\frac{r_d\,n_d}{360}\right)-1\right]

The period’s rate compounds the daily fixings, so it is known on the last day.

rdr_d
SOFR for business day d
ndn_d
calendar days it applies
EURIBOR caplet
Xi=N τi max⁡(Li−K, 0),Li fixed at ts,i−2 days\displaystyle X_i = N\,\tau_i\,\max(L_i - K,\,0), \quad L_i \text{ fixed at } t_{s,i} - 2 \text{ days}

A term rate fixes before the period starts and pays at its end: the payment is known three months ahead.

Floorlet
Yi=N τi max⁡(K−Ri, 0)\displaystyle Y_i = N\,\tau_i\,\max(K - R_i,\,0)

What a floor pays for a period whose rate ends below the strike.

Risk and the hedge

DV01
DV01=∑iN τi P(0,Tip) Φ(di)×0.0001\displaystyle \text{DV01} = \sum_{i} N\,\tau_i\,P(0,T^{p}_i)\,\Phi(d_i)\times 0.0001

The change in the cap’s value for a 1 bp rise in all forward rates.

Worked example USD 13,363 per bp for the client

Futures hedge
nSR3=DV01USD 25\displaystyle n_{\text{SR3}} = \frac{\text{DV01}}{\text{USD } 25}

The number of three-month SOFR futures (USD 25 per bp) the bank sells across the strip, short the cap.

Worked example about 535 contracts

Swap equivalent
Nswap=DV01A3y×0.0001\displaystyle N_{\text{swap}} = \frac{\text{DV01}}{A_{3y}\times 0.0001}

The notional of a 3-year swap paying fixed with the same DV01.

A3yA_{3y}
annuity of the 3-year swap

Worked example about USD 47,600,000

Vega
ν=∑iN τi P(0,Tip)Tie φ(di)×0.0001\displaystyle \nu = \sum_{i} N\,\tau_i\,P(0,T^{p}_i)\sqrt{T^{e}_i}\,\varphi(d_i)\times 0.0001

The change in value for a 1 bp rise of all caplet volatilities.

Worked example USD 12,496 per bp

Payoffs and the borrower’s rate

Borrower’s rate with a cap
rloan=min⁡(R, K)+m+π\displaystyle r_{\text{loan}} = \min(R,\,K) + m + \pi

The loan pays the floating rate plus its margin; the cap removes everything above the strike.

mm
loan margin
π\pi
premium as a running cost

Worked example at most 5.00% + 2.50% + 0.49% = 7.99%

Collar
max⁡(R−Kc, 0)−max⁡(Kf−R, 0)\displaystyle \max(R - K_c,\,0) - \max(K_f - R,\,0)

A bought cap and a sold floor; the floor strike is set so the premiums cancel.

Kc, KfK_c,\ K_f
cap and floor strikes

Worked example 5.50% / 4.51% at zero cost

Borrower’s rate with a collar
rloan=min⁡(max⁡(R, Kf), Kc)+m\displaystyle r_{\text{loan}} = \min\left(\max(R,\,K_f),\,K_c\right) + m

The loan rate stays in a band.

Worked example 7.01% to 8.00%

Investor’s rate with a floor
rasset=max⁡(R, Kf)−πf\displaystyle r_{\text{asset}} = \max(R,\,K_f) - \pi_f

A floating-rate asset with a bought floor earns at least the floor less its premium.

Worked example 4.50% − 0.27%

Conventions of the worked example (USD, SOFR compounded in arrears, Act/360, start Oct 30, 2026): quarterly forwards from 4.85% rising 2.5 bp a quarter (illustrative), one flat normal volatility per price (100.8 / 108.8 bp for the 1.25% / 1.35% market, 111.2 bp for the client). The other jurisdictions change the benchmark, the day count and the payment lag, as in the card.
Rates›Forward Rate Agreement

Formulas

Settlement

Overnight FRA settlement
X=N (K−R) τpaid after T2\displaystyle X = N\,(K - R)\,\tau \qquad \text{paid after } T_2

The receiver of fixed gets the fixed rate less the period’s compounded rate, on the notional and for the days of the period; a negative amount is paid.

NN
notional
KK
fixed rate
RR
overnight rate compounded over the period
τ\tau
days/360; days/365 for SONIA and CORRA

Worked example 50,000,000 × (4.19% − 4.05%) × 91/360 = USD 17,694 on May 4, 2027

Compounded rate
R=[∏d=1D(1+rd nd360)−1]1τ\displaystyle R = \left[\prod_{d=1}^{D}\left(1+\frac{r_d\,n_d}{360}\right)-1\right]\frac{1}{\tau}

The period’s rate compounds the daily fixings, so it is known on the last day of the period.

rd, ndr_d,\ n_d
fixing of business day d and the days it applies
Classic FRA settlement (EURIBOR)
XT1=N (K−L) τ1+L τ\displaystyle X_{T_1} = \frac{N\,(K - L)\,\tau}{1 + L\,\tau}

The EURIBOR FRA pays on the first day of the period, so the difference is discounted at the fixing for the days of the period.

LL
EURIBOR fixed two TARGET days before T₁

Worked example EUR 17,694 / (1 + 4.05% × 91/360) = EUR 17,515 on Jan 29, 2027

Locked rate
N R τ+X=N K τ\displaystyle N\,R\,\tau + X = N\,K\,\tau

Deposit interest at the period’s rate plus the FRA settlement gives the fixed rate on the notional, whatever the rate turns out to be.

Worked example 511,875 + 17,694 = USD 529,569 = 4.19% × 50,000,000 × 91/360

Valuation

Discount factors
P(0,T)=∏k(1+fk τk)−1\displaystyle P(0,T) = \prod_k \left(1+f_k\,\tau_k\right)^{-1}

The overnight index curve discounts the payment; here two segments, to the start and over the period.

Worked example P(T₁) = 0.98901, P(T₂) = 0.97860, P(T_pay) = 0.97814

Forward rate
F=1τ(P(0,T1)P(0,T2)−1)\displaystyle F = \frac{1}{\tau}\left(\frac{P(0,T_1)}{P(0,T_2)} - 1\right)

The rate the curve implies for the period: the fair FRA rate before margin.

Worked example 4.21%, the mid of the 4.20 / 4.22% market

Forward from futures
F≈∑kwk (100−Pk)−CA\displaystyle F \approx \sum_k w_k\,(100 - P_k) - CA

The FRA period overlaps two IMM quarters, so the dealer weights the two contracts by the days of each in the period and deducts the convexity adjustment.

wkw_k
share of the period in contract k
CACA
convexity adjustment

Worked example the strip prices the period at 95.79, 4.21%; CA 0.05 bp

FRA value
V=N τ (K−F) P(0,Tpay)\displaystyle V = N\,\tau\,(K - F)\,P(0,T_{\text{pay}})

The value to the receiver of fixed before the period: its rate less the forward, on the period, discounted.

Worked example 50,000,000 × 91/360 × (4.19% − 4.21%) × 0.97814 = −USD 2,473 at mid

Swap from FRAs
S=∑jτj Fj P(0,Tj)∑jτj P(0,Tj)\displaystyle S = \frac{\sum_j \tau_j\,F_j\,P(0,T_j)}{\sum_j \tau_j\,P(0,T_j)}

A swap rate is the average of the forward rates of its periods, weighted by their discounted accruals.

Risk and hedging

DV01
DV01=N τ P(0,Tpay)×0.0001\displaystyle \text{DV01} = N\,\tau\,P(0,T_{\text{pay}})\times 0.0001

The change in value for a one basis point move of the forward rate.

Worked example 50,000,000 × 91/360 × 0.97814 × 0.0001 = USD 1,236

Futures hedge
n=DV01BPV\displaystyle n = \frac{\text{DV01}}{\text{BPV}}

Contracts that offset the FRA; the bank paying fixed buys futures, which gain when rates fall.

Worked example 1,236 / 25 = 49.5: about 50 SR3 bought

Dealer margin
margin=N τ (Kbid−K)\displaystyle \text{margin} = N\,\tau\,(K_{\text{bid}} - K)

The gap between the market bid and the client’s rate, on the period.

Worked example 50,000,000 × 91/360 × (4.20% − 4.19%) = USD 1,264

Convexity adjustment
CA≈12 σ2 t1 t2\displaystyle CA \approx \tfrac{1}{2}\,\sigma^2\,t_1\,t_2

The futures rate exceeds the forward rate by an amount that is negligible three months out and grows with maturity.

Worked example σ = 0.90%, t₁ = 93/365, t₂ = 184/365: 0.05 bp

Counterparty and margin

Variation margin
VMt=Vt−Vt−1\displaystyle VM_t = V_t - V_{t-1}

Under a CSA or at the CCP the change in value moves as collateral each day; for a three-month FRA the amounts are small.

Exposure before settlement
EE(t)≈N τ σ t  φ(0) P(0,Tpay)\displaystyle EE(t) \approx N\,\tau\,\sigma\,\sqrt{t}\;\varphi(0)\,P(0,T_{\text{pay}})

The expected positive exposure of an at-the-money FRA grows with the square root of time until the rate is set.

φ(0)\varphi(0)
0.399, the normal density at zero
σ\sigma
normal volatility of the forward rate

Conventions of the US worked example (SOFR, Act/360, paid two business days after the period end); SONIA and CORRA use Act/365F and the EURIBOR FRA pays at the start, discounted. Discount curve on the trade date, illustrative: 4.30% to the start of the period, 4.21% over it.
Rates›Basis Swap

Formulas

Cash flows

Compounded overnight rate
Rj=1τj[∏d=1Dj(1+rd nd360)−1]\displaystyle R_j = \frac{1}{\tau_j}\left[\prod_{d=1}^{D_j}\left(1+\frac{r_d\,n_d}{360}\right)-1\right]

Each leg compounds its daily fixings over the quarter; both rates are known at the end of the period.

rdr_d
SOFR or the effective fed funds rate of business day d
ndn_d
calendar days it applies
τj\tau_j
days/360 of quarter j
Spread leg
CjFF=N (RjFF+s) τj\displaystyle C^{\text{FF}}_j = N\,(R^{\text{FF}}_j + s)\,\tau_j

The fed funds leg carries the spread, added to the compounded rate.

ss
basis spread, 4.5 bp

Worked example 500,000,000 × (4.25% + 0.045%) × 91/360 = USD 5,428,403

SOFR leg
CjSOFR=N RjSOFR τj\displaystyle C^{\text{SOFR}}_j = N\,R^{\text{SOFR}}_j\,\tau_j

The other leg pays the compounded SOFR flat.

Worked example 500,000,000 × 4.30% × 91/360 = USD 5,434,722

Net payment
Xj=N (RjFF+s−RjSOFR)τj\displaystyle X_j = N\,\left(R^{\text{FF}}_j + s - R^{\text{SOFR}}_j\right)\tau_j

Only the difference moves: the spread received less the basis realized over the quarter.

Worked example (4.295% − 4.30%) × 500,000,000 × 91/360 = −USD 6,319

Valuation

Discount factor
P(0,Tj)=∏k≤j(1+FkSOFR τk)−1\displaystyle P(0,T_j) = \prod_{k\le j}\left(1+F^{\text{SOFR}}_k\,\tau_k\right)^{-1}

Both legs are discounted on the SOFR curve.

Worked example P = 0.95908 for the fourth payment, 0.92359 for the last

Annuity
A=∑j=1nτj P(0,Tj)\displaystyle A = \sum_{j=1}^{n}\tau_j\,P(0,T_j)

The present value of one unit of spread paid every quarter: the scale of the swap’s spread risk.

Worked example A = 1.9394

Par basis
s∗=∑jτj(FjSOFR−FjFF)P(0,Tj)A\displaystyle s^{*} = \frac{\sum_j \tau_j\left(F^{\text{SOFR}}_j - F^{\text{FF}}_j\right)P(0,T_j)}{A}

The spread that makes the swap worth zero: the quoted basis.

Worked example s* = 5.0 bp on the example curves, the bid

Mark to market
Vrec(t)=N (s−st∗)At\displaystyle V_{\text{rec}}(t) = N\,\left(s - s^{*}_t\right)A_t

For the receiver of the spread leg: its spread less today’s par basis, times the annuity.

Worked example at the start: 500,000,000 × (4.5 − 5.0) bp × 1.9394 = −USD 48,484

Basis widening
ΔV=−N Δs∗ A\displaystyle \Delta V = -N\,\Delta s^{*}\,A

A wider basis makes the paid SOFR leg dearer against the fed funds leg received, so the client loses.

Worked example basis to 8.0 bp: 500,000,000 × (4.5 − 8.0) bp × 1.9394 = −USD 339,390

Risk and hedging

Basis DV01
DV01s=N A×0.0001\displaystyle \text{DV01}_s = N\,A\times 0.0001

The change in value for a one basis point move of the par basis.

Worked example 500,000,000 × 1.9394 × 0.0001 = USD 96,968

Dealer margin
margin=N (sbid−s) A\displaystyle \text{margin} = N\,(s_{\text{bid}} - s)\,A

The desk receives fed funds + 5.0 bp on its hedge and pays 4.5 bp to the client.

Worked example 500,000,000 × 0.5 bp × 1.9394 = USD 48,484

Futures hedge
nSR3=DV01j25nZQ=DV01j3×41.67\displaystyle n_{\text{SR3}} = \frac{\text{DV01}_j}{25} \qquad n_{\text{ZQ}} = \frac{\text{DV01}_j}{3\times 41.67}

A quarter of basis risk is hedged with SOFR futures against fed funds futures, one ZQ contract per month of the quarter (USD 41.67 per bp).

Worked example first quarter, DV01 USD 12,494: about 500 SR3 against 100 ZQ in each month

Fed funds future
PZQ=100−1Dm∑d=1DmrdEFFRBPV=5,000,000×0.0001×30360\displaystyle P^{\text{ZQ}} = 100 - \frac{1}{D_m}\sum_{d=1}^{D_m} r^{\text{EFFR}}_d \qquad \text{BPV} = 5{,}000{,}000\times 0.0001\times\frac{30}{360}

The 30-day fed funds future settles to the simple average of the effective fed funds rate over its calendar month, weekends included; one basis point is USD 41.67.

DmD_m
calendar days in the month

Margin

Variation margin
VMt=Vt−Vt−1\displaystyle VM_t = V_t - V_{t-1}

The CCP marks the swap to its curves each day and moves the change in cash.

Worked example basis from 5.0 to 8.0 bp: −USD 290,905 for the client

Price alignment interest
PAIt=Mt−1 rt−1 n360\displaystyle PAI_t = M_{t-1}\,r_{t-1}\,\frac{n}{360}

Interest at the overnight rate on the cumulative variation margin, so that margining leaves the value unchanged.

Mt−1M_{t-1}
cumulative variation margin
rt−1r_{t-1}
SOFR

Conventions of the worked example (USD, SOFR against effective fed funds, quarterly, Act/360, paid two business days after each period); forward rates of the trade date, illustrative, with fed funds 5.0 bp below SOFR in every quarter.
Rates›Inflation Swap

Formulas

Cash flows

Inflation leg
Cinfl=N(I(T−L)I(T0−L)−1)\displaystyle C^{\text{infl}} = N\left(\frac{I(T-L)}{I(T_0-L)} - 1\right)

The inflation receiver gets the growth of the index between the base and the final reference dates, each read with the lag.

NN
notional
I(⋅)I(\cdot)
reference index
LL
lag: three months for CPI-U, HICP ex tobacco and Canada’s CPI, two for RPI
T0, TT_0,\ T
start and maturity

Worked example USD 100,000,000 × (1.3050 − 1) = USD 30,500,000

Fixed leg
Cfix=N[(1+K)T−1]\displaystyle C^{\text{fix}} = N\left[(1+K)^{T} - 1\right]

The fixed rate compounds annually over the life, with each year counted as a whole period.

KK
fixed rate
TT
years to maturity

Worked example 100,000,000 × (1.0248^10 − 1) = USD 27,758,901

Net payment at maturity
X=Cinfl−Cfix\displaystyle X = C^{\text{infl}} - C^{\text{fix}}

One net amount moves on the maturity date, here seen from the inflation receiver.

Worked example 30,500,000 − 27,758,901 = +USD 2,741,099 for the client

Reference index, interpolated (CPI-U, Canada)
Iref(d)=Im−3+d−1Dm(Im−2−Im−3)\displaystyle I_{\text{ref}}(d) = I_{m-3} + \frac{d-1}{D_m}\left(I_{m-2} - I_{m-3}\right)

The index for day d of month m interpolates between the monthly indexes of three and two months earlier, as for TIPS; the RPI and HICP swaps use the index of the lagged month, without interpolation.

Im−3, Im−2I_{m-3},\ I_{m-2}
indexes of the months three and two months before
DmD_m
days in month m

Worked example Oct 30, 2026: CPI-U(Jul) + 29/31 × (CPI-U(Aug) − CPI-U(Jul))

Realized inflation
π=(I(T−L)I(T0−L))1/T−1\displaystyle \pi = \left(\frac{I(T-L)}{I(T_0-L)}\right)^{1/T} - 1

The average annual inflation that the index ratio stands for.

Worked example 1.3050^(1/10) − 1 = 2.698% a year

Valuation

Forward index
I(0,T)=I0 (1+ZT)T\displaystyle I(0,T) = I_0\,(1 + Z_T)^{T}

The zero-coupon swap rate of each maturity gives the market’s forward value of the index at that date: the inflation curve.

I0I_0
base index
ZTZ_T
zero-coupon inflation swap rate for T years

Worked example mid 2.475%: index ratio 1.02475^10 = 1.27697

Swap value
V=N[(1+ZT)T−(1+K)T]Pn(0,T)\displaystyle V = N\left[(1+Z_T)^{T} - (1+K)^{T}\right]P_n(0,T)

The inflation receiver holds the projected inflation leg and owes the fixed leg, both paid at T and discounted on the nominal curve.

Pn(0,T)P_n(0,T)
nominal discount factor on the overnight curve

Worked example 100,000,000 × (1.27697 − 1.27759) × 0.6854 = −USD 42,713: the dealer’s 0.5 bp

Seasoned swap
Vt=N[IknownI0 (1+Zt,T)T−t−(1+K)T]Pn(t,T)\displaystyle V_t = N\left[\frac{I_{\text{known}}}{I_0}\,(1+Z_{t,T})^{T-t} - (1+K)^{T}\right]P_n(t,T)

After the start, the inflation already published is locked in and only the remaining years are projected from the curve.

IknownI_{\text{known}}
latest published reference index
Zt,TZ_{t,T}
swap rate for the remaining years
Seasonal adjustment
I(0,t)=Itrend(0,t)  sm(t)∏m=112sm=1\displaystyle I(0,t) = I^{\text{trend}}(0,t)\;s_{m(t)} \qquad \prod_{m=1}^{12} s_m = 1

The index is not seasonally adjusted and rises faster in some months than in others; the curve between the annual maturities carries monthly factors estimated from past prints.

sms_m
seasonal factor of calendar month m
Breakeven and the Fisher relation
1+ynom=(1+yreal)(1+πBE)πBE≈ynom−yreal\displaystyle 1 + y_{\text{nom}} = (1 + y_{\text{real}})(1 + \pi^{\text{BE}}) \qquad \pi^{\text{BE}} \approx y_{\text{nom}} - y_{\text{real}}

The inflation priced by the bond market is the gap between the nominal and the real yield of the same maturity; the swap rate trades close to it, the difference being the swap-cash basis.

ynom, yrealy_{\text{nom}},\ y_{\text{real}}
yields of the nominal and the inflation-linked bond

Risk

Inflation sensitivity (IE01)
IE01=∂V∂ZT×0.0001=N T (1+ZT)T−1 Pn(0,T)×0.0001\displaystyle \text{IE01} = \frac{\partial V}{\partial Z_T}\times 0.0001 = N\,T\,(1+Z_T)^{T-1}\,P_n(0,T)\times 0.0001

The change in value for a one basis point rise of the breakeven to maturity.

Worked example 100,000,000 × 10 × 1.02475^9 × 0.6854 × 0.0001 = USD 85,408 per bp

Nominal rate sensitivity
∂V∂y=−T1+y V\displaystyle \frac{\partial V}{\partial y} = -\frac{T}{1+y}\,V

With both legs paid on one date the nominal rate risk is proportional to the swap’s value: close to zero at the start, growing as inflation accrues.

yy
nominal zero rate to T, annual
Breakeven hedge with linkers
nlink RDV01link=IE01nnom DV01nom=nlink RDV01link\displaystyle n_{\text{link}}\,\text{RDV01}_{\text{link}} = \text{IE01} \qquad n_{\text{nom}}\,\text{DV01}_{\text{nom}} = n_{\text{link}}\,\text{RDV01}_{\text{link}}

A dealer paying inflation buys linkers sized on the inflation sensitivity and sells nominal bonds of the same maturity to remove the rate risk that comes with them.

RDV01link\text{RDV01}_{\text{link}}
real yield DV01 of the linker
DV01nom\text{DV01}_{\text{nom}}
DV01 of the nominal bond
nn
quantities
Variation margin
VMt=Vt−Vt−1\displaystyle VM_t = V_t - V_{t-1}

With no coupons, the daily margin is the only cash that moves before maturity on a cleared swap; a monthly print moves it in one step.

Year-on-year and LPI

Year-on-year payment
Ci=N τi(I(Ti−L)I(Ti−1−L)−1−KYoY)\displaystyle C_i = N\,\tau_i\left(\frac{I(T_i - L)}{I(T_{i-1} - L)} - 1 - K_{\text{YoY}}\right)

Each year the receiver gets that year’s inflation against a fixed rate, so payments are spread over the life and the exposure stays small.

τi\tau_i
accrual fraction, 1 for an annual period
KYoYK_{\text{YoY}}
fixed rate of the year-on-year swap
Convexity of the year-on-year leg
ETi ⁣[I(Ti)I(Ti−1)]=I(0,Ti)I(0,Ti−1) eCi\displaystyle \mathbb{E}^{T_i}\!\left[\frac{I(T_i)}{I(T_{i-1})}\right] = \frac{I(0,T_i)}{I(0,T_{i-1})}\,e^{C_i}

A year-on-year rate paid at the end of its year differs from the ratio of two forward indexes by a convexity adjustment, set by the volatilities of real and nominal rates and their correlation (Jarrow-Yildirim).

CiC_i
convexity adjustment, small, positive or negative
LPI (UK pension increases)
LPIi=LPIi−1 min⁡ ⁣(max⁡ ⁣(RPIiRPIi−1, 1+f), 1+c)\displaystyle \text{LPI}_i = \text{LPI}_{i-1}\,\min\!\left(\max\!\left(\frac{\text{RPI}_i}{\text{RPI}_{i-1}},\,1+f\right),\,1+c\right)

The pension increase follows the RPI each year within a floor and a cap, so an LPI swap is an inflation swap with a strip of year-on-year floors and caps.

f, cf,\ c
floor and cap: 0% and 5%, or 0% and 2.5%

Net at maturity

Net against realized inflation
X(π)=N[(1+π)T−(1+K)T]\displaystyle X(\pi) = N\left[(1+\pi)^{T} - (1+K)^{T}\right]

The receiver’s net payment as a function of the average inflation realized: zero at the fixed rate, slightly convex.

π\pi
realized average inflation a year

Worked example π = 2.698%: +USD 2,741,099; π = 0%: −USD 27,758,901

Conventions of the worked example (USD, CPI-U, three-month lag, interpolated); the other jurisdictions change the index and the lag, as in the card. Nominal discount factor to 10 years 0.6854 (3.85% a year, illustrative).
Rates›Cross-Currency Swap

Formulas

Cash flows

Initial exchange
client pays NEURreceives NUSD=NEUR S0\displaystyle \text{client pays } N_{\text{EUR}} \qquad \text{receives } N_{\text{USD}} = N_{\text{EUR}}\,S_0

The two principals are exchanged at the spot rate of the trade date, on the start date.

S0S_0
spot on the trade date, USD per EUR

Worked example EUR 500,000,000 × 1.0850 = USD 542,500,000 on Oct 30, 2026

EUR interest
CjEUR=NEUR(Rj€STR+b)τj\displaystyle C^{\text{EUR}}_j = N_{\text{EUR}}\left(R^{\text{€STR}}_j + b\right)\tau_j

The client receives the compounded €STR of the quarter plus the basis on the EUR principal it gave.

Rj€STRR^{\text{€STR}}_j
€STR compounded over quarter j
bb
basis, −21 bp for the client
τj\tau_j
days/360

Worked example 500,000,000 × (2.40% − 0.21%) × 90/360 = EUR 2,737,500

USD interest
CjUSD=NUSD RjSOFR τj\displaystyle C^{\text{USD}}_j = N_{\text{USD}}\,R^{\text{SOFR}}_j\,\tau_j

The client pays compounded SOFR flat on the USD principal it received.

Worked example 542,500,000 × 4.30% × 90/360 = USD 5,831,875

Overnight rate compounded in arrears
Rj=1τj[∏d=1Dj(1+rd nd360)−1]\displaystyle R_j = \frac{1}{\tau_j}\left[\prod_{d=1}^{D_j}\left(1+\frac{r_d\,n_d}{360}\right)-1\right]

Each leg compounds the daily fixings of its own currency, as an overnight index swap does (over 365 days for SONIA and CORRA).

rdr_d
overnight fixing of business day d
ndn_d
calendar days it applies
Final exchange
client receives NEURpays NUSD=NEUR S0\displaystyle \text{client receives } N_{\text{EUR}} \qquad \text{pays } N_{\text{USD}} = N_{\text{EUR}}\,S_0

The principal goes back at the rate of the trade date, whatever spot is at maturity.

Worked example Oct 30, 2031: EUR 500,000,000 against USD 542,500,000

Valuation

Forward exchange rate
F(t)=S0 PEUR(0,t)PUSD(0,t)\displaystyle F(t) = S_0\,\frac{P_{\text{EUR}}(0,t)}{P_{\text{USD}}(0,t)}

The rate at which EUR paid at t are worth USD today: spot carried by the two discount curves, the EUR curve including the basis.

PUSDP_{\text{USD}}
SOFR discount factor
PEURP_{\text{EUR}}
EUR discount factor for collateral in USD: €STR plus the basis

Worked example F(5 years) = 1.0850 × 0.89469 / 0.80499 = 1.2059

Covered interest parity and the basis
FS0=1+rUSD τ1+(rEUR+b) τ\displaystyle \frac{F}{S_0} = \frac{1 + r_{\text{USD}}\,\tau}{1 + (r_{\text{EUR}} + b)\,\tau}

Without the basis the forward would follow the two overnight rates; with a negative basis a euro holder borrowing dollars through the swap pays more than SOFR in effect.

bb
cross-currency basis, negative when dollars are scarce

Worked example first quarter: 1.0850 × (1 + 4.30% × 91/360) / (1 + 2.20% × 91/360) = 1.0907

USD leg
PVUSD=NUSD[−PUSD(0,T0)+∑jRjτjPUSD(0,Tj)+PUSD(0,Tn)]=0\displaystyle PV_{\text{USD}} = N_{\text{USD}}\left[-P_{\text{USD}}(0,T_0) + \sum_j R_j\tau_j P_{\text{USD}}(0,T_j) + P_{\text{USD}}(0,T_n)\right] = 0

Discounted on SOFR, a SOFR leg with both exchanges of principal is worth zero: the coupons telescope.

EUR annuity
AEUR=∑j=1nτj PEUR(0,Tj)\displaystyle A_{\text{EUR}} = \sum_{j=1}^{n}\tau_j\,P_{\text{EUR}}(0,T_j)

The value of one unit of spread paid every quarter on the EUR leg: the scale of the basis risk.

Worked example A = 4.787 over the 20 quarters

Swap value to the client
V=S0 NEUR (b−bmkt) AEUR−PVUSD\displaystyle V = S_0\,N_{\text{EUR}}\,(b - b^{\text{mkt}})\,A_{\text{EUR}} - PV_{\text{USD}}

The EUR leg is worth the gap between the client’s spread and the market basis over the life, converted at spot.

bmktb^{\text{mkt}}
market basis on the client’s side, −20 bp

Worked example 1.0850 × 500,000,000 × (−1 bp) × 4.787 = −USD 259,686: the dealer’s 1 bp

Mark-to-market resets

Reset notional
NUSD,k=NEUR Sk\displaystyle N_{\text{USD},k} = N_{\text{EUR}}\,S_k

At each reset the USD notional is set again from the EUR notional at the spot fixing of the day.

SkS_k
spot fixing at reset k

Worked example spot 1.1000 at the first reset: USD 550,000,000

Reset payment
Δk=NUSD,k−NUSD,k−1\displaystyle \Delta_k = N_{\text{USD},k} - N_{\text{USD},k-1}

The change in USD notional is paid at the reset: the party that will get the EUR back receives it when the euro has risen, so the FX gain is settled each quarter.

Worked example 550,000,000 − 542,500,000 = USD 7,500,000 to the USD borrower

FX exposure left on a resetting swap
ΔV≈NEUR (St−Sk)\displaystyle \Delta V \approx N_{\text{EUR}}\,(S_t - S_k)

Only the move since the last reset is unsettled, so the exposure stays at about one quarter’s FX move.

Risk

FX delta after the initial exchange
ΔV≈NEUR ΔS\displaystyle \Delta V \approx N_{\text{EUR}}\,\Delta S

Once the principals are exchanged, the EUR leg (floating interest and the EUR principal to come) is worth about its notional, so the swap moves with spot on the whole principal.

Worked example 1% of 1.0850 on EUR 500,000,000: USD 5,425,000

Basis sensitivity
Basis01=S0 NEUR AEUR×0.0001\displaystyle \text{Basis01} = S_0\,N_{\text{EUR}}\,A_{\text{EUR}}\times 0.0001

The change in value for a one basis point move of the cross-currency basis.

Worked example 1.0850 × 500,000,000 × 4.787 × 0.0001 = USD 259,686 per bp

Rate risk of a floating leg
DV01leg≈N τ1 P(0,T1)×0.0001\displaystyle \text{DV01}_{\text{leg}} \approx N\,\tau_1\,P(0,T_1)\times 0.0001

A floating leg with its principal behaves like a bond that resets every quarter, so its rate risk is one period long and a floating-for-floating swap carries little rate risk.

Worked example USD leg: 542,500,000 × 91/360 × 0.9892 × 0.0001 = USD 13,566

Counterparty and margin

Potential exposure on the principal, no collateral
PFE95%(t)≈NUSD(e1.645 σt−1)\displaystyle \text{PFE}_{95\%}(t) \approx N_{\text{USD}}\left(e^{1.645\,\sigma\sqrt{t}} - 1\right)

The FX move on the principal exceeded only one time in twenty by date t, for an exchange rate without drift.

σ\sigma
EUR/USD volatility, 7% illustrative

Worked example at 5 years: 542,500,000 × (e^(1.645 × 0.07 × √5) − 1) = USD 159,316,036

Exposure with daily variation margin
PFE95%≈NUSD×1.645 σMPoR250\displaystyle \text{PFE}_{95\%} \approx N_{\text{USD}}\times 1.645\,\sigma\sqrt{\frac{\text{MPoR}}{250}}

With daily margin only the move over the margin period of risk is exposed.

MPoR\text{MPoR}
margin period of risk, 10 business days

Worked example 542,500,000 × 1.645 × 0.07 × √(10/250) = USD 12,493,775

SA-CCR exposure, FX part
EAD=1.4(RC+AddOn)AddOnFX=4%×NUSD×MF\displaystyle \text{EAD} = 1.4\left(RC + \text{AddOn}\right) \qquad \text{AddOn}_{\text{FX}} = 4\%\times N_{\text{USD}}\times MF

The standardized exposure of the bank capital rules: replacement cost plus an add-on of 4% of the notional for FX, an interest rate add-on on top, scaled down for a margined trade.

RCRC
replacement cost
MFMF
maturity factor: 1 unmargined

Worked example unmargined, at the start: 1.4 × 4% × 542,500,000 = USD 30,380,000

The final exchange

Value of the final exchange
VT=NEUR ST−NUSD\displaystyle V_T = N_{\text{EUR}}\,S_T - N_{\text{USD}}

What the final exchange is worth to the client at maturity, against spot.

STS_T
spot at maturity

Worked example S_T = 1.2000: +USD 57,500,000; at the forward 1.2059: +USD 60,449,865

Market data of the worked example, illustrative: spot 1.0850, SOFR 4.30% and €STR 2.40% held flat, basis −20 bp on the client’s side, EUR discounting at €STR plus the basis (collateral in USD); EUR annuity 4.787; EUR/USD volatility 7% for the exposure figures.
Fixed Income›Government Bond

Formulas

Cash flows and settlement

Coupon
C=N cf\displaystyle C = N\,\frac{c}{f}

Each coupon is the face amount times the annual rate divided by the number of payments a year.

NN
face amount
cc
coupon rate
ff
payments a year: 2 for Treasuries, gilts and Canadas, 1 for Bunds and OATs

Worked example USD 50,000,000 × 4.250% / 2 = USD 1,062,500

Price in 32nds
P=a+b32  (+164 for a plus)\displaystyle P = a + \frac{b}{32} \;\left(+\tfrac{1}{64}\text{ for a plus}\right)

US Treasuries are quoted in points and 32nds of a point; a plus adds half a 32nd.

a, ba,\ b
the points and the 32nds of the quote

Worked example 99-16+ = 99 + 16.5/32 = 99.515625

Accrued interest, Act/Act
AI=cf×dD\displaystyle AI = \frac{c}{f}\times\frac{d}{D}

The share of the current coupon the buyer pays the seller, by actual days elapsed over actual days in the period; the Canadian convention uses c × d / 365.

dd
days from the last coupon to settlement
DD
days in the coupon period

Worked example 2.125 × 75 / 184 = 0.866168 per 100

Dirty price
Pdirty=Pclean+AI\displaystyle P_{\text{dirty}} = P_{\text{clean}} + AI

The quote is clean; the cash that moves on the settlement date is clean plus accrued.

Worked example 99.515625 + 0.866168 = 100.381793

Settlement amount
S=N Pdirty100\displaystyle S = N\,\frac{P_{\text{dirty}}}{100}

The amount paid against delivery of the bonds.

Worked example USD 49,757,812.50 + USD 433,084.24 = USD 50,190,896.74

Price and yield

Price from the yield
Pdirty=∑i=1nCFi(1+y/f) w+i−1w=dnextD\displaystyle P_{\text{dirty}} = \sum_{i=1}^{n} \frac{CF_i}{(1+y/f)^{\,w+i-1}} \qquad w = \frac{d_{\text{next}}}{D}

The street convention: each coupon and the principal discounted at the yield, compounded f times a year, with the first period cut to the fraction left.

CFiCF_i
coupon i, plus the principal on the last date
ww
fraction of the current period left
nn
coupons left

Worked example w = 0.5924, n = 20: y = 4.3105% gives 100.381793

Discount factor at the yield
dfi=(1+y/f)−(w+i−1)\displaystyle df_i = (1+y/f)^{-(w+i-1)}

The weight of each flow in the price; on the card’s ladder the discounted flows add up to the dirty price.

Worked example first coupon 0.98745, principal 0.65851

Yield to maturity
y:  Pdirty(y)=Pclean+AI\displaystyle y:\; P_{\text{dirty}}(y) = P_{\text{clean}} + AI

The yield is solved from the price, by iteration; it assumes every coupon is reinvested at the same yield.

Worked example 99-16 gives 4.312%, 99-16+ gives 4.310%: a 64th is worth about 0.2 bp

Price on a curve
Pdirty=∑iCFi P(0,ti)\displaystyle P_{\text{dirty}} = \sum_{i} CF_i\,P(0,t_i)

With a fitted zero curve every bond is priced on the same discount factors; its own yield then shows whether it is rich or cheap.

P(0,t)P(0,t)
discount factor of the fitted curve
Spread to the curve
Pdirty=∑iCFi e−(z(ti)+s) ti\displaystyle P_{\text{dirty}} = \sum_{i} CF_i\,e^{-(z(t_i)+s)\,t_i}

The constant spread over the zero curve that reprices the bond: positive when the issue is cheap.

z(t)z(t)
zero rate
ss
the bond’s spread

Risk

DV01
DV01=−∂P∂y×0.0001≈P(y−1bp)−P(y+1bp)2\displaystyle \text{DV01} = -\frac{\partial P}{\partial y}\times 0.0001 \approx \frac{P(y-1\text{bp})-P(y+1\text{bp})}{2}

The price change for one basis point of yield, per 100 or on the position.

Worked example 0.07901 per 100: USD 39,505 per bp on USD 50,000,000

Modified duration
Dmod=DV01Pdirty×104\displaystyle D_{\text{mod}} = \frac{\text{DV01}}{P_{\text{dirty}}}\times 10^4

The percentage price change for a 1% change in yield.

Worked example 7.871

Macaulay duration
Dmac=Dmod (1+y/f)=∑iti PV(CFi)Pdirty\displaystyle D_{\text{mac}} = D_{\text{mod}}\,(1+y/f) = \frac{\sum_i t_i\,PV(CF_i)}{P_{\text{dirty}}}

The PV-weighted average life of the flows, in years.

Worked example 8.041 years

Convexity
C=P(y+Δy)+P(y−Δy)−2P(y)P(y) Δy2\displaystyle C = \frac{P(y+\Delta y) + P(y-\Delta y) - 2P(y)}{P(y)\,\Delta y^2}

The curvature of the price against the yield: the duration lengthens as yields fall.

Worked example 74.2

Price change
ΔPP≈−Dmod Δy+12 C Δy2\displaystyle \frac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta y + \tfrac{1}{2}\,C\,\Delta y^2

Duration gives the first-order move, convexity the correction.

Worked example +50 bp: −3.843% by the formula, −3.844% repriced

Hedge ratio
Nhedge=N DV01bondDV01hedge\displaystyle N_{\text{hedge}} = N\,\frac{\text{DV01}_{\text{bond}}}{\text{DV01}_{\text{hedge}}}

The face amount of another bond, or futures contracts on the DV01 of the cheapest-to-deliver over its conversion factor, that offsets the DV01.

The auction

Single-price auction (US)
y∗=min⁡{ y:∑b: yb≤yQb≥A−Qnc}\displaystyle y^{*} = \min\Big\{\,y : \sum_{b:\,y_b \le y} Q_b \ge A - Q_{\text{nc}}\Big\}

Competitive bids are accepted from the lowest yield up until the offering, less the noncompetitive bids, is covered; every accepted bid pays the stop, and bids at the stop are prorated.

y∗y^{*}
stop (high) yield
Qb, ybQ_b,\ y_b
amount and yield of bid b
AA
offering amount
QncQ_{\text{nc}}
noncompetitive bids

Worked example stop 4.322%

Bid-to-cover
BTC=∑bQbA\displaystyle BTC = \frac{\sum_b Q_b}{A}

Total bids over the amount offered, the first measure of demand.

Worked example 2.5 × USD 39 billion = USD 97.5 billion of bids

Tail
tail=y∗−yWI\displaystyle \text{tail} = y^{*} - y_{\text{WI}}

The stop against the when-issued yield at the bid deadline: a positive tail is a weak auction.

Worked example 4.322% − 4.320% = 0.2 bp

Coupon of a new issue
c=18⌊8 y∗⌋\displaystyle c = \tfrac{1}{8}\left\lfloor 8\,y^{*} \right\rfloor

A new note’s coupon is the stop rounded down to an eighth, so it is issued at or below par; a reopening keeps its coupon.

Worked example 4.322% gives 4.250%, price 99.4204 (cat_lib bond_price, DV01 0.0802 per 100)

Bid-price auction (UK, EU, Canada)
bidder b pays Pb   for Pb≥P∗\displaystyle \text{bidder } b \text{ pays } P_b \;\text{ for } P_b \ge P^{*}

In a multiple-price auction each winning bidder pays its own price (or yield); the gilt conventionals, the Bunds, the OATs and the Canadas are sold this way.

P∗P^{*}
lowest accepted price
Award limit (US)
Qmax⁡=35%×A\displaystyle Q_{\max} = 35\%\times A

No bidder may be awarded more than 35% of the offering at the stop, its own and its reported position included.

Worked example USD 13.65 billion on USD 39 billion

Financing and carry

Carry per day
carry≈N cf D−S rrepo360\displaystyle \text{carry} \approx N\,\frac{c}{f\,D} - S\,\frac{r_{\text{repo}}}{360}

A position financed in repo earns its coupon accrual and pays the repo rate on the cash borrowed.

SS
settlement amount
rrepor_{\text{repo}}
repo rate

Worked example USD 5,774 − USD 5,507 = USD 267 a day at 3.95% repo

Forward price
Pfwd=(Pclean+AI0)(1+rrepo t360)−AIt\displaystyle P_{\text{fwd}} = (P_{\text{clean}} + AI_0)\left(1 + r_{\text{repo}}\,\frac{t}{360}\right) - AI_t

The price at which a repo-financed position breaks even at a forward date (no coupon in between): the basis of the futures and of when-issued trades.

Fails charge (US)
charge per day=N max⁡(3%−rFF, 0)360\displaystyle \text{charge per day} = N\,\frac{\max(3\% - r_{\text{FF}},\ 0)}{360}

A seller that fails to deliver Treasuries owes this charge, so a fail costs money even when rates are low.

rFFr_{\text{FF}}
fed funds target, lower bound

Worked example zero with fed funds at 3.75%; at 1.00% it would be USD 2,778 a day on USD 50 million

Numbers of the US worked example: the 4.250% note due Aug 15, 2036 bought at 99-16+ for settlement on Oct 29, 2026, USD 50,000,000; the overnight repo rate of 3.95% and the fed funds rate of 3.75% are illustrative. The gilt, the Bund and the Government of Canada bond use the same formulas with their own frequency and accrued convention.
Fixed Income›Inflation-Linked Bond

Formulas

Indexation and cash flows

Reference index
Iref(d)=Im−3+d−1Dm(Im−2−Im−3)\displaystyle I_{\text{ref}}(d) = I_{m-3} + \frac{d-1}{D_m}\left(I_{m-2} - I_{m-3}\right)

The index for a day of month m is interpolated between the index of three months and of two months earlier: the lag that lets the index be published before it is used.

Im−3I_{m-3}
index of the third month before
dd
day of the month
DmD_m
days in month m

Worked example Oct 29: 332.100 + 28/31 × (332.700 − 332.100) = 332.64194; Jul 15: 330.71613

Index ratio
IR(d)=Iref(d)Iref(d0)\displaystyle IR(d) = \frac{I_{\text{ref}}(d)}{I_{\text{ref}}(d_0)}

The growth of the index since the dated date, rounded to five decimals; it scales every cash amount.

d0d_0
dated date of the bond

Worked example 332.64194 / 330.71613 = 1.00582

Settlement amount
S=N Preal+AIreal100 IR(ds)\displaystyle S = N\,\frac{P_{\text{real}} + AI_{\text{real}}}{100}\,IR(d_s)

The real price and the real accrued interest, both times the index ratio of the settlement date.

Worked example USD 25,000,000 × (98.3681 + 0.504076) / 100 × 1.00582 = USD 24,861,915

Coupon
Ci=N c2 IR(ti)\displaystyle C_i = N\,\frac{c}{2}\,IR(t_i)

The fixed real rate on the adjusted principal of the payment date: coupons grow with the index and have no floor.

Worked example at today’s ratio: 25,000,000 × 0.875% × 1.00582 = USD 220,023

Principal at maturity
R=N max⁡(IR(T), 1)  (TIPS, OAT€i)R=N IR(T)  (gilts, RRBs)\displaystyle R = N\,\max\left(IR(T),\ 1\right) \;\text{(TIPS, OAT€i)} \qquad R = N\,IR(T) \;\text{(gilts, RRBs)}

The deflation floor applies to the principal only, and only on the bonds that have one.

Real yield and breakeven

Real price
Preal+AIreal=∑iCFireal(1+r/f) w+i−1\displaystyle P_{\text{real}} + AI_{\text{real}} = \sum_{i} \frac{CF^{\text{real}}_i}{(1+r/f)^{\,w+i-1}}

The real flows (coupons and par before indexation) discounted at the real yield: the formula of a nominal bond.

rr
real yield

Worked example r = 1.935%: 98.3681 clean

Breakeven inflation
BE=ynom−r\displaystyle BE = y_{\text{nom}} - r

The inflation rate that makes the linker and the nominal bond of the same maturity return the same, before risk and liquidity premiums.

Worked example 4.310% − 1.935% = 2.375%

Fisher relation
(1+ynom/f)=(1+r/f)(1+π/f)\displaystyle (1+y_{\text{nom}}/f) = (1+r/f)(1+\pi/f)

Projected nominal flows discounted at the nominal rate equal real flows discounted at the real rate, the identity behind the card’s ladder.

π\pi
projected inflation

Worked example the ladder projects the index at 2.375% and its PV equals the settlement amount

Projected index ratio
IR(t)=IR(ds) (1+π/f) f t\displaystyle IR(t) = IR(d_s)\,(1+\pi/f)^{\,f\,t}

The index ratio of a future date at a flat projected inflation; a desk uses the inflation swap curve with seasonality.

Risk

Real DV01
DV01r=−∂S∂r×0.0001\displaystyle \text{DV01}_r = -\frac{\partial S}{\partial r}\times 0.0001

The change in the settlement value for one basis point of real yield, on the indexed principal.

Worked example 0.08736 per 100 × 250,000 × 1.00582 = USD 21,967 per bp

Breakeven exposure
ΔS≈−DV01r(Δynom−ΔBE)×104\displaystyle \Delta S \approx -\text{DV01}_r\left(\Delta y_{\text{nom}} - \Delta BE\right)\times 10^4

A long linker is long real rates falling and long breakevens rising: hedging the nominal yield leaves the breakeven exposure.

Hedge with nominals
Nnom=β DV01rDV01nom\displaystyle N_{\text{nom}} = \beta\,\frac{\text{DV01}_r}{\text{DV01}_{\text{nom}}}

Desks hedge the real rate with nominal bonds or futures in proportion to the observed yield beta of real yields to nominal yields.

β\beta
regression of real yield changes on nominal yield changes
Carry with the index
carry≈S[IR(t+δ)IR(t)−1]+N cf δD−S rrepo δ360\displaystyle \text{carry} \approx S\left[\frac{IR(t+\delta)}{IR(t)} - 1\right] + N\,\frac{c}{f}\,\frac{\delta}{D} - S\,r_{\text{repo}}\,\frac{\delta}{360}

The accretion of the index ratio over the holding period, known a few months ahead because of the lag, plus the real accrual less the financing.

The deflation floor

Floor at maturity
N max⁡(1−IR(T), 0)\displaystyle N\,\max\left(1 - IR(T),\ 0\right)

The extra paid at maturity when the index has fallen below its level at the dated date: a zero-coupon put on the cumulative index.

Value of the floor
Vfloor=N P(0,T) ET[max⁡(1−IR(T), 0)]\displaystyle V_{\text{floor}} = N\,P(0,T)\,\mathbb{E}^{T}\left[\max\left(1 - IR(T),\ 0\right)\right]

Priced with an option model on the cumulative index; small when the ratio is well above 1.

Worked example TIPS today 1.00582; the RRB at 1.63314 has no floor

Numbers of the US worked example: USD 25,000,000 of the 1.750% TIPS due Jul 15, 2036 at a real yield of 1.935% for settlement on Oct 29, 2026. The CPI-U levels are illustrative: April 2026 330.400, May 331.100, July 332.100, August 332.700.
Fixed Income›Agency Bond

Formulas

Cash flows

Coupon
C=N c2\displaystyle C = N\,\frac{c}{2}

Semiannual coupons on US agency debt.

Worked example USD 50,000,000 × 4.250% / 2 = USD 1,062,500

Accrued interest, 30/360
AI=c 360 (Y2−Y1)+30 (M2−M1)+(D2−D1)360\displaystyle AI = c\,\frac{360\,(Y_2-Y_1) + 30\,(M_2-M_1) + (D_2-D_1)}{360}

Agency, corporate and municipal bonds count every month as 30 days.

Y, M, DY,\ M,\ D
year, month and day of the last coupon (1) and of settlement (2)

Worked example 4.250 × 7 / 360 = 0.082639 per 100: USD 41,319

Settlement amount
S=N P+AI100\displaystyle S = N\,\frac{P + AI}{100}

Worked example USD 49,925,000 + USD 41,319 = USD 49,966,319

Price, yield and spread

Price from the yield
P+AI=∑iCFi(1+y/2) w+i−1w=d30/360180\displaystyle P + AI = \sum_{i} \frac{CF_i}{(1+y/2)^{\,w+i-1}} \qquad w = \frac{d_{30/360}}{180}

The street formula with 30/360 days for the first fraction.

Worked example 99.85 gives 4.304%

Spread to Treasuries
s=y−yUST\displaystyle s = y - y_{\text{UST}}

US agencies are quoted as a spread over the Treasury of the nearest maturity (interpolated for off-the-run maturities).

Worked example 4.304% − 4.159% = 14.5 bp

Margin in spread
margin=Δs×DV01\displaystyle \text{margin} = \Delta s \times \text{DV01}

A spread margin is worth the spread difference times the position’s DV01.

Worked example 0.5 bp × USD 13,836 = USD 6,918

Risk

DV01 and duration
DV01=−∂P∂y×0.0001Dmod=DV01P+AI×104\displaystyle \text{DV01} = -\frac{\partial P}{\partial y}\times 0.0001 \qquad D_{\text{mod}} = \frac{\text{DV01}}{P+AI}\times 10^4

For a bullet the rate DV01 and the spread DV01 are the same number.

Worked example 0.02767 per 100, USD 13,836 per bp; D = 2.77

Effective duration
Deff=P(y−Δy)−P(y+Δy)2 P Δy\displaystyle D_{\text{eff}} = \frac{P(y-\Delta y) - P(y+\Delta y)}{2\,P\,\Delta y}

For a callable the curve is bumped in the model and the call decision allowed to change.

Worked example ±10 bp: 2.13 for the callable, against 4.37 for the bullet

Effective convexity
Ceff=P(y+Δy)+P(y−Δy)−2PP Δy2\displaystyle C_{\text{eff}} = \frac{P(y+\Delta y) + P(y-\Delta y) - 2P}{P\,\Delta y^2}

Negative for a callable near the money: the investor is short the issuer’s option.

Worked example −34, against 23 for the bullet

The callable

Decomposition
Pcallable=Pbullet−Vcall\displaystyle P_{\text{callable}} = P_{\text{bullet}} - V_{\text{call}}

The investor holds a bullet and has sold the issuer a Bermudan call at par: economically a receiver swaption.

Worked example 103.5465 − 3.6558 = 99.8907

Coupon pickup
Δc≈VcallAA=∑i1/2(1+y/2)i\displaystyle \Delta c \approx \frac{V_{\text{call}}}{A} \qquad A = \sum_i \frac{1/2}{(1+y/2)^{i}}

The option premium spread over the life as extra coupon.

AA
annuity of 1% a year on the bullet curve

Worked example 3.656 / 4.433 = 82 bp: 4.50% + 0.82% ≈ 5.30%

Ho-Lee lattice
ri,j=ai+σΔt (2j−i)\displaystyle r_{i,j} = a_i + \sigma\sqrt{\Delta t}\,(2j - i)

A normal short rate on a recombining tree, the drifts a_i fitted so that the lattice reprices the bullet curve.

σ\sigma
normal volatility
Δt\Delta t
step, a quarter
jj
up moves after i steps

Worked example σ = 102 bp; the lattice reprices the bullet at 103.5465

Call rule
Vi,j=min⁡(Vi,jcont, 100+AIi)\displaystyle V_{i,j} = \min\left(V^{\text{cont}}_{i,j},\ 100 + AI_i\right)

On each call date the issuer calls when the value of continuing exceeds the call price: backward induction from maturity.

Structuring margin
M=N 100−Pcallable100\displaystyle M = N\,\frac{100 - P_{\text{callable}}}{100}

The investor pays par for a note the model values below par.

Worked example USD 109,292, about 11 cents

Call and yield to worst

Yield to call
P+AI=∑ti≤tcCFi(1+yc/2) w+i−1+100(1+yc/2) w+nc−1\displaystyle P + AI = \sum_{t_i \le t_c} \frac{CF_i}{(1+y_c/2)^{\,w+i-1}} + \frac{100}{(1+y_c/2)^{\,w+n_c-1}}

The yield if the bond is called at par on the call date.

Worked example at 100.50 on the issue date: 4.782% to the first call, 5.185% to maturity

Yield to worst
yworst=min⁡(yc1, …, yck, ymat)\displaystyle y_{\text{worst}} = \min\left(y_{c_1},\ \dots,\ y_{c_k},\ y_{\text{mat}}\right)

The lowest yield over every call date and maturity: the quote for a premium callable.

Worked example 4.782%

Amount on a call
N(1+c d30/360360)\displaystyle N\left(1 + c\,\frac{d_{30/360}}{360}\right)

Par plus the interest accrued since the last coupon.

Worked example USD 100,000,000 + USD 1,325,000 on Feb 4, 2028

Numbers of the US worked example: USD 50,000,000 FHLB 4.250% bullet due Oct 22, 2029 at 99.85 for settlement on Oct 29, 2026 (route B); USD 100,000,000 5nc1 callable at 5.30% issued on Nov 4, 2026 against a flat 4.50% bullet curve, priced on a Ho-Lee lattice at a normal volatility of 102 bp (route C), illustrative.
Fixed Income›Sovereign and SSA Bond

Formulas

Cash flows

Coupon
C=N cf\displaystyle C = N\,\frac{c}{f}

Semiannual for USD global bonds and Canadian Maple bonds, annual for euro and most sterling issues.

Worked example USD 25,000,000 × 3.875% / 2 = USD 484,375

Accrued interest
AI=c d30/360360  (USD)AI=c dD  (EUR, GBP, Act/Act annual)\displaystyle AI = c\,\frac{d_{30/360}}{360} \;\text{(USD)} \qquad AI = c\,\frac{d}{D} \;\text{(EUR, GBP, Act/Act annual)}

The day count follows the currency’s market.

Worked example 3.875 × 15 / 360 = 0.161458 per 100: USD 40,365

Settlement amount
S=N P+AI100\displaystyle S = N\,\frac{P + AI}{100}

Worked example route S: USD 24,987,865; route E: USD 24,349,219

Spread, yield and price

Swap rate on the bond’s basis
MSsa=2[(1+MS 365360)1/2−1]\displaystyle MS_{\text{sa}} = 2\left[\left(1 + MS\,\tfrac{365}{360}\right)^{1/2} - 1\right]

The SOFR swap rate (annual, Act/360) restated as a semiannual 30/360 yield before the spread is added.

Worked example 3.70% gives 3.7169%

Yield from the spread
y=MSsa+s\displaystyle y = MS_{\text{sa}} + s

SSA bonds are priced at a spread to mid-swaps; the yield sets the price.

Worked example 3.7169% + 0.205% = 3.9219%, price 99.79

Bid and offer in price
Pbid=P(MSsa+sbid)Poffer=P(MSsa+soffer)\displaystyle P_{\text{bid}} = P(MS_{\text{sa}} + s_{\text{bid}}) \qquad P_{\text{offer}} = P(MS_{\text{sa}} + s_{\text{offer}})

A wider spread is a lower price: the dealer bids the wider spread.

Worked example +22 bp: 99.722; +20 bp: 99.812

Spread to Treasuries
s=y−yUST\displaystyle s = y - y_{\text{UST}}

Emerging market sovereigns are quoted in price with the spread to the Treasury of similar maturity.

Worked example 96.60 gives 6.915%: 6.915% − 4.31% = 260 bp

Margin
M=N Pclient−Pmid100M=Δs×DV01\displaystyle M = N\,\frac{P_{\text{client}} - P_{\text{mid}}}{100} \qquad M = \Delta s \times \text{DV01}

The dealer’s margin against the mid of the market, in price or in spread.

Worked example route E: (96.60 − 96.50) × 250,000 = USD 25,000; route S: 0.5 bp × USD 11,154 = USD 5,577

Risk

DV01
DV01=−∂P∂y×0.0001\displaystyle \text{DV01} = -\frac{\partial P}{\partial y}\times 0.0001

The rate risk, hedged with a swap (SSA) or Treasury futures (emerging markets).

Worked example route S: 0.04461 per 100, USD 11,154 per bp

Spread DV01 (CS01)
CS01=−∂P∂s×0.0001\displaystyle \text{CS01} = -\frac{\partial P}{\partial s}\times 0.0001

The loss for one basis point of wider spread; for a fixed coupon bond it equals the DV01.

Worked example route E: USD 15,367 per bp

CDS hedge
NCDS=CS01bondRPV01×0.0001\displaystyle N_{\text{CDS}} = \frac{\text{CS01}_{\text{bond}}}{RPV01 \times 0.0001}

The notional of protection whose value moves as much as the bond for a parallel spread move.

RPV01RPV01
risky annuity of the CDS, in years

Worked example USD 15,367 / (4.45 × 0.0001) = USD 34,500,000 of protection

Jump to default
JTD=N(P+AI100−R)\displaystyle JTD = N\left(\frac{P + AI}{100} - R\right)

The loss if the sovereign defaults today and the bond is worth its recovery.

RR
recovery rate

Worked example USD 18,099,219 at 25%

Credit

Hazard rate
λ≈sCDS1−R\displaystyle \lambda \approx \frac{s_{\text{CDS}}}{1 - R}

The default intensity implied by the CDS spread, by the credit triangle.

Worked example 180 bp / 0.75 = 2.40% a year

Default probability
PD(T)=1−e−λT\displaystyle PD(T) = 1 - e^{-\lambda T}

The probability of default before T at a constant hazard rate.

Worked example 5 years: 11.3%

Bond-CDS basis
basis=sCDS−sbond\displaystyle \text{basis} = s_{\text{CDS}} - s_{\text{bond}}

Negative when the bond pays more than the protection costs: a buyer of the bond and of protection earns the gap.

sbonds_{\text{bond}}
the bond’s z-spread or asset swap spread

Numbers of the US worked example: USD 25,000,000 of the supranational 3.875% due Oct 14, 2031 at mid-swaps + 20.5 bp for settlement on Oct 29, 2026 (route S) and USD 25,000,000 of the sovereign 6.375% due Mar 15, 2035 at 96.60 for settlement on Oct 30, 2026 (route E); the 10-year Treasury at 4.31%, the 5-year CDS at 180 bp with a 25% recovery and a risky annuity of 4.45, illustrative.
Fixed Income›Corporate Bond

Formulas

Cash flows and price

Coupon
C=N cf\displaystyle C = N\,\frac{c}{f}

Each coupon is the annual rate divided by the number of payments a year.

NN
face amount
cc
coupon rate
ff
payments a year: 2 for US and Canadian bonds, 1 for euro and most sterling bonds

Worked example 10,000,000 × 5.425% / 2 = USD 271,250

Accrued interest
AI=N c τ(tprev, tsettle)\displaystyle AI = N\,c\,\tau(t_{\text{prev}},\,t_{\text{settle}})

The buyer pays the seller the coupon earned since the last coupon date.

τ\tau
accrual fraction: 30/360 in the US, Act/Act (ICMA) for Eurobonds, Actual/365 in Canada

Worked example 10,000,000 × 5.425% × 164/360 = USD 247,139

Dirty and clean price
Pdirty=∑i=1nCFi(1+y/f) w+i−1Pclean=Pdirty−AI\displaystyle P_{\text{dirty}} = \sum_{i=1}^{n}\frac{CF_i}{(1+y/f)^{\,w+i-1}} \qquad P_{\text{clean}} = P_{\text{dirty}} - AI

The street convention discounts every flow at the yield, with a fractional first period; the quote is the clean price.

CFiCF_i
coupon i, plus the principal at maturity
ww
fraction of the coupon period left to the next coupon date
yy
yield

Worked example y = 5.250%: dirty 103.8214, clean 101.35

Settlement amount
cash=N Pclean100+AI\displaystyle \text{cash} = N\,\frac{P_{\text{clean}}}{100} + AI

What moves against the bonds at the depository on the settlement date.

Worked example 10,135,000 + 247,139 = USD 10,382,139

Discount factor at the yield
DFi=(1+y/f)−(w+i−1)\displaystyle DF_i = (1+y/f)^{-(w+i-1)}

The discount factor of each coupon date in the cash flow table: with it, the coupons and principal are worth the dirty price.

Spreads

Benchmark spread
sbench=y−ybench\displaystyle s_{\text{bench}} = y - y_{\text{bench}}

The quote: the yield over the on-the-run government bond of similar maturity.

Worked example 5.250% − 4.310% = 94 bp

G-spread
sG=y−ygov(T)\displaystyle s_G = y - y_{\text{gov}}(T)

The yield over the government curve interpolated at the bond’s maturity.

TT
the bond’s maturity

Worked example Treasury at 10.06 years 4.313%: 94 bp

I-spread
sI=y−S(T)\displaystyle s_I = y - S(T)

The yield over the swap rate interpolated at the bond’s maturity; wider than the G-spread when swap rates are below Treasury yields.

S(T)S(T)
par SOFR swap rate at the maturity

Worked example 5.250% − 3.850% = 140 bp

Z-spread
Pdirty=∑iCFi P(0,ti) e−z ti\displaystyle P_{\text{dirty}} = \sum_i CF_i\,P(0,t_i)\,e^{-z\,t_i}

The constant spread over the zero curve that reprices the bond: each flow discounted at its own zero rate plus z.

P(0,t)P(0,t)
SOFR discount factor
zz
Z-spread, continuously compounded

Worked example z = 135 bp over the SOFR zero curve

Option-adjusted spread
Pdirty=EQ ⁣[∑iCFi e−∫0ti(ru+OAS) du]\displaystyle P_{\text{dirty}} = \mathbb{E}^{\mathbb{Q}}\!\left[\sum_i CF_i\,e^{-\int_0^{t_i}(r_u + \text{OAS})\,du}\right]

The spread over simulated short rates that reprices a callable bond once the issuer’s call is valued in each path; close to the Z-spread for a bond with only a make-whole call.

rur_u
short rate in a model calibrated to the swap curve and to swaption volatilities
Make-whole call price
KMW=max⁡ ⁣(100, ∑iCFi(1+(yT+m)/2)ki)\displaystyle K_{\text{MW}} = \max\!\left(100,\ \sum_i \frac{CF_i}{(1+(y_{\text{T}}+m)/2)^{k_i}}\right)

An early redemption pays the remaining flows discounted at the Treasury yield plus a small spread, so the issuer rarely calls for a lower rate.

mm
make-whole spread, 15 to 25 bp
yTy_{\text{T}}
Treasury yield of the remaining life

Risk and hedging

Modified duration
D=−1Pdirty∂P∂y\displaystyle D = -\frac{1}{P_{\text{dirty}}}\frac{\partial P}{\partial y}

The percentage price change for a change in yield.

Worked example 7.51

DV01
DV01=D Pdirty N×10−4\displaystyle \text{DV01} = D\,P_{\text{dirty}}\,N \times 10^{-4}

The value change for one basis point of yield; the dealer’s margin is quoted in basis points of spread, so it is a number of DV01s.

Worked example USD 7,794 per bp; the 2 bp margin is about USD 15,600

Convexity
ΔPP≈−D Δy+12 C Δy2\displaystyle \frac{\Delta P}{P} \approx -D\,\Delta y + \tfrac{1}{2}\,C\,\Delta y^2

The second-order term: a bullet bond gains more when yields fall than it loses when they rise.

Worked example C = 70.6

Spread duration (CS01)
CS01=−∂P∂s×10−4\displaystyle \text{CS01} = -\frac{\partial P}{\partial s} \times 10^{-4}

The loss for a 1 bp widening of the credit spread with the government curve unchanged; close to the DV01 for a fixed-rate bullet bond.

Hedge ratio
h=DV01bondDV01hedge\displaystyle h = \frac{\text{DV01}_{\text{bond}}}{\text{DV01}_{\text{hedge}}}

The face amount of the hedge that offsets the bond’s rate risk, leaving the spread risk.

Worked example 10-year note (4.25%, 99-16+): DV01 0.0803 per 100; USD 9.70 million of notes

Credit

Expected loss
EL=PD×LGD×EAD\displaystyle EL = PD \times LGD \times EAD

The average loss from default over a horizon.

PDPD
probability of default
LGDLGD
loss given default, 1 − recovery
EADEAD
exposure at default
Credit triangle
s≈λ (1−R)\displaystyle s \approx \lambda\,(1-R)

A credit spread pays for the default intensity times the loss given default.

λ\lambda
hazard rate, a year
RR
recovery rate

Worked example Z-spread 135 bp at 40% recovery: λ ≈ 2.25% a year, an upper bound since spreads also pay for liquidity

Bond-CDS basis
basis=sCDS−sZ\displaystyle \text{basis} = s_{\text{CDS}} - s_{\text{Z}}

The issuer’s CDS spread less the bond’s spread over the swap curve; a negative basis is bought with the bond and the protection.

Worked example in US dollars: USD 10,000,000 of a 5.425% bond due Nov 15, 2036, bought at 101.35 for settlement on Oct 29, 2026; Treasury yields 7 years 4.13% and 10 years 4.31%, SOFR swaps 7 years 3.777% and 10 years 3.850% (illustrative).
Fixed Income›High Yield Bond

Formulas

Cash flows and yields

Coupon
C=N c2\displaystyle C = N\,\frac{c}{2}

The semiannual coupon.

cc
coupon rate

Worked example 10,000,000 × 7.25% / 2 = USD 362,500

Accrued interest
AI=N c d30/360360\displaystyle AI = N\,c\,\frac{d_{30/360}}{360}

The interest accrued since the last coupon date; zero once the bond trades flat.

Worked example 10,000,000 × 7.25% × 44/360 = USD 88,611

Yield to maturity
Pdirty=∑i=1nCFi(1+yM/2) w+i−1\displaystyle P_{\text{dirty}} = \sum_{i=1}^{n}\frac{CF_i}{(1+y_M/2)^{\,w+i-1}}

The yield that discounts all the coupons and the principal at maturity to the dirty price.

Worked example 92.25 clean: 9.10%

Yield to call
Pdirty=∑ti≤tcC(1+yc/2) w+i−1+Kc(1+yc/2) w+nc−1\displaystyle P_{\text{dirty}} = \sum_{t_i \le t_c}\frac{C}{(1+y_c/2)^{\,w+i-1}} + \frac{K_c}{(1+y_c/2)^{\,w+n_c-1}}

The same with the bond redeemed at the call price on a call date.

KcK_c
call price
tct_c
call date

Worked example to the par call of Mar 15, 2029: 11.04%

Yield to worst
yW=min⁡(yM, yc1, yc2, … )\displaystyle y_W = \min\left(y_M,\ y_{c_1},\ y_{c_2},\ \dots\right)

The lowest yield the bond can earn without a default; the market quotes it with the price.

Worked example 9.10%: maturity, since the bond trades below every call price

Valuation

Spread to worst
sW=yW−ygov(tW)\displaystyle s_W = y_W - y_{\text{gov}}(t_W)

The yield to worst over the government curve at the workout date.

tWt_W
the date that gives the worst yield

Worked example 9.10% − 3.94% = 516 bp

Callable bond
Pcallable=Pstraight−Cissuer\displaystyle P_{\text{callable}} = P_{\text{straight}} - C_{\text{issuer}}

The holder owns a straight bond and has sold the issuer an option to buy it back at the call prices.

Call decision in a lattice
Vt=min⁡ ⁣(Kt, Et ⁣[Vt+Δt+CFt+Δt]e−(rt+OAS) Δt)\displaystyle V_t = \min\!\left(K_t,\ \mathbb{E}_t\!\left[V_{t+\Delta t} + CF_{t+\Delta t}\right]e^{-(r_t + \text{OAS})\,\Delta t}\right)

Backward induction: on each node the issuer calls when the bond is worth more than the call price.

KtK_t
call price in force at t
Z-spread
Pdirty=∑iCFi P(0,ti) e−z ti\displaystyle P_{\text{dirty}} = \sum_i CF_i\,P(0,t_i)\,e^{-z\,t_i}

The spread over the SOFR zero curve that reprices the flows to maturity.

Worked example 518 bp

Risk

Duration to worst
DW=−1P∂P∂yW\displaystyle D_W = -\frac{1}{P}\frac{\partial P}{\partial y_W}

The duration of the flows to the workout date; it jumps when the workout date changes.

Worked example 4.27

DV01
DV01=DW Pdirty N×10−4\displaystyle \text{DV01} = D_W\,P_{\text{dirty}}\,N \times 10^{-4}

The value change for one basis point of yield.

Worked example USD 3,979 per bp

Negative convexity
∂2Pcallable∂y2<0near the call price\displaystyle \frac{\partial^2 P_{\text{callable}}}{\partial y^2} < 0 \quad \text{near the call price}

Near the call price the bond gains less from a fall in yields than it loses from a rise.

Default and recovery

Credit triangle
λ≈s1−R\displaystyle \lambda \approx \frac{s}{1-R}

The default intensity implied by the spread and the recovery.

RR
recovery rate

Worked example 516 bp at 40% recovery: about 8.6% a year

Recovery waterfall
Rk=min⁡ ⁣(1, max⁡ ⁣(0, EV−∑j≻kDjDk))\displaystyle R_k = \min\!\left(1,\ \max\!\left(0,\ \frac{EV - \sum_{j \succ k} D_j}{D_k}\right)\right)

Each class recovers what is left of the enterprise value after the claims ranking ahead of it.

EVEV
enterprise value
DkD_k
claims of class k
j≻kj \succ k
classes senior to k
Distressed return
Vplan−PP\displaystyle \frac{V_{\text{plan}} - P}{P}

The return of a claim bought flat and exchanged under the plan.

Worked example (52 − 45.50) / 45.50 = 14.3%: USD 650,000 on USD 10,000,000 face

Flat trading
cash=N P100(AI=0)\displaystyle \text{cash} = N\,\frac{P}{100} \quad (AI = 0)

A defaulted bond changes hands with no accrued interest.

Worked example 10,000,000 × 45.50% = USD 4,550,000

Worked example: USD 10,000,000 of a 7.25% bond due Mar 15, 2032 bought at 92.25 for settlement on Oct 29, 2026; calls at 103.625 (2027), 101.8125 (2028) and par (2029); Treasury yields 5 years 3.90%, 7 years 4.13% (illustrative).
Fixed Income›Credit Default Swap

Formulas

Premium leg

Coupon
Ci=N c di360\displaystyle C_i = N\,c\,\frac{d_i}{360}

Each quarterly coupon is the standard rate on the notional for the days of the period.

cc
standard coupon, 100 or 500 bp
did_i
days in period i; the last period includes the maturity date

Worked example 1% × 91/360 × 25,000,000 = USD 63,194

Accrued at the trade
AI=N c tstep-in−tstart360\displaystyle AI = N\,c\,\frac{t_{\text{step-in}} - t_{\text{start}}}{360}

The coupon accrued from the last IMM date to the step-in date (T+1), refunded by the seller since the first coupon is paid in full.

Worked example 1% × 38/360 × 25,000,000 = USD 26,389

Risky annuity
RPV01=∑idi360 P(0,ti) Q(ti)+accrual on default\displaystyle \text{RPV01} = \sum_i \frac{d_i}{360}\,P(0,t_i)\,Q(t_i) + \text{accrual on default}

The value of 1 a year paid quarterly until default or maturity.

Q(t)Q(t)
survival probability to t
P(0,t)P(0,t)
SOFR discount factor

Worked example 4.372 years, net of the accrued at the trade

Premium leg
PVprem=N c RPV01\displaystyle PV_{\text{prem}} = N\,c\,\text{RPV01}

What the coupons are worth to the seller, default risk included.

Valuation and the upfront

Survival probability
Q(t)=e−λt\displaystyle Q(t) = e^{-\lambda t}

With a constant hazard rate, the probability that the name has not defaulted by t.

λ\lambda
hazard rate

Worked example λ = 3.23%: Q to maturity = 84.7%, a default probability of 15.3% over 5.15 years

Credit triangle
λ≈s1−R\displaystyle \lambda \approx \frac{s}{1-R}

The spread pays for the default intensity times the loss given default.

RR
recovery, 40%

Worked example 1.92% / 0.60 = 3.20%; exact fit 3.23%

Protection leg
PVprot=(1−R) N∫0TP(0,t) (−dQ(t))\displaystyle PV_{\text{prot}} = (1-R)\,N\int_0^T P(0,t)\,\bigl(-dQ(t)\bigr)

The loss given default, weighted by the probability of default in each instant and discounted.

Worked example 8.39% of notional: USD 2,098,616

Par spread
s=PVprotN RPV01\displaystyle s = \frac{PV_{\text{prot}}}{N\,\text{RPV01}}

The running spread that makes the two legs equal: the quote.

Worked example 192 bp by construction

Upfront
U=N (s−c) RPV01(s)\displaystyle U = N\,(s - c)\,\text{RPV01}(s)

The standard coupon is paid instead of the quoted spread; the difference over the life is settled up front.

Worked example (1.92% − 1.00%) × 4.372 = 4.02%: USD 1,005,587

Cash on T+3
cash=U−AI\displaystyle \text{cash} = U - AI

The upfront less the accrued coupon refunded by the seller.

Worked example 1,005,587 − 26,389 = USD 979,198

Risk

CS01
CS01≈N RPV01×10−4\displaystyle \text{CS01} \approx N\,\text{RPV01} \times 10^{-4}

The value change for 1 bp of spread.

Worked example USD 10,930 on USD 25 million; USD 46,306 on USD 100 million of CDX IG

Dealer margin
margin≈(sclient−shedge) CS01\displaystyle \text{margin} \approx (s_{\text{client}} - s_{\text{hedge}})\,\text{CS01}

The margin in basis points of spread, turned into present value.

Worked example B: (192 − 185) × 10,930 = USD 76,512; K: 0.25 × 46,306 = USD 11,576

Jump-to-default
JTDbuyer=N (1−R)−V\displaystyle \text{JTD}_{\text{buyer}} = N\,(1-R) - V

The gain of the protection buyer if the name defaults now: the payment less the current value of the contract.

VV
value of the protection to the buyer

Worked example 25,000,000 × 60% − 1,005,587 = USD 13,994,413

Index spread
sindex≈∑kwk sk RPV01k∑kwk RPV01k\displaystyle s_{\text{index}} \approx \frac{\sum_k w_k\,s_k\,\text{RPV01}_k}{\sum_k w_k\,\text{RPV01}_k}

The index trades close to the duration-weighted average of its names; the gap is the index basis.

wkw_k
weight of name k

Credit event

Protection payment
N(1−Pfinal100)\displaystyle N\left(1 - \frac{P_{\text{final}}}{100}\right)

The seller pays par less the auction’s final price on the notional.

PfinalP_{\text{final}}
auction final price

Worked example 25,000,000 × (1 − 0.35) = USD 16,250,000

Accrued to the event
N c tEDD−tlast360\displaystyle N\,c\,\frac{t_{\text{EDD}} - t_{\text{last}}}{360}

The buyer pays the coupon accrued from the last coupon date to the event determination date.

Worked example 1% × 75/360 × 25,000,000 = USD 52,083

Index after a default
Nnew=N (1−wk)payment=N wk (1−R)\displaystyle N^{\text{new}} = N\,(1 - w_k) \qquad \text{payment} = N\,w_k\,(1-R)

An index position continues on the other names with the notional reduced by the defaulted name’s weight.

Worked example one name of 125 in CDX IG: USD 100,000,000 becomes USD 99,200,000

Payoffs

Buyer at a credit event
N (1−R)−premiums paid\displaystyle N\,(1-R) - \text{premiums paid}

The protection payment less what the buyer paid before the event.

Worked example 16,250,000 − 1,295,170 − 52,083 = USD 14,902,747

Mark to market
V(s)=N (s−c) RPV01(s)\displaystyle V(s) = N\,(s - c)\,\text{RPV01}(s)

The value of a standard contract at today’s spread, for the buyer; its change since the trade is the P&L.

Worked example spread to 292 bp: +USD 1,009,894

Worked example, route B: USD 25,000,000, 5 years to Dec 20, 2031, 100 bp coupon, traded at 192 bp on Oct 28, 2026; recovery 40%; SOFR discounting; one flat hazard rate calibrated to the quoted spread, the standard model in its simplest form (illustrative).
Fixed Income›Leveraged Loan

Formulas

Interest and settlement

All-in rate
r=max⁡(RSOFR, F)+m\displaystyle r = \max(R_{\text{SOFR}},\ F) + m

The base rate, floored, plus the margin, fixed at the start of each interest period.

FF
floor
mm
margin

Worked example max(3.90%, 0.50%) + 3.50% = 7.40%

Interest for a period
I=B r d360\displaystyle I = B\,r\,\frac{d}{360}

Paid by the borrower through the agent at the end of the interest period.

BB
outstanding amount
dd
days in the period

Worked example Nov 18, 2026 to Dec 31, 2026: 10,000,000 × 7.40% × 43/360 = USD 88,389

Amortization
A=0.25%×N0 a quarter\displaystyle A = 0.25\%\times N_0 \ \text{a quarter}

A small repayment each quarter; the balance at maturity.

N0N_0
original amount

Worked example USD 25,000 a quarter on USD 10,000,000

Purchase price
cash=N P100\displaystyle \text{cash} = N\,\frac{P}{100}

Par trades settle without accrued interest: the agent pays each lender the interest for the days it held the loan.

Worked example 10,000,000 × 99.00% = USD 9,900,000

Delayed compensation
DC=N r dlate360−N P100 rcarry dlate360\displaystyle DC = N\,r\,\frac{d_{\text{late}}}{360} - N\,\frac{P}{100}\,r_{\text{carry}}\,\frac{d_{\text{late}}}{360}

From T+7 to settlement the seller passes on the loan’s interest and the buyer pays the cost of carry on the price.

dlated_{\text{late}}
days from T+7 to settlement

Worked example 24,667 − 12,870 = USD 11,797 for 12 days

Valuation

Discount factor
DFk=∏j≤k(1+(Rj+DM) τj)−1\displaystyle DF_k = \prod_{j \le k}\left(1 + (R_j + DM)\,\tau_j\right)^{-1}

Each period is discounted at the base rate plus the discount margin.

DMDM
discount margin
Discount margin
N P100=∑k(Ik+Ak) DFk(DM)\displaystyle N\,\frac{P}{100} = \sum_k (I_k + A_k)\,DF_k(DM)

The spread over the base rate that discounts the projected flows to the price.

Worked example 371 bp to maturity

Discount margin to a 3-year life
N P100=∑tk≤3(Ik+Ak) DFk+B3 DF3\displaystyle N\,\frac{P}{100} = \sum_{t_k \le 3} (I_k + A_k)\,DF_k + B_3\,DF_3

The market convention: the loan is assumed repaid at par after three years.

B3B_3
balance at 3 years

Worked example 386 bp: the 1-point discount earned over 3 years instead of 6

Quick discount margin
DM≈m+100−PL\displaystyle DM \approx m + \frac{100 - P}{L}

A rule of thumb: the margin plus the discount spread over the expected life.

LL
expected life, years

Worked example 350 + 100/3 = 383 bp, against 386 bp exactly

The floor

Floored base rate
max⁡(R, F)=R+max⁡(F−R, 0)\displaystyle \max(R,\ F) = R + \max(F - R,\ 0)

The floor adds a put on the base rate to the coupon.

Floor value
Floor=∑kBk τk P(0,tk) [(F−fk) N(−dk)+σNtk  φ(dk)]\displaystyle \text{Floor} = \sum_k B_k\,\tau_k\,P(0,t_k)\,\Bigl[(F - f_k)\,N(-d_k) + \sigma_N\sqrt{t_k}\;\varphi(d_k)\Bigr]

A strip of puts on the base rate in the normal model, one for each interest period.

fkf_k
forward base rate
dkd_k
(f_k − F)/(σ√t_k)
σN\sigma_N
normal volatility

Worked example USD 13,460, 2.5 bp a year

Risk

Spread duration
Ds=−1P∂P∂DM\displaystyle D_s = -\frac{1}{P}\frac{\partial P}{\partial DM}

Close to the expected life, since the coupon resets and the spread is fixed.

Worked example 4.8 years to maturity: USD 4,743 per bp

Rate duration
Dr≈treset1+r treset\displaystyle D_r \approx \frac{t_{\text{reset}}}{1 + r\,t_{\text{reset}}}

The rate risk lasts only until the next reset.

tresett_{\text{reset}}
time to the next reset

Worked example 43 days: 0.12 years

Expected loss
EL=PD×(1−R)\displaystyle EL = PD \times (1 - R)

First-lien loans recover more than unsecured bonds of the same borrower.

CLO tests

Overcollateralization test
OCc=par of the loans−haircutspar of class c and the classes above it ≥ trigger\displaystyle OC_c = \frac{\text{par of the loans} - \text{haircuts}}{\text{par of class } c \text{ and the classes above it}} \ \ge\ \text{trigger}

When a test fails, interest is diverted from the junior notes to repay the senior ones.

Interest coverage test
ICc=interest received−senior feesinterest due on class c and above\displaystyle IC_c = \frac{\text{interest received} - \text{senior fees}}{\text{interest due on class } c \text{ and above}}

The same test on interest.

CCC excess
haircut=∑excess CCC(100−Pmarket)\displaystyle \text{haircut} = \sum_{\text{excess CCC}} (100 - P_{\text{market}})

CCC loans above the limit (typically 7.5% of the portfolio) count at market value in the OC test.

Worked example: USD 10,000,000 of a term loan B bought at 99.00 on Oct 28, 2026, Term SOFR + 350 bp with a 0.50% floor, maturity Dec 31, 2032; Term SOFR projected flat at 3.90%, 100 bp of normal volatility for the floor (illustrative).
Fixed Income›Municipal Bond

Formulas

Cash flows

Short first coupon
C1=N c d30/360(dated, t1)360\displaystyle C_1 = N\,c\,\frac{d_{30/360}(\text{dated},\ t_1)}{360}

A new issue dated between coupon dates pays a short (or long) first coupon.

Worked example USD 5,000,000 × 5% × 169 / 360 = USD 117,361.11

Regular coupon
C=N c2\displaystyle C = N\,\frac{c}{2}

Worked example USD 125,000

Delivery at closing
S=N P100\displaystyle S = N\,\frac{P}{100}

New issue bonds dated at closing carry no accrued interest.

Worked example USD 5,000,000 × 116.656 / 100 = USD 5,832,800

Accrued interest in the secondary
AI=c d30/360360\displaystyle AI = c\,\frac{d_{30/360}}{360}

Munis accrue on 30/360.

Worked example route S, Nov 13, 2026: 5 × 1 / 360 = 0.013889 per 100

Price and yield

Price from the yield (MSRB Rule G-33)
P+AI=∑iCFi(1+y/2) w+i−1\displaystyle P + AI = \sum_{i} \frac{CF_i}{(1+y/2)^{\,w+i-1}}

The dollar price of a muni is computed from the yield to the call or to maturity, 30/360, semiannual.

Worked example 3.05% gives 116.656; 3.10% gives 116.185

Yield to worst
P=min⁡kP(y; tk)⇔yworst=min⁡ky(P; tk)\displaystyle P = \min_k P(y;\ t_k) \quad\Leftrightarrow\quad y_{\text{worst}} = \min_k y(P;\ t_k)

The price at the yield to the call date or to maturity that is lowest: for a premium bond, the call.

tkt_k
each call date and the maturity

Worked example the 2046 maturity: 3.78% to the 2036 call, price 110.057

Tax-equivalent yield
TEY=y1−t\displaystyle TEY = \frac{y}{1 - t}

The taxable yield that leaves the same after tax as a tax-exempt yield.

tt
marginal tax rate

Worked example 3.10% / (1 − 40.8%) = 5.24%

Muni to Treasury ratio
ρ=yMMDyUST\displaystyle \rho = \frac{y_{\text{MMD}}}{y_{\text{UST}}}

The relative value of the muni market: the AAA muni yield over the Treasury yield of the same maturity.

Worked example 2.90% / 4.310% = 67%

The competitive sale

True interest cost
∑tDSt(1+TIC/2)2t=Par+Premium−UW\displaystyle \sum_{t} \frac{DS_t}{(1+TIC/2)^{2t}} = \text{Par} + \text{Premium} - \text{UW}

The issuer’s cost of funds: the rate that discounts the debt service to the proceeds it receives. The lowest TIC wins a competitive sale.

DStDS_t
coupons and principal due at t, all maturities
UW\text{UW}
underwriting spread

Worked example proceeds USD 332,600,850: TIC 3.68%

Net interest cost
NIC=∑coupons−Premium+UW∑kPark Tk\displaystyle NIC = \frac{\sum \text{coupons} - \text{Premium} + \text{UW}}{\sum_k \text{Par}_k\,T_k}

The older measure: interest over bond-years, without discounting.

TkT_k
years to maturity k

Worked example 3.96%

Underwriting spread
UW=∑kPark Pkreoffer−Pkbid100\displaystyle \text{UW} = \sum_k \text{Par}_k\,\frac{P^{\text{reoffer}}_k - P^{\text{bid}}_k}{100}

The bank buys at its bid and reoffers to investors; the gap pays the syndicate, quoted in dollars per USD 1,000 bond.

Worked example USD 5.00 per bond: USD 25,000 on the allotment, USD 1,500,000 on the deal

Risk and markup

DV01
DV01=−∂P∂y×0.0001\displaystyle \text{DV01} = -\frac{\partial P}{\partial y}\times 0.0001

Worked example 0.09358 per 100: USD 4,679 per bp on USD 5,000,000

Duration to worst
Dworst=Dmod(cash flows to tworst)\displaystyle D_{\text{worst}} = D_{\text{mod}}(\text{cash flows to } t_{\text{worst}})

For a premium callable the duration is that of a bond maturing on the call date.

Worked example the 2046 maturity at 3.78%: 7.93 to the 2036 call, against 12.94 if it ran to 2046

Markup
M=N Psold−Pcost100\displaystyle M = N\,\frac{P_{\text{sold}} - P_{\text{cost}}}{100}

The dealer’s markup on a principal trade; disclosed in dollars and percent on retail confirmations.

Worked example (116.185 − 115.720) × 50,000 = USD 23,250

Price to worst

Price to worst
Pworst(y)=min⁡(Pcall(y), Pmat(y))\displaystyle P_{\text{worst}}(y) = \min\left(P_{\text{call}}(y),\ P_{\text{mat}}(y)\right)

Below the coupon the call price is lower, above it the maturity price: the kink is at the coupon rate.

Numbers of the US worked example: the 5% general obligation bond due Nov 1, 2036, dated and delivered on Nov 12, 2026 at 3.05% (route P), and the same bond at 3.10% for settlement on Nov 13, 2026 (route S). The scale of the deal (USD 15,000,000 a year, yields 2.55% in 2027 to 3.78% in 2046, the 2037 to 2046 maturities priced to the 2036 call) is illustrative.
Fixed Income›Asset Swap

Formulas

The package

Package price
N P100+AI+U=N+AIU=N(1−P100)\displaystyle N\,\frac{P}{100} + AI + U = N + AI \qquad U = N\left(1 - \frac{P}{100}\right)

The investor pays par plus accrued: the bond at its price and the upfront into the swap.

UU
upfront

Worked example 24,375,000 + 625,000 = USD 25,000,000, plus USD 137,500 accrued

Fixed leg
Xi=N c τi\displaystyle X_i = N\,c\,\tau_i

The swap pays the bond’s coupon on the bond’s dates; the first period runs from the start date, so it returns the accrued paid on the bond.

Worked example first payment 4.50% × 136/360 × 25,000,000 = USD 425,000; then USD 562,500

Floating leg
Yj=N (Rj+A) dj360\displaystyle Y_j = N\,(R_j + A)\,\frac{d_j}{360}

SOFR compounded over each quarter plus the asset swap spread.

AA
asset swap spread
Net income
coupon−Xi+Yj  ≈  N (Rj+A) τj\displaystyle \text{coupon} - X_i + Y_j \;\approx\; N\,(R_j + A)\,\tau_j

The coupon passes through; the investor keeps the floating rate plus the spread.

Worked example SOFR + 108 bp

The asset swap spread

Bond on the swap curve
Pswap=∑iCFi P(0,ti)−AI\displaystyle P_{\text{swap}} = \sum_i CF_i\,P(0,t_i) - AI

The bond’s clean value if its flows were discounted at the riskless swap curve.

P(0,t)P(0,t)
SOFR discount factor

Worked example 104.23

Par asset swap spread
A=Pswap−P100⋅PV01flt\displaystyle A = \frac{P_{\text{swap}} - P}{100 \cdot \text{PV01}_{\text{flt}}}

The spread that makes the package worth par: the bond’s discount to its riskless value, spread over the floating leg’s annuity.

PV01flt\text{PV01}_{\text{flt}}
annuity of the floating leg, years

Worked example (104.23 − 97.50) / 6.133 = 110 bp mid

Annuity of the floating leg
PV01flt=∑jdj360 P(0,tj)\displaystyle \text{PV01}_{\text{flt}} = \sum_j \frac{d_j}{360}\,P(0,t_j)

The value of 1 a year paid on the floating dates.

Worked example 6.133

Upfront as the swap’s value
U=N[PVflt+A PV01flt−PVfix]\displaystyle U = N\left[PV_{\text{flt}} + A\,\text{PV01}_{\text{flt}} - PV_{\text{fix}}\right]

The investor pays upfront what its side of the swap is worth: the floating leg and the spread it receives less the coupons it pays away.

PVflt, PVfixPV_{\text{flt}},\ PV_{\text{fix}}
present values of the legs, per unit of notional

Worked example 25,000,000 × (0.2282 + 1.10% × 6.133 − 0.2707) = USD 625,000 at the mid

Market asset swap
Amkt=AparP/100\displaystyle A_{\text{mkt}} = \frac{A_{\text{par}}}{P/100}

When the swap notional is the bond’s price, the same spread is earned on less notional, so the quote is higher for a discount bond.

Worked example 110 / 0.975 = 112.8 bp (approximation)

I-spread and Z-spread
sI=y−S(T)Pdirty=∑iCFi P(0,ti) e−z ti\displaystyle s_I = y - S(T) \qquad P_{\text{dirty}} = \sum_i CF_i\,P(0,t_i)\,e^{-z\,t_i}

The other two spreads over the swap curve; all three are close for a bond near par.

Worked example I-spread 116 bp, Z-spread 111 bp, asset swap 110 bp

Risk

Margin
margin=ΔA×PV01flt×N\displaystyle \text{margin} = \Delta A \times \text{PV01}_{\text{flt}} \times N

The margin in basis points of spread, in present value.

Worked example 2 bp × 6.133 × 25,000,000 = USD 30,666

Spread risk
∂V∂A≈−N PV01flt×10−4\displaystyle \frac{\partial V}{\partial A} \approx -N\,\text{PV01}_{\text{flt}} \times 10^{-4}

The package loses this much for each basis point the asset swap spread widens.

Worked example USD 15,333 per bp

Package at default
Vdefault=N R+Vswap\displaystyle V_{\text{default}} = N\,R + V_{\text{swap}}

The bond pays its recovery; the swap keeps its market value, positive or negative.

RR
recovery rate

The CDS-bond basis

Basis
basis=sCDS−A\displaystyle \text{basis} = s_{\text{CDS}} - A

The CDS spread of the issuer less the asset swap spread of its bond, for the same maturity.

Negative basis trade
carry≈A−sCDS−funding spread\displaystyle \text{carry} \approx A - s_{\text{CDS}} - \text{funding spread}

Buy the package and the protection: the investor earns the gap less the cost of funding the bond.

Worked example in US dollars: USD 25,000,000 of a 4.50% bond due Sep 15, 2033 at 97.50, settlement Oct 29, 2026; SOFR curve with 7-year swaps at 3.777% and 10-year at 3.850% (illustrative).
Securitized›Agency MBS TBA

Formulas

Pool cash flows

PSA speed
CPRt=PSA100×6%×min⁡(t, 30)30\displaystyle \text{CPR}_t = \frac{\text{PSA}}{100}\times 6\%\times\frac{\min(t,\,30)}{30}

The benchmark prepayment path: 0.2% CPR a month of loan age for 30 months, then flat; 150 PSA is one and a half times that path.

tt
loan age in months
CPR\text{CPR}
conditional prepayment rate, annualized

Worked example 150 PSA: 1.2% CPR in December (age 4), 9.0% from month 30

Monthly prepayment rate
SMM=1−(1−CPR)1/12\displaystyle \text{SMM} = 1 - (1 - \text{CPR})^{1/12}

The share of the balance, after scheduled principal, that borrowers repay early in one month.

Worked example 1.2% CPR gives an SMM of 0.1006%

Scheduled payment
MPt=Bt−1 g1−(1+g)−ntg=WAC12\displaystyle MP_t = B_{t-1}\,\frac{g}{1 - (1+g)^{-n_t}} \qquad g = \frac{\text{WAC}}{12}

The borrowers’ level monthly payment at the mortgage rate over the remaining term.

Bt−1B_{t-1}
balance at the start of the month
ntn_t
months remaining
WAC\text{WAC}
weighted average mortgage rate, 6.30%
Interest and principal to the holder
It=Bt−1 c12Pt=(MPt−Bt−1 g)+SMMt(Bt−1−(MPt−Bt−1 g))\displaystyle I_t = B_{t-1}\,\frac{c}{12} \qquad P_t = \left(MP_t - B_{t-1}\,g\right) + \text{SMM}_t\left(B_{t-1} - \left(MP_t - B_{t-1}\,g\right)\right)

The holder receives the coupon on the opening balance (servicing and the guarantee fee take the rest of the WAC) and all the principal, scheduled and prepaid.

cc
pass-through coupon, 5.5%

Worked example December on USD 100,000,000: interest 458,333, principal 196,190 (95,732 scheduled, 100,458 prepaid)

Factor
Ft=BtB0current face=original face×Ft\displaystyle F_t = \frac{B_t}{B_0} \qquad \text{current face} = \text{original face}\times F_t

The share of the original face still outstanding, published by the agency for each pool every month; trades settle on current face.

Worked example after December: 0.99804; after twelve months: 0.95996

Payment delay
interest of month m paid on the 25th of month m+1\displaystyle \text{interest of month } m \text{ paid on the 25th of month } m+1

The stated delay of 55 days counts from the first day of the accrual month: the payment comes 24 days after the month ends (the 20th for Ginnie Mae II).

Worked example December’s payment on Jan 25, 2027

Settlement and the roll

Price in 32nds
P=a+b32a-b+=a+b+0.532\displaystyle P = a + \frac{b}{32} \qquad a\text{-}b{+} = a + \frac{b + 0.5}{32}

MBS prices are quoted in points and 32nds of a point; a plus adds a 64th.

Worked example 99-25+ = 99 + 25.5/32 = 99.796875

Settlement amount
S=Fcur(P100+c d−1360)\displaystyle S = F_{\text{cur}}\left(\frac{P}{100} + c\,\frac{d-1}{360}\right)

Current face times the price, plus the interest accrued from the 1st of the month to the settlement day d (30/360).

FcurF_{\text{cur}}
current face
dd
day of the month of settlement

Worked example 100,000,000 × (99.609375% + 5.5% × 13/360) = USD 99,807,986

Good delivery variance
∣∑kFk−Ftrade∣≤0.01%×Ftrade\displaystyle \left|\textstyle\sum_k F_k - F_{\text{trade}}\right| \le 0.01\%\times F_{\text{trade}}

The current face of the pools delivered must match the trade within the variance.

FkF_k
current face of pool k

Worked example USD 100,000,000 ± 10,000

Variation margin
VMt=F Pt−Pt−1100\displaystyle VM_t = F\,\frac{P_t - P_{t-1}}{100}

The daily mark to market of the forward, paid under the MSFTA.

Worked example 1 point on USD 100,000,000 = USD 1,000,000

Implied financing rate of the roll
rroll=[P2+A2+100 c12+p (100−P2)P1+A1−1]360D\displaystyle r_{\text{roll}} = \left[\frac{P_2 + A_2 + 100\,\tfrac{c}{12} + p\,(100 - P_2)}{P_1 + A_1} - 1\right]\frac{360}{D}

The rate at which the roll seller borrows: it receives P1 now and pays P2 a month later, giving up the month’s coupon and the paydown returned at par.

P1, P2P_1,\ P_2
front and back month prices
A1, A2A_1,\ A_2
accrued interest at the two settlements
pp
principal paid down in the month, share of face
DD
days between the two settlements

Worked example (99.6094 + 0.1986 + 0.4583 + 0.00171 × 0.3906) / (99.7969 + 0.1681) − 1, × 360/32 = 3.40%

Breakeven drop
drop∗=P1−(P1+A1)(1+r D360)−A2−100 c12−100 p1−p\displaystyle \text{drop}^{*} = P_1 - \frac{(P_1 + A_1)\left(1 + r\,\tfrac{D}{360}\right) - A_2 - 100\,\tfrac{c}{12} - 100\,p}{1 - p}

The drop at which rolling costs the same as financing the pools in repo at r; a larger drop means the roll is special.

rr
repo rate for agency MBS

Worked example repo 3.95%: 4.4/32 against the 6/32 traded

Valuation

Price from projected cash flows
P+A=∑k=1NCFk(1+y/12) tk\displaystyle P + A = \sum_{k=1}^{N}\frac{CF_k}{(1 + y/12)^{\,t_k}}

The price plus accrued equals the cash flows projected at a speed and discounted at the mortgage yield; the payment delay sits in t_k.

tkt_k
months from settlement to the k-th payment
yy
mortgage yield, monthly compounding

Worked example 99-19+ for December at 150 PSA: y = 5.501%

Bond-equivalent yield
yBE=2[(1+y12)6−1]\displaystyle y_{\text{BE}} = 2\left[\left(1 + \frac{y}{12}\right)^{6} - 1\right]

Converts the monthly mortgage yield for comparison with semiannual bonds.

Worked example 5.501% gives 5.565%

Weighted average life
WAL=∑ktk Pk12∑kPk\displaystyle \text{WAL} = \frac{\sum_k t_k\,P_k}{12\sum_k P_k}

The average time to the return of principal, in years: the maturity MBS are compared on.

PkP_k
principal of payment k

Worked example 9.1 years at 150 PSA

Nominal spread
s=yBE−swap(WAL)\displaystyle s = y_{\text{BE}} - \text{swap}(\text{WAL})

The yield over the swap curve at the WAL: simple, and blind to the option.

Worked example 5.56% − 3.82% = +174 bp

Option-adjusted spread
P+A=1M∑j=1M∑kCFk(j)∏i≤k(1+ri(j)+OAS12)\displaystyle P + A = \frac{1}{M}\sum_{j=1}^{M}\sum_{k}\frac{CF_k^{(j)}}{\prod_{i \le k}\left(1 + \frac{r_i^{(j)} + \text{OAS}}{12}\right)}

Cash flows projected on each simulated rate path and discounted along it; the OAS is the spread that recovers the market price.

MM
number of rate paths
ri(j)r_i^{(j)}
one-month rate on path j
Option cost
option cost=Z-spread−OAS\displaystyle \text{option cost} = Z\text{-spread} - \text{OAS}

The part of the spread that pays for the borrowers’ prepayment option; it grows with rate volatility.

Price against rates

Price change
ΔPP≈−Deff Δy+12 Ceff Δy2\displaystyle \frac{\Delta P}{P} \approx -D_{\text{eff}}\,\Delta y + \tfrac{1}{2}\,C_{\text{eff}}\,\Delta y^2

Duration and convexity together; for a pass-through near the money the convexity is negative, so the price rises less than it falls.

Worked example −100 bp: +3.09 points; +100 bp: -7.46 points

S-curve of prepayments
CPR(I)=turnover(I)+CPRmax⁡1+e−(I−I0)/sI=WAC−m\displaystyle \text{CPR}(I) = \text{turnover}(I) + \frac{\text{CPR}_{\max}}{1 + e^{-(I - I_0)/s}} \qquad I = \text{WAC} - m

Home sales give a base speed; a refinancing wave starts once the incentive is large enough (illustrative parameters).

mm
mortgage rate offered to borrowers
I0, sI_0,\ s
center and width of the wave

Worked example 9.0% today, 31.6% at −100 bp, 5.1% at +100 bp

A bond less a call
PMBS=Pbond−Vcall\displaystyle P_{\text{MBS}} = P_{\text{bond}} - V_{\text{call}}

A pass-through is a bond with fixed cash flows less the call the borrowers hold, which gains value as rates fall.

Worked example at −100 bp: 106.64 − 3.75 = 102.89

Risk

Effective duration
Deff=P−−P+2 P Δy\displaystyle D_{\text{eff}} = \frac{P_{-} - P_{+}}{2\,P\,\Delta y}

Prices from the model with the curve shifted down and up, prepayments re-projected each time.

Δy\Delta y
the shift, 25 bp here

Worked example 6.34 years, against 6.48 for the same cash flows held fixed

Effective convexity
Ceff=P++P−−2PP Δy2\displaystyle C_{\text{eff}} = \frac{P_{+} + P_{-} - 2P}{P\,\Delta y^2}

Negative for a pass-through at the money: the duration shortens in a rally and extends in a sell-off.

Worked example -428

Hedge with a swap
Nswap=Deff P100 FAswap\displaystyle N_{\text{swap}} = \frac{D_{\text{eff}}\,\tfrac{P}{100}\,F}{A_{\text{swap}}}

The notional of a pay-fixed swap with the same DV01 as the position.

AswapA_{\text{swap}}
annuity of the swap, 8.30 for 10 years

Worked example 6.34 × 0.9980 × 100,000,000 / 8.30 = USD 76.3 million of 10-year swap

Illustrative inputs of the worked example: UMBS 30-year 5.5% pools with a 6.30% WAC, 357 months remaining, 3 months seasoned, projected at 150 PSA; the SOFR swap curve of the interest rate swap card; agency MBS repo at 3.95%; a primary mortgage rate of 6.30% for the S-curve.
Securitized›Specified Pool

Formulas

Settlement

Price of a pool
Ppool=PTBA+pay-up\displaystyle P_{\text{pool}} = P_{\text{TBA}} + \text{pay-up}

Pools are bid as the TBA of the same coupon and month plus a pay-up in 32nds.

Worked example 97-12 + 1-08 = 98-20 (98.6250); 97-12 + 0-12 = 97-24 (97.7500)

Current face
Fcur=Forig×f\displaystyle F_{\text{cur}} = F_{\text{orig}}\times f

The face outstanding today, from the latest factor.

ff
pool factor

Worked example 50,000,000 × 0.83 = USD 41,500,000

Settlement amount
S=∑kForig,k fk Pk100+Fcur c d−1360\displaystyle S = \sum_k F_{\text{orig},k}\,f_k\,\frac{P_k}{100} + F_{\text{cur}}\,c\,\frac{d-1}{360}

Each pool at its own price, plus the interest accrued from the 1st of the month (30/360).

cc
coupon, 5.0%
dd
settlement day of the month, the 29th

Worked example 24,557,625 + 16,226,500 + 161,389 = USD 40,945,514

Average pay-up
pay-up‾=∑kFcur,k pay-upk∑kFcur,k\displaystyle \overline{\text{pay-up}} = \frac{\sum_k F_{\text{cur},k}\,\text{pay-up}_k}{\sum_k F_{\text{cur},k}}

The pay-up of the lot, weighted by current face.

Worked example (24.9 × 40 + 16.6 × 12) / 41.5 = 28.8/32

Next factor
ft+1=ft−scheduledt+prepaidtForig\displaystyle f_{t+1} = f_t - \frac{\text{scheduled}_t + \text{prepaid}_t}{F_{\text{orig}}}

October’s principal paid down sets the factor published in November.

Worked example 0.83000 − 250,078 / 50,000,000 = 0.82500

Pay-up and valuation

Pay-up
pay-up=Ppool(OAS)−PTBA(OAS)\displaystyle \text{pay-up} = P_{\text{pool}}(\text{OAS}) - P_{\text{TBA}}(\text{OAS})

At the same OAS, the pay-up is the value of the slower prepayments of the pool against the cheapest pools deliverable into TBA.

Yield given up
Δy=yTBA−ypool\displaystyle \Delta y = y_{\text{TBA}} - y_{\text{pool}}

At their own projected speeds, the pool yields less than the TBA: the price of the protection today.

Worked example 5.337% − 5.149% = 19 bp

Price of each
P=∑kCFk(CPRpool)(1+y/12)tk\displaystyle P = \sum_k \frac{CF_k(\text{CPR}_{\text{pool}})}{(1 + y/12)^{t_k}}

The same discounting as the TBA, with the pool’s own speeds.

Worked example WAL 10.0 years for the pool, 9.8 for the TBA

Spread to swaps
s=yBE−swap(WAL)\displaystyle s = y_{\text{BE}} - \text{swap}(\text{WAL})

Nominal spread at each one’s WAL.

Worked example pool +136 bp, TBA +155 bp

Against rates

Pay-up across scenarios
pay-up(Δ)=Ppool(yp+Δ, CPRp(Δ))−PTBA(yT+Δ, CPRT(Δ))\displaystyle \text{pay-up}(\Delta) = P_{\text{pool}}(y_p + \Delta,\ \text{CPR}_p(\Delta)) - P_{\text{TBA}}(y_T + \Delta,\ \text{CPR}_T(\Delta))

Each price re-projected on its S-curve after a parallel shift, at an unchanged spread.

Δ\Delta
parallel shift

Worked example 83/32 at −100 bp, 40/32 today, 41/32 at +100 bp

Two S-curves
CPRLLB(I)=turnover(I)+20%1+e−(I−1.5%)/0.4%\displaystyle \text{CPR}_{\text{LLB}}(I) = \text{turnover}(I) + \frac{20\%}{1 + e^{-(I - 1.5\%)/0.4\%}}

The low loan balance wave needs a larger incentive and stays smaller.

Worked example at −100 bp: 8.5% against 15.8%

Risk and hedge

Effective duration
Deff=P−−P+2 P Δy\displaystyle D_{\text{eff}} = \frac{P_{-} - P_{+}}{2\,P\,\Delta y}

With speeds re-projected, 25 bp each way.

Worked example pool 7.18, TBA 7.29

Hedge ratio
h=Dpool PpoolDTBA PTBA\displaystyle h = \frac{D_{\text{pool}}\,P_{\text{pool}}}{D_{\text{TBA}}\,P_{\text{TBA}}}

Face of TBA to sell per unit of pool face for the same DV01.

Worked example 1.00 today: USD 41.3 million of TBA against USD 41.5 million of pools; 3.67 after a 100 bp rally

Position P&L
ΔΠ=Fcur Δ pay-up100+carry\displaystyle \Delta\Pi = F_{\text{cur}}\,\frac{\Delta\,\text{pay-up}}{100} + \text{carry}

Hedged, the desk earns the change in the pay-up and the carry of the pools over the TBA.

Carry against the roll
carry=Fcur(c12−p P−100100)−Fcur P100 rrepo D360\displaystyle \text{carry} = F_{\text{cur}}\left(\tfrac{c}{12} - p\,\tfrac{P - 100}{100}\right) - F_{\text{cur}}\,\tfrac{P}{100}\,r_{\text{repo}}\,\tfrac{D}{360}

A month of coupon and the paydown at par, less the repo cost of the pools; the short TBA earns or pays the drop.

Illustrative inputs: UMBS 30-year 5.0% pools with a 5.80% WAC, 331 months remaining, 26 months seasoned; a primary mortgage rate of 6.30%; two S-curves with the same turnover and different refinancing waves; the SOFR swap curve of the interest rate swap card.
Securitized›Agency CMO

Formulas

Structure

PAC schedule
St=min⁡(Pt 100 PSA, Pt 250 PSA)\displaystyle S_t = \min\left(P_t^{\,100\,\text{PSA}},\ P_t^{\,250\,\text{PSA}}\right)

The principal the PACs can count on each month: the lower of the collateral principal at the two band speeds.

Pt xP_t^{\,x}
collateral principal in month t at speed x

Worked example the schedule adds up to USD 355.1 million, the room for USD 350 million of PACs

PAC principal
PtPAC=min⁡(Pt, St+shortfallt−1)\displaystyle P_t^{\text{PAC}} = \min\left(P_t,\ S_t + \text{shortfall}_{t-1}\right)

The PACs receive their schedule (and any earlier shortfall) first, PAC 1 before PAC 2.

Support principal
Ptsup=Pt−PtPAC\displaystyle P_t^{\text{sup}} = P_t - P_t^{\text{PAC}}

The support receives everything above the schedule; once it is retired, the PACs take all the principal.

Sequential pay
Pt(j)=min⁡(Bt−1(j), Pt−∑i<jPt(i))\displaystyle P_t^{(j)} = \min\left(B_{t-1}^{(j)},\ P_t - \textstyle\sum_{i<j} P_t^{(i)}\right)

Each class takes the principal left after the classes ahead of it, until it is retired.

Conservation
∑j(It(j)+Pt(j))=It+Pt\displaystyle \sum_j \left(I_t^{(j)} + P_t^{(j)}\right) = I_t + P_t

The classes always add up to the collateral: structuring moves the risk between investors, it does not remove it.

Pricing and the arbitrage

Class price
P+A=∑kCFk(j)(1+y/12) tk\displaystyle P + A = \sum_k \frac{CF_k^{(j)}}{(1 + y/12)^{\,t_k}}

Each class priced off its projected cash flows at the pricing speed.

Worked example PAC 2 at 100-08: 5.421% monthly, 5.483% bond-equivalent

WAL
WAL=∑ktk Pk12∑kPk\displaystyle \text{WAL} = \frac{\sum_k t_k\,P_k}{12\sum_k P_k}

Average time to the return of principal.

Worked example PAC 1 3.01, PAC 2 10.67, support 12.85 years

Spread to swaps
s=yBE−swap(WAL)\displaystyle s = y_{\text{BE}} - \text{swap}(\text{WAL})

The quote each investor sees, against the swap curve at the class’s WAL.

Worked example PAC 1 +141, PAC 2 +163, support +182, collateral +174 bp

Deal arbitrage
Π=∑jFj Pj100−Fcoll Pcoll100−fees\displaystyle \Pi = \sum_j F_j\,\frac{P_j}{100} - F_{\text{coll}}\,\frac{P_{\text{coll}}}{100} - \text{fees}

The desk issues when the classes sell for more than the pools and the costs.

Worked example 499,750,000 − 498,125,000 − 625,000 = USD 1,000,000

Settlement amount
S=F(P100+c d−1360)\displaystyle S = F\left(\frac{P}{100} + c\,\frac{d-1}{360}\right)

Price plus accrued from the 1st of the month.

Worked example 200,000,000 × (100.25% + 5.5% × 29/360) = USD 201,386,111

Against prepayment speed

WAL against speed
WAL(j)(PSA)\displaystyle \text{WAL}^{(j)}(\text{PSA})

Run the structure at each constant speed; the PACs are flat inside the band.

Worked example PAC 2: 11.9 years at 75 PSA, 10.7 from 100 to 250, 7.6 at 400

Effective band
bandt=[PSAmin⁡,t, PSAmax⁡,t]\displaystyle \text{band}_t = \left[\text{PSA}_{\min,t},\ \text{PSA}_{\max,t}\right]

The range of speeds that still keeps the PACs on schedule given the support left; it narrows as the support pays down.

Risk

Effective duration
Deff=P−−P+2 P Δy\displaystyle D_{\text{eff}} = \frac{P_{-} - P_{+}}{2\,P\,\Delta y}

Each class repriced with the collateral re-projected after a shift.

Support convexity
Ceffsup≪Ceffcoll<0\displaystyle C_{\text{eff}}^{\text{sup}} \ll C_{\text{eff}}^{\text{coll}} < 0

The support concentrates the negative convexity the PACs give up.

Interest of the classes
It(j)=Bt−1(j) cj12\displaystyle I_t^{(j)} = B_{t-1}^{(j)}\,\frac{c_j}{12}

Each class earns its coupon on its own balance; here all classes carry the collateral coupon.

Worked example PAC 2 in its first month: 200,000,000 × 5.5% / 12 = USD 916,667

Illustrative inputs: USD 500,000,000 of UMBS 30-year 5.5% pools (6.30% WAC, 357 months remaining, 3 months seasoned); classes with the 5.5% coupon of the collateral; settlement on Nov 30, 2026 with 29 days of accrued interest; pricing speed 150 PSA; the SOFR swap curve of the interest rate swap card.
Securitized›Asset-Backed Security

Formulas

Collections and the waterfall

ABS prepayment speed
SMMt=ABS1−ABS (t−1)\displaystyle \text{SMM}_t = \frac{\text{ABS}}{1 - \text{ABS}\,(t-1)}

Auto deals measure prepayments as a share of the original number of loans each month; as a share of the remaining loans it rises over time.

ABS\text{ABS}
absolute prepayment speed, 1.3%

Worked example month 1: 1.30%; month 24: 1.85%

Collections
Ct=Bt−1 APR12+schedt+prepaidt+recoveriest\displaystyle C_t = B_{t-1}\,\frac{\text{APR}}{12} + \text{sched}_t + \text{prepaid}_t + \text{recoveries}_t

What the servicer collects in a month.

Priority of payments
Ct→servicing→IA→IB→IC→reserve→Pnotes→residual\displaystyle C_t \rightarrow \text{servicing} \rightarrow I^{A} \rightarrow I^{B} \rightarrow I^{C} \rightarrow \text{reserve} \rightarrow P^{\text{notes}} \rightarrow \text{residual}

The order in which the trustee pays; principal goes sequentially A1, A2, A3, A4, B, C.

Principal to the notes
Ptnotes=max⁡(Nt−1−(Bt−OCt∗), 0)OCt∗=max⁡(5.5% Bt, 1.25% B0)\displaystyle P_t^{\text{notes}} = \max\left(N_{t-1} - \left(B_t - OC_t^{*}\right),\ 0\right) \qquad OC_t^{*} = \max(5.5\%\,B_t,\ 1.25\%\,B_0)

Enough principal to bring the notes down to the pool less the target overcollateralization; until the target is met, excess spread turbos the notes.

NN
notes outstanding
BB
pool balance
Excess spread
XS=APR−servicing−∑jNj cjB\displaystyle XS = \text{APR} - \text{servicing} - \frac{\sum_j N_j\,c_j}{B}

The yearly income of the pool left after the fees and the coupons: the first line of defense against losses.

Worked example about 1.60% a year at the start

Cumulative net loss
CNLt=∑s≤t(defaultss−recoveriess)B0\displaystyle \text{CNL}_t = \frac{\sum_{s \le t}\left(\text{defaults}_s - \text{recoveries}_s\right)}{B_0}

The deal’s key performance measure, tested against the triggers.

Worked example 0.80% in the base case

Pricing

Price from cash flows
P=∑kCFk(1+y/12) tk\displaystyle P = \sum_k \frac{CF_k}{(1 + y/12)^{\,t_k}}

The class priced off its projected flows at the pricing speed and loss.

Worked example A2 at 99.98 with a 4.55% coupon: yield 4.614%

Spread to the I-curve
s=y−swap(WAL)\displaystyle s = y - \text{swap}(\text{WAL})

Fixed-rate ABS are quoted against the swap curve at their WAL.

Worked example 4.614% − 3.959% = +65 bp

First coupon
I1=F c days30/360360\displaystyle I_1 = F\,c\,\frac{\text{days}_{30/360}}{360}

The first period runs from closing to the first payment date.

Worked example 30,000,000 × 4.55% × 39/360 = USD 147,875

Secondary settlement
S=Forig f(P100+c d360)\displaystyle S = F_{\text{orig}}\,f\left(\frac{P}{100} + c\,\frac{d}{360}\right)

Original face times factor at the price, plus accrued since the last payment date.

Worked example 10,000,000 × 0.9921 × (100.032% + 0.0000%) = USD 9,923,884

Credit enhancement and tranche loss

Hard credit enhancement
CEA=B−NA+RB\displaystyle CE_A = \frac{B - N_A + R}{B}

Everything below class A as a share of the pool: the junior classes, the overcollateralization and the reserve.

NAN_A
class A notes outstanding
RR
reserve account

Worked example (1,250,000,000 − 1,140,000,000 + 3,125,000) / 1,250,000,000 = 9.05%

Tranche loss (static)
Lj=min⁡(max⁡(L−aj, 0), dj−aj)dj−aj\displaystyle L_j = \frac{\min\left(\max(L - a_j,\,0),\ d_j - a_j\right)}{d_j - a_j}

A class loses once pool losses exceed its attachment point and is wiped out at its detachment point; excess spread and timing move both in practice.

aj, dja_j,\ d_j
attachment and detachment, % of the pool
Loss multiple
m=breakeven CNLexpected CNL\displaystyle m = \frac{\text{breakeven CNL}}{\text{expected CNL}}

Rating agencies compare the loss a class can absorb with the expected loss.

Worked example A4: about 14 times the 0.80% base case

Risk

Spread duration
ΔPP≈−Ds Δs\displaystyle \frac{\Delta P}{P} \approx -D_s\,\Delta s

Price sensitivity to the class spread, close to the WAL for a short amortizing class.

Weighted average life
WAL=∑ktk Pk12∑kPk\displaystyle \text{WAL} = \frac{\sum_k t_k\,P_k}{12\sum_k P_k}

From closing, in years.

Worked example A2: 1.01 years

Trigger test
CNLt≤triggert\displaystyle \text{CNL}_t \le \text{trigger}_t

A breach redirects excess spread to the senior classes until the test is cured.

Illustrative deal: a USD 1,250,000,000 pool of 60,000 prime auto loans at a 6.90% APR with 54 months remaining, 1.3% ABS, defaults at a constant monthly rate giving a 0.80% cumulative net loss with 50% recoveries, a 1.00% servicing fee, a 0.25% reserve, an overcollateralization target of 5.5% of the pool; class spreads to the I-curve illustrative; the SOFR swap curve of the interest rate swap card.
Securitized›Commercial Mortgage-Backed Security

Formulas

Loans

Debt service coverage
DSCR=NOIdebt service\displaystyle \text{DSCR} = \frac{\text{NOI}}{\text{debt service}}

Net operating income of the property over the loan’s annual payments; conduit loans start around 1.4 to 2.0.

NOI\text{NOI}
net operating income
Loan-to-value and debt yield
LTV=LVDY=NOIL\displaystyle \text{LTV} = \frac{L}{V} \qquad \text{DY} = \frac{\text{NOI}}{L}

Leverage against the appraised value, and the income per dollar of loan, which does not depend on the appraisal.

Amortization and balloon
B120=B0 (1+g)360−(1+g)120(1+g)360−1\displaystyle B_{120} = B_0\,\frac{(1+g)^{360} - (1+g)^{120}}{(1+g)^{360} - 1}

The balance left after ten years of a 30-year schedule: the balloon.

gg
monthly loan rate

Worked example at 4.75%: 80.7% of the original loan

Class interest
It(j)=Bt−1(j) cj12\displaystyle I_t^{(j)} = B_{t-1}^{(j)}\,\frac{c_j}{12}

Each class earns its coupon on its balance; a WAC-capped class earns the lower of its coupon and the loans’ net rate.

Worked example 15,000,000 × 4.402% / 12 = USD 55,025 a month

Price and the trade

Price from yield
P+A=∑kCFk(1+y/12) tk\displaystyle P + A = \sum_k \frac{CF_k}{(1 + y/12)^{\,t_k}}

Monthly flows discounted at the yield; the AAA class is close to a bullet at the balloons.

Worked example swaps + 92 at a 5.9-year WAL: 3.731% + 0.92% = 4.651%, price 98.75

Settlement amount
S=Fcur(P100+c d−1360)\displaystyle S = F_{\text{cur}}\left(\frac{P}{100} + c\,\frac{d-1}{360}\right)

Current face at the price, plus accrued from the 1st (30/360).

Worked example 15,000,000 × (98.75% + 4.402% × 28/360) = USD 14,863,857

Dealer margin
Π≈F Presale−Pbid100≈F Ds Δs\displaystyle \Pi \approx F\,\frac{P_{\text{resale}} - P_{\text{bid}}}{100} \approx F\,D_s\,\Delta s

The spread gained between bid and resale, times the spread duration.

Worked example 15,000,000 × 5.08 × 0.04% ≈ USD 29,800

Swap hedge
Nswap=DV01bondA7y×10−4\displaystyle N_{\text{swap}} = \frac{\text{DV01}_{\text{bond}}}{A_{7y}\times 10^{-4}}

Pay-fixed notional with the bond’s DV01.

Worked example 7,554 / (6.135 × 0.0001) = USD 12,312,629

Prepayment protection

Defeasance cost
D=∑kCFk(1+yT/12)k−B\displaystyle D = \sum_{k} \frac{CF_k}{(1 + y_T/12)^{k}} - B

The price of the Treasuries that replicate every remaining payment and the balloon, less the balance repaid.

yTy_T
Treasury yield

Worked example 5.04% of USD 23,340,875 at 3.75%

Yield maintenance
YM=max⁡(1% B, ∑kCFk(1+yT/12)k−B)\displaystyle YM = \max\left(1\%\,B,\ \sum_k \frac{CF_k}{(1 + y_T/12)^{k}} - B\right)

A prepayment premium that leaves the lender indifferent, with a floor of 1% of the balance.

Risk

Spread duration
Ds=−1P∂P∂s\displaystyle D_s = -\frac{1}{P}\frac{\partial P}{\partial s}

Sensitivity to the class spread; close to the modified duration for a fixed-rate class.

Worked example 5.08 years

DV01
DV01=Ds P+A100 F×10−4\displaystyle \text{DV01} = D_s\,\frac{P + A}{100}\,F\times 10^{-4}

Value of one basis point on the position.

Worked example USD 7,554 per bp

Credit support
CSj=∑i below jB(i)∑iB(i)\displaystyle CS_j = \frac{\sum_{i \text{ below } j} B^{(i)}}{\sum_i B^{(i)}}

The share of the deal below a class that absorbs losses first.

Worked example 30% for the AAA classes

Appraisal reduction
AR=max⁡(0, B+arrears−90% Vnew)\displaystyle AR = \max\left(0,\ B + \text{arrears} - 90\%\,V_{\text{new}}\right)

Cuts the servicer’s advances on a defaulted loan in proportion, so the most junior classes lose interest first.

VnewV_{\text{new}}
new appraised value

Illustrative inputs: the class pays a fixed 4.402% monthly (30/360) and receives its principal 30%, 40% and 30% in August, September and October 2032; distribution dates on the 15th; the SOFR swap curve of the interest rate swap card; a 6-year Treasury yield of 3.75% for the defeasance.
Equities›Share

Formulas

Settlement and costs

Settlement amount of a purchase
A=q Pˉ+q c+τ q Pˉ\displaystyle A = q\,\bar P + q\,c + \tau\,q\,\bar P

What the buyer pays on the settlement date: the consideration, the commission and any transaction tax on the consideration.

qq
number of shares
Pˉ\bar P
average price
cc
commission per share
τ\tau
tax rate: 0.5% UK, 0.4% France, none in the US and Canada

Worked example 500,000 × 80.12 + 500,000 × 0.02 = USD 40,070,000; in London GBP 40,270,300 with the tax

Transaction tax
T=τ q Pˉ\displaystyle T = \tau\,q\,\bar P

Stamp duty reserve tax in the UK and the financial transaction taxes in the EU are a percentage of the consideration, paid by the buyer.

Worked example UK 0.5% × 40,060,000 = GBP 200,300; France 0.4% = EUR 160,240

VWAP
VWAP=∑kpk vk∑kvk\displaystyle \text{VWAP} = \frac{\sum_k p_k\,v_k}{\sum_k v_k}

The volume-weighted average price of the day’s trades: the benchmark of a VWAP algorithm.

pk, vkp_k,\ v_k
price and volume of trade k in the market

Worked example 80.12 against 80.10: 2 cents, 2.5 bp, USD 10,000 on 500,000 shares

Implementation shortfall
IS=Pˉ−P0P0+cP0(purchase)\displaystyle IS = \frac{\bar P - P_0}{P_0} + \frac{c}{P_0} \quad \text{(purchase)}

The cost of the execution against the price when the order arrived, commission included; a sale takes the opposite sign.

P0P_0
arrival price

Worked example (80.12 − 80.00) / 80.00 = 15.0 bp, plus 2.5 bp of commission: 17.5 bp, USD 70,000

Risk bid discount
d=Plast−PbidPlast\displaystyle d = \frac{P_{\text{last}} - P_{\text{bid}}}{P_{\text{last}}}

The price of immediacy on a principal block: the dealer keeps the discount less what it costs to work out of the position.

Worked example (80.00 − 79.60) / 80.00 = 50 bp, USD 200,000; the bank pays USD 39,800,000

Dividends and corporate actions

Ex-date
Dex=Drec−(s−1) business days\displaystyle D_{\text{ex}} = D_{\text{rec}} - (s - 1)\ \text{business days}

A buyer is entitled to the dividend only if the purchase settles by the record date, so the ex-date follows the settlement cycle.

ss
settlement cycle in business days
DrecD_{\text{rec}}
record date

Worked example T+1: ex-date = record date, November 20; T+2: one business day earlier

Price on the ex-date
Pex≈Pcum−D\displaystyle P_{\text{ex}} \approx P_{\text{cum}} - D

The share opens lower by about the dividend, all else equal.

DD
dividend per share

Worked example about USD 0.50 lower at the open of November 20

Dividend received
Div=q D (1−w)\displaystyle \text{Div} = q\,D\,(1 - w)

The cash the holder of record receives, net of the withholding tax at source.

ww
withholding rate

Worked example 500,000 × 0.50 = USD 250,000 for a US fund; USD 212,500 for a non-US fund at the 15% treaty rate

Split
qnew=k qPadj=Pk\displaystyle q_{\text{new}} = k\,q \qquad P_{\text{adj}} = \frac{P}{k}

A k-for-1 split multiplies the shares and divides the price; past prices are adjusted by the same factor.

kk
split ratio
Theoretical ex-rights price
TERP=N Pcum+Nnew PsubN+Nnew\displaystyle TERP = \frac{N\,P_{\text{cum}} + N_{\text{new}}\,P_{\text{sub}}}{N + N_{\text{new}}}

The share price expected after a rights issue at a discount; the right is worth the difference to the subscription price.

N, NnewN,\ N_{\text{new}}
existing and new shares
PsubP_{\text{sub}}
subscription price

Valuation and return

Mark to market
V=q Pclose\displaystyle V = q\,P_{\text{close}}

A listed share is marked at the official closing price of its primary market (Level 1).

Worked example 500,000 × 84.00 = USD 42,000,000 a year later

Total return
TR=P1−P0+∑DP0\displaystyle TR = \frac{P_1 - P_0 + \sum D}{P_0}

Price change plus the dividends received, over the price paid.

Worked example (84.00 − 80.12 + 2.00) / 80.12 = 7.34%; 7.29% on the cash flows, after the commissions

Dividend yield
y=DannualP\displaystyle y = \frac{D_{\text{annual}}}{P}

The year’s dividends over the price.

Worked example 2.00 / 80.12 = 2.50%

Dividend discount model
P0=D1k−g\displaystyle P_0 = \frac{D_1}{k - g}

A share is worth its future dividends discounted at the required return; with constant growth the sum reduces to this ratio.

D1D_1
next year’s dividend
kk
required return
gg
growth rate of the dividend

Worked example 2.00 / (8.5% − 6.0%) = USD 80.00

Risk and liquidity

Beta
β=cov⁡(Ri, Rm)var⁡(Rm)\displaystyle \beta = \frac{\operatorname{cov}(R_i,\,R_m)}{\operatorname{var}(R_m)}

The sensitivity of the share’s return to the market’s: the number used to hedge it with index futures.

Ri, RmR_i,\ R_m
returns of the share and of the index
Value at risk, one day
VaR99%=2.326 σ252 V\displaystyle \text{VaR}_{99\%} = 2.326\,\frac{\sigma}{\sqrt{252}}\,V

The loss exceeded on one day in a hundred under a normal distribution of returns.

σ\sigma
annual volatility

Worked example 2.326 × 1.89% × USD 40,060,000 = USD 1,760,928 at 30% volatility

Days to liquidate
days=qu ADV\displaystyle \text{days} = \frac{q}{u\,\text{ADV}}

How long a position takes to sell without taking more than a share u of the daily volume.

uu
participation rate
ADV\text{ADV}
average daily volume, shares

Worked example 500,000 / (20% × 3,333,333) = 0.75 day

Market impact, square-root rule
ΔPP≈Y σ252qADV\displaystyle \frac{\Delta P}{P} \approx Y\,\frac{\sigma}{\sqrt{252}}\sqrt{\frac{q}{\text{ADV}}}

An empirical rule for the price move caused by trading a block at once; Y is close to 1 and is calibrated by each desk.

YY
impact coefficient, 0.8 here

Worked example 0.8 × 1.89% × √0.15 = 59 bp, the impact the desk expects for an immediate sale of the block

Worked example of the US route A unless stated: 500,000 shares at an average of USD 80.12; the dividend, the volatility (30% a year) and the sale at USD 84.00 a year later are illustrative.
Equities›Listed Option

Formulas

Pricing (Black-Scholes)

Forward
F=S e(r−q) T\displaystyle F = S\,e^{(r - q)\,T}

The index level the options are priced off: spot carried at the financing rate less the dividend yield.

SS
index level
rr
USD rate
qq
dividend yield
TT
time to expiry, years

Worked example 5,000 × e^((3.90% − 1.40%) × 79/365) = 5,027.1

Call and put
C=e−rT[F N(d1)−K N(d2)]P=e−rT[K N(−d2)−F N(−d1)]\displaystyle C = e^{-rT}\left[F\,N(d_1) - K\,N(d_2)\right] \qquad P = e^{-rT}\left[K\,N(-d_2) - F\,N(-d_1)\right]

The Black-Scholes value of a European option on an index paying a continuous dividend yield.

NN
standard normal distribution
KK
strike

Worked example P(4,750) = 95.00 points, × USD 100 × 2,000 = USD 19,000,000

d₁ and d₂
d1,2=ln⁡(F/K)±12 σ2TσT\displaystyle d_{1,2} = \frac{\ln(F/K) \pm \tfrac{1}{2}\,\sigma^2 T}{\sigma\sqrt{T}}

The distance between the forward and the strike in standard deviations of the index at expiry.

σ\sigma
implied volatility of the strike

Worked example σ = 22.6%: d₁ = 0.591, d₂ = 0.486

Put-call parity
C−P=e−rT (F−K)\displaystyle C - P = e^{-rT}\,(F - K)

A bought call and a sold put at the same strike make a forward: the link that keeps call and put prices consistent.

Worked example 4,750 call 369.80 − put 95.00 = e^(−rT) × (5,027.13 − 4,750)

Implied volatility
σimp: BS(S,K,T,r,q,σimp)=market price\displaystyle \sigma_{\text{imp}}:\ \text{BS}(S, K, T, r, q, \sigma_{\text{imp}}) = \text{market price}

The volatility that makes the model price equal to the traded price; the market quotes and compares options in this unit.

Worked example 95.00 for the 4,750 put gives 22.64%

Skew (illustrative)
σ(K)=σATM+s(1−KS)\displaystyle \sigma(K) = \sigma_{\text{ATM}} + s\left(1 - \frac{K}{S}\right)

Index puts trade at higher volatilities than calls: the price of crash protection.

ss
skew slope

Worked example 18.5% + 0.829 × (1 − K/5,000): 14.4% at 5,250

Greeks

Delta
ΔC=e−qT N(d1)ΔP=−e−qT N(−d1)\displaystyle \Delta_C = e^{-qT}\,N(d_1) \qquad \Delta_P = -e^{-qT}\,N(-d_1)

The index exposure of one option per point of the index.

Worked example put -0.276: the 2,000 puts offset about USD 276 million of the portfolio at the start

Gamma
Γ=e−qT φ(d1)S σT\displaystyle \Gamma = \frac{e^{-qT}\,\varphi(d_1)}{S\,\sigma\sqrt{T}}

How fast the delta changes with the index: largest at the money and close to expiry.

φ\varphi
standard normal density

Worked example a 1% fall adds about USD 31.7 million of protection

Vega
ν=S e−qT φ(d1)T\displaystyle \nu = S\,e^{-qT}\,\varphi(d_1)\sqrt{T}

The change in value for a change in volatility, quoted per volatility point.

Worked example about USD 1,554,000 per vol point on the 2,000 puts

Theta
ΘP=−Se−qTφ(d1) σ2T+rKe−rTN(−d2)−qSe−qTN(−d1)\displaystyle \Theta_P = -\frac{S e^{-qT}\varphi(d_1)\,\sigma}{2\sqrt{T}} + rK e^{-rT} N(-d_2) - qS e^{-qT} N(-d_1)

The value lost each day with the market unchanged: the running cost of holding protection.

Worked example about USD 202,000 a day on the position at the start

Hedged P&L
dΠ≈12 Γ S2(σrealized2−σimplied2)dt\displaystyle d\Pi \approx \tfrac{1}{2}\,\Gamma\,S^2\left(\sigma_{\text{realized}}^2 - \sigma_{\text{implied}}^2\right)dt

A delta-hedged bought option gains when the volatility realized exceeds the implied volatility paid; the market maker who sold it has the opposite position.

Payoffs at expiry

Put
max⁡(K−ST, 0)×m×n\displaystyle \max(K - S_T,\,0) \times m \times n

What the puts pay at expiry, in cash for an index option.

STS_T
settlement level (the SOQ)
mm
multiplier, USD 100
nn
contracts

Worked example (4,750 − 4,600) × 100 × 2,000 = USD 30,000,000

Protective put
(ST−S0)+max⁡(K−ST, 0)−P\displaystyle (S_T - S_0) + \max(K - S_T,\,0) - P

The portfolio plus the put: the loss stops at the strike, the premium is the cost.

Worked example floor −(250 + 95) × USD 200,000 = −USD 69,000,000; breakeven 5,095

Covered call
(ST−S0)−max⁡(ST−K, 0)+C\displaystyle (S_T - S_0) - \max(S_T - K,\,0) + C

The portfolio with a sold call: income now, the upside sold above the strike.

Worked example 5,250 call at 52.90: USD 10,580,000 received

Bull call spread
max⁡(ST−K1, 0)−max⁡(ST−K2, 0)−(C1−C2)\displaystyle \max(S_T - K_1,\,0) - \max(S_T - K_2,\,0) - (C_1 - C_2)

A bought call financed in part by a sold call at a higher strike.

Worked example 184.40 − 52.90 = 131.50; at most 118.50 points of gain

Straddle
max⁡(ST−K, 0)+max⁡(K−ST, 0)−(C+P)\displaystyle \max(S_T - K,\,0) + \max(K - S_T,\,0) - (C + P)

A call and a put at the same strike: a position on the size of the move.

Worked example 184.40 + 157.50 = 341.90; breakevens 4,658.10 and 5,341.90

Butterfly
max⁡(ST−K1,0)−2max⁡(ST−K2,0)+max⁡(ST−K3,0)\displaystyle \max(S_T - K_1,0) - 2\max(S_T - K_2,0) + \max(S_T - K_3,0)

Pays the most if the index ends at the middle strike; a low-cost view that the index will stay in a range.

K2K_2
middle strike, halfway between K₁ and K₃

Worked example 369.80 − 2 × 184.40 + 52.90 = 53.90; up to 250 points at 5,000

Settlement and margin

Premium paid
P×m×n+commission\displaystyle P \times m \times n + \text{commission}

The whole premium is paid on T+1; a bought option has no further margin.

Worked example 95.00 × 100 × 2,000 = USD 19,000,000, plus USD 2,000 of commission

Exercise by exception
exercised if K−ST≥0.01 (put)ST−K≥0.01 (call)\displaystyle \text{exercised if } K - S_T \ge 0.01 \text{ (put)} \qquad S_T - K \ge 0.01 \text{ (call)}

OCC exercises every option in the money by one cent or more at expiry, unless the holder gives contrary instructions.

Worked example 4,750 − 4,600 = 150 points: exercised

Margin on a sold index option (customer)
M=P m+max⁡(15% S m−OTM m,  10% K m)\displaystyle M = P\,m + \max\left(15\%\,S\,m - \text{OTM}\,m,\ \ 10\%\,K\,m\right)

The strategy-based minimum for a sold broad-based index put: the premium plus a share of the index value less the amount out of the money.

OTM\text{OTM}
amount out of the money

Worked example a sold 4,750 put: 95 + max(750 − 250, 475) = 595 points, USD 59,500 a contract

Market data of the worked example, illustrative: S&P 500 at 5,000.00, USD rate 3.90%, dividend yield 1.40%, 79 days to the January 15, 2027 expiry; volatility 22.6% at 4,750, 18.5% at 5,000 and 14.4% at 5,250 (a linear skew solved from the 95.00 put).
Equities›Index Future

Formulas

Fair value and basis

Dividends to expiry
D=S q t365\displaystyle D = S\,q\,\frac{t}{365}

The dividends the index stocks are expected to pay before expiry, in index points.

SS
index level
qq
dividend yield
tt
days to expiry

Worked example 4,982.00 × 1.30% × 51/365 = 9.05 points

Fair value
F∗=S(1+r t360)−D\displaystyle F^{*} = S\left(1 + r\,\frac{t}{360}\right) - D

Buying the stocks and selling the future locks the financing cost and earns the dividends, so the future is worth the index carried at the rate, less the dividends.

rr
overnight rate to expiry (SOFR; SONIA, €STR or CORRA elsewhere, Act/365 for GBP and CAD)

Worked example 4,982.00 × (1 + 3.90% × 51/360) − 9.05 = 5,000.48

Carry in continuous time
F∗=S e(r−q) T\displaystyle F^{*} = S\,e^{(r - q)\,T}

The same relation with continuous compounding and a dividend yield.

TT
time to expiry, years
Basis
B=F−SB∗=F∗−S\displaystyle B = F - S \qquad B^{*} = F^{*} - S

The gap between the future and the index; it falls to zero at expiry. Negative when dividends exceed financing, as often for the S&P/TSX 60.

Worked example traded 5,000.25 − 4,982.00 = 18.25; fair 18.48; the fill is 0.23 point cheap, USD 11,303 on 1,000 contracts

Implied financing rate
rimpl=(F+DS−1)360t\displaystyle r_{\text{impl}} = \left(\frac{F + D}{S} - 1\right)\frac{360}{t}

The rate a buyer of the stocks earns by selling the future: compared with the cost of funding, it says whether to buy the basket or the future.

Worked example ((5,000.25 + 9.05) / 4,982.00 − 1) × 360/51 = 3.87%, against SOFR at 3.90%

Notional and margin

Notional
Notional=n m F\displaystyle \text{Notional} = n\,m\,F

The exposure the contracts carry.

nn
contracts
mm
multiplier, currency per point

Worked example 1,000 × 50 × 5,000.25 = USD 250,012,500

Variation margin, short position
VMt=n m (Ft−1−Ft)\displaystyle VM_t = n\,m\,(F_{t-1} - F_t)

Each day’s change in the settlement price, paid in cash: the short receives when the index falls.

FtF_t
settlement price of day t

Worked example day 1: 1,000 × 50 × (5,000.25 − 4,991.75) = USD 425,000

Cumulative variation margin
∑s≤tVMs=n m (F0−Ft)\displaystyle \sum_{s \le t} VM_s = n\,m\,(F_0 - F_t)

The margin paid and received adds up to the P&L since the trade: nothing is left to settle at the end.

Worked example after five days: 50,000 × (5,000.25 − 4,986.25) = USD 700,000

Initial margin and leverage
IM=n ML=n m FIM\displaystyle IM = n\,M \qquad L = \frac{n\,m\,F}{IM}

The deposit the CCP requires per contract, and the exposure it carries per unit of margin.

MM
margin per contract

Worked example 1,000 × 12,000 = USD 12,000,000, 4.8% of the notional, leverage 20.8 times

Roll and expiry

Fair roll
F2−F1≈S(r t12360−q t12365)\displaystyle F_2 - F_1 \approx S\left(r\,\frac{t_{12}}{360} - q\,\frac{t_{12}}{365}\right)

The calendar spread between two expiries is the carry over the days between them.

t12t_{12}
days between the two expiries

Worked example 4,982.00 × (3.90% × 91/360 − 1.30% × 91/365) = 32.97 points, December to March

Final settlement
VMT=n m (FT−1−SSOQ)\displaystyle VM_T = n\,m\,(F_{T-1} - S_{\text{SOQ}})

On expiry the last margin payment is against the special settlement level of the index.

SSOQS_{\text{SOQ}}
special opening quotation (EDSP, auction average or opening level elsewhere)
Convergence
FT=ST\displaystyle F_T = S_T

At expiry the future equals the index level it settles to: the basis is zero.

Hedging with futures

Hedge ratio
n=β Vm F\displaystyle n = \beta\,\frac{V}{m\,F}

The number of contracts that offsets the market exposure of a portfolio of value V and beta β.

VV
portfolio value
β\beta
beta of the portfolio against the index

Worked example 1.0 × 250,000,000 / (50 × 5,000.25) = 999.95: 1,000 contracts

Hedged P&L at expiry
Π=V(STS0−1)+Div+n m (F0−ST)≈V r t360\displaystyle \Pi = V\left(\frac{S_T}{S_0} - 1\right) + \text{Div} + n\,m\,(F_0 - S_T) \approx V\,r\,\frac{t}{360}

Stocks plus a short future earn the financing rate: the index move cancels and the basis converges.

Worked example USD 1.37 million over 51 days, against USD 1.38 million at SOFR

Equitization
n=Cm F\displaystyle n = \frac{C}{m\,F}

A fund with cash C buys futures to be invested at once, then sells them as it buys the stocks.

CC
cash to invest

Risk

Residual risk of the hedge
σΠ=V TE t/365\displaystyle \sigma_{\Pi} = V\,TE\,\sqrt{t/365}

What is left after the hedge: the tracking error of the portfolio against the index.

TETE
annual tracking error

Worked example USD 250,000,000 × 2% × √(51/365) = USD 1,868,997 one standard deviation

Margin call in a move
VM=n m F Δ\displaystyle VM = n\,m\,F\,\Delta

The cash due the next morning for an index move Δ against the position.

Δ\Delta
index move, %

Worked example a 5% rise calls USD 12,500,625 from the short

US worked example: index 4,982.00, SOFR 3.90%, dividend yield 1.30%, 51 days from Oct 28, 2026 to the December expiry, all illustrative. The other jurisdictions use their own contract size, rate and day count, as in the card.
Equities›OTC Option

Formulas

Pricing

Basket
BT=∑i=1nwi Si,TSi,0\displaystyle B_T = \sum_{i=1}^{n} w_i\,\frac{S_{i,T}}{S_{i,0}}

The basket is the weighted average of each stock’s performance from the trade date.

wiw_i
weight of stock i
Si,TS_{i,T}
closing price on the valuation date

Worked example n = 10, w_i = 10%, strike 100%

Basket volatility
σB2=∑i∑jwi wj ρij σi σjσB=σ1n+(1−1n)ρ\displaystyle \sigma_B^2 = \sum_{i}\sum_{j} w_i\,w_j\,\rho_{ij}\,\sigma_i\,\sigma_j \qquad \sigma_B = \sigma\sqrt{\tfrac{1}{n} + \left(1 - \tfrac{1}{n}\right)\rho}

The basket varies less than its stocks unless they move together; the second form holds for equal weights, volatilities and correlations.

ρij\rho_{ij}
correlation between stocks i and j

Worked example 31.0% × √(0.1 + 0.9 × 0.45) = 22.0%

Call on the basket
C=e−rT[FB N(d1)−K N(d2)]FB=B0 e(r−q)T\displaystyle C = e^{-rT}\left[F_B\,N(d_1) - K\,N(d_2)\right] \qquad F_B = B_0\,e^{(r - q)T}

Black-Scholes on the basket, treated as lognormal with the basket volatility (moment matching).

qq
dividend yield of the basket

Worked example 13.80% at 22.0%; the client pays 14.2% at 22.8%, USD 7,100,000

Dealer margin
margin=C(σB+0.8%)−C(σB)\displaystyle \text{margin} = C(\sigma_B + 0.8\%) - C(\sigma_B)

The margin is quoted in volatility points and paid inside the premium.

Worked example 14.2% − 13.80% = 0.40%, about USD 202,000

Greeks and correlation

Delta
Δ=e−qTN(d1)hedge in stock i=wi Δ Nnotional\displaystyle \Delta = e^{-qT}N(d_1) \qquad \text{hedge in stock } i = w_i\,\Delta\,N_{\text{notional}}

The amount of each stock the dealer holds against the option.

Worked example 0.594: about USD 29,700,000 of stocks, USD 2,970,000 of each

Vega
ν=B0 e−qT φ(d1)T\displaystyle \nu = B_0\,e^{-qT}\,\varphi(d_1)\sqrt{T}

The change in value for one point of basket volatility.

Worked example about USD 261,000 per vol point

Correlation sensitivity
∂C∂ρ=ν ∂σB∂ρ=ν (1−1/n) σ22 σB\displaystyle \frac{\partial C}{\partial \rho} = \nu\,\frac{\partial \sigma_B}{\partial \rho} = \nu\,\frac{(1 - 1/n)\,\sigma^2}{2\,\sigma_B}

A higher correlation raises the basket volatility and the call’s value: the buyer is long correlation.

Worked example about USD 51,000 per correlation point

Dividend sensitivity
∂C∂q=−T B0 e−qT N(d1)\displaystyle \frac{\partial C}{\partial q} = -T\,B_0\,e^{-qT}\,N(d_1)

Higher expected dividends lower the forward and the call.

Worked example about −USD 59,000 for 10 bp more dividend yield

Implied correlation
ρimp=σB2−∑iwi2 σi2∑i≠jwi wj σi σj\displaystyle \rho_{\text{imp}} = \frac{\sigma_B^2 - \sum_i w_i^2\,\sigma_i^2}{\sum_{i \ne j} w_i\,w_j\,\sigma_i\,\sigma_j}

The correlation that reconciles an index or basket volatility with the volatilities of its stocks; dispersion trades buy or sell it.

Worked example (22.0%² − 0.1 × 31.0%²) / (0.9 × 31.0%²) = 45%

Payoffs

Call on the basket
N max⁡(BT−K, 0)\displaystyle N\,\max(B_T - K,\,0)

Cash at valuation on the basket performance above the strike.

NN
notional

Worked example USD 50,000,000 × (128% − 100%) = USD 14,000,000

Worst-of call
N max⁡ ⁣(min⁡iSi,TSi,0−K, 0)\displaystyle N\,\max\!\left(\min_i \frac{S_{i,T}}{S_{i,0}} - K,\ 0\right)

Pays only on the worst performer: cheap, and long correlation.

Worked example 0.84% of notional at 45% correlation

Best-of call
N max⁡ ⁣(max⁡iSi,TSi,0−K, 0)\displaystyle N\,\max\!\left(\max_i \frac{S_{i,T}}{S_{i,0}} - K,\ 0\right)

Pays on the best performer: expensive, and short correlation.

Worked example 62.6% of notional at 45% correlation

One-factor model
P ⁣(min⁡iXi>x∣M)=∏i=1nP(Xi>x∣M)=N ⁣(μM−ln⁡xσT(1−ρ))n\displaystyle P\!\left(\min_i X_i > x \mid M\right) = \prod_{i=1}^{n} P(X_i > x \mid M) = N\!\left(\frac{\mu_M - \ln x}{\sigma\sqrt{T(1-\rho)}}\right)^{n}

With equal correlations each stock is a common factor plus its own noise, so the worst and the best of n stocks have closed conditional distributions.

MM
common factor, standard normal
μM\mu_M
mean log return given M

Settlement and collateral

Cash settlement
X=N max⁡(BT−K, 0)paid T+2 business days\displaystyle X = N\,\max(B_T - K,\,0) \quad \text{paid } T + 2 \text{ business days}

One payment from the bank after the calculation agent’s statement.

Worked example USD 14,000,000 on November 1, 2028

Variation margin
VMt=Vt−Vt−1\displaystyle VM_t = V_t - V_{t-1}

The option’s value is collateralized daily under the CSA; the seller posts.

Worked example one year on, basket at 110%: V = 16.1%, about USD 8,050,000 posted by the bank

Exposure of the buyer
EE(t)=E[max⁡(Vt, 0)]=E[Vt]\displaystyle EE(t) = \mathbb{E}\left[\max(V_t,\,0)\right] = \mathbb{E}\left[V_t\right]

A bought option is always an asset of the buyer, so the exposure to the dealer is its whole value, net of the collateral held.

Market data of the worked example, illustrative: 10 stocks with the same 31.0% volatility and a 1.70% dividend yield, pairwise correlation 45%, USD rate 3.80% for 2 years; the basket volatility is 22.0%, 22.8% with the dealer’s margin.
Equities›Variance Swap

Formulas

Payoff

Realized variance
σR2=252N∑t=1N(ln⁡StSt−1)2×1002\displaystyle \sigma_R^2 = \frac{252}{N}\sum_{t=1}^{N}\left(\ln\frac{S_t}{S_{t-1}}\right)^{2} \times 100^2

The annualized mean of the squared daily log returns, with no mean subtracted, in volatility points squared.

StS_t
official close on day t
NN
number of returns, about 252

Worked example realized volatility 16.0: variance 256

Settlement
X=Nvar(σR2−K2)\displaystyle X = N_{\text{var}}\left(\sigma_R^2 - K^2\right)

What the buyer of variance receives (or pays when negative).

KK
variance strike, in volatility points

Worked example 13,158 × (256 − 361) = −USD 1,381,590: the bank pays the client

Variance notional
Nvar=Nvega2K\displaystyle N_{\text{var}} = \frac{N_{\text{vega}}}{2K}

Sets the variance notional so that a small move of realized volatility around the strike is worth the vega notional.

Worked example 500,000 / (2 × 19.0) = 13,158 (rounded in the confirmation)

Cap
X=Nvar(min⁡(σR2, (2.5K)2)−K2)\displaystyle X = N_{\text{var}}\left(\min\left(\sigma_R^2,\,(2.5K)^2\right) - K^2\right)

Limits the seller’s loss in a crash.

Worked example 47.5² = 2,256.25: the client’s loss stops at USD 24,938,000

In volatility points
Nvar(σR2−K2)=Nvega (σR−K)(σR+K)2K≈Nvega (σR−K)\displaystyle N_{\text{var}}\left(\sigma_R^2 - K^2\right) = N_{\text{vega}}\,\frac{(\sigma_R - K)(\sigma_R + K)}{2K} \approx N_{\text{vega}}\,(\sigma_R - K)

Near the strike the swap behaves like a position in volatility; further away the convexity dominates.

Worked example linear: 500,000 × 3.0 = USD 1,500,000; exact: USD 1,381,590

Volatility swap
X=Nvega(σR−Kvol)\displaystyle X = N_{\text{vega}}\left(\sigma_R - K_{\text{vol}}\right)

The linear version: simpler to read, harder to hedge, since no option strip replicates it.

Valuation and replication

Fair strike (replication)
Kvar2=2 erTT[∫0FP(K)K2 dK+∫F∞C(K)K2 dK]\displaystyle K_{\text{var}}^2 = \frac{2\,e^{rT}}{T}\left[\int_0^{F}\frac{P(K)}{K^2}\,dK + \int_F^{\infty}\frac{C(K)}{K^2}\,dK\right]

The price of the log contract: a strip of out-of-the-money puts and calls weighted by 1/K², independent of any model of volatility.

P(K), C(K)P(K),\ C(K)
put and call prices at strike K
FF
forward

Worked example 19.5 with the skew; 18.0 with a flat 18.0% volatility

Skew of the strip (illustrative)
σ(K)=σATM−β ln⁡(K/F)\displaystyle \sigma(K) = \sigma_{\text{ATM}} - \beta\,\ln(K/F)

The skew solved so that the strip gives the 19.5 fair strike.

β\beta
skew slope

Worked example β = 0.184: 1.9 volatility points at 90% of the forward

VIX
σ2=2T∑iΔKiKi2 erT Q(Ki)−1T(FK0−1)2\displaystyle \sigma^2 = \frac{2}{T}\sum_i \frac{\Delta K_i}{K_i^2}\,e^{rT}\,Q(K_i) - \frac{1}{T}\left(\frac{F}{K_0} - 1\right)^2

Cboe’s index applies the same replication to 30-day S&P 500 options; VIX futures and options trade it on the exchange.

Q(Ki)Q(K_i)
mid price of the out-of-the-money option at K_i
K0K_0
first strike below the forward
Mark to market
Vt=Nvar e−r(T−t)[tT σR,t2+T−tT Kt2−K2]\displaystyle V_t = N_{\text{var}}\,e^{-r(T-t)}\left[\frac{t}{T}\,\sigma_{R,t}^2 + \frac{T-t}{T}\,K_t^2 - K^2\right]

The variance already realized and the fair variance of the remaining period, weighted by time, against the strike.

σR,t\sigma_{R,t}
volatility realized to date
KtK_t
fair strike for the remaining period

Worked example six months: (0.5 × 14.5² + 0.5 × 17.5²) = 258.25: USD 1,326,000 to the client

Convexity adjustment
Kvol≈Kvar2−Var⁡(σR)\displaystyle K_{\text{vol}} \approx \sqrt{K_{\text{var}}^2 - \operatorname{Var}(\sigma_R)}

The fair volatility swap strike sits below the variance strike by the uncertainty of realized volatility.

Worked example √(19.5² − 5²) = 18.85

Risk

Vega to implied volatility
∂V∂Kt=2 Nvar Kt T−tT\displaystyle \frac{\partial V}{\partial K_t} = 2\,N_{\text{var}}\,K_t\,\frac{T-t}{T}

The swap’s sensitivity to the fair strike of the remaining period, falling as variance is realized.

Worked example at the start: 2 × 13,158 × 19.5 = USD 513,162 per vol point

Daily accrual
ΔVday≈NvarN(252×1002 rt2−K2)\displaystyle \Delta V_{\text{day}} \approx \frac{N_{\text{var}}}{N}\left(252 \times 100^2\,r_t^2 - K^2\right)

Each day adds the squared return to the realized variance: big days help the buyer.

Worked example a 2% day: 13,158/252 × (1,008 − 361) = USD 33,783 to the buyer

Constant dollar gamma
S22 ∂2Π∂S2=1Tfor Π=2T(S−FF−ln⁡SF)\displaystyle \frac{S^2}{2}\,\frac{\partial^2 \Pi}{\partial S^2} = \frac{1}{T} \quad \text{for } \Pi = \frac{2}{T}\left(\frac{S - F}{F} - \ln\frac{S}{F}\right)

The 1/K² weights make the strip’s gamma in dollars independent of the index level, so the hedge earns realized variance wherever the index goes.

Settlement and credit

Variation margin
VMt=Vt−Vt−1\displaystyle VM_t = V_t - V_{t-1}

The swap is collateralized daily under the CSA.

Worked example six months: about USD 1,326,000 posted by the bank

Exposure on the seller
PFE≤Nvar((2.5K)2−K2)\displaystyle PFE \le N_{\text{var}}\left((2.5K)^2 - K^2\right)

The cap bounds what a variance seller can owe, and the credit line is sized on it.

Worked example 13,158 × (2,256.25 − 361) = USD 24,938,000

Data of the worked example, illustrative: S&P 500, 1 year, vega notional USD 500,000, strike 19.0, at-the-money implied volatility 18.0%, USD rate 3.90%, dividend yield 1.40%; the skew of the strip is 1.9 volatility points for each 10% below the forward.
Equities›Total Return Swap

Formulas

Resets

Price return
Ei=Ni−1(PiPi−1−1)\displaystyle E_i = N_{i-1}\left(\frac{P_i}{P_{i-1}} - 1\right)

The equity leg pays the change in value of the shares over the period; negative, the client pays it.

Ni−1N_{i-1}
notional at the start of period i
PiP_i
basket level at reset i

Worked example 100,000,000 × (101.80 / 100.00 − 1) = USD 1,800,000

Dividend amount
Divi=Ni−1 diPi−1 ρ\displaystyle \text{Div}_i = N_{i-1}\,\frac{d_i}{P_{i-1}}\,\rho

The dividends with an ex-date in the period, passed through at the agreed rate.

did_i
dividends per basket unit
ρ\rho
pass-through: 100% to a US fund, 70% to a non-US fund under Section 871(m)
Financing amount
Fi=Ni−1 (ri+s) τi\displaystyle F_i = N_{i-1}\,(r_i + s)\,\tau_i

The client pays the overnight rate plus the spread on the notional for the days of the period.

rir_i
overnight rate for the period (SOFR compounded)
ss
spread
τi\tau_i
days/360 (days/365 for SONIA and CORRA)

Worked example 100,000,000 × (3.90% + 0.60%) × 32/360 = USD 400,000

Net payment
Xi=Ei+Divi−Fi\displaystyle X_i = E_i + \text{Div}_i - F_i

At each reset only the net moves, seen from the client.

Worked example reset 1: 1,800,000 − 400,000 = USD 1,400,000

Notional reset
Ni=Ni−1 PiPi−1\displaystyle N_i = N_{i-1}\,\frac{P_i}{P_{i-1}}

The notional follows the shares, so the number of shares of exposure stays the same.

Worked example USD 101,800,000 after reset 1

Valuation

Value between resets
Vt=Ni−1(PtPi−1−1)+Divt−Ni−1 (r+s) τ(t)\displaystyle V_t = N_{i-1}\left(\frac{P_t}{P_{i-1}} - 1\right) + \text{Div}_t - N_{i-1}\,(r + s)\,\tau(t)

The swap is worth the gain on the shares since the last reset and the dividends accrued, less the financing accrued; variation margin settles it daily.

Value at a reset
Vti=0after the payment Xi\displaystyle V_{t_i} = 0 \quad \text{after the payment } X_i

Each reset pays the accrued value in cash, so the swap starts the next period at zero.

The bank’s spread
s≈f+k+m−ℓ\displaystyle s \approx f + k + m - \ell

The spread covers the bank’s funding of the hedge over the overnight rate, the cost of its capital and its margin, less what lending the hedge shares earns.

ff
funding spread
kk
capital cost
mm
margin
ℓ\ell
stock lending income

Worked example 60 bp = 35 + 20 + 15 − 10, illustrative

Swap against cash purchase
costswap−costcash≈(r+s) N−rf N+τtx N\displaystyle \text{cost}_{\text{swap}} - \text{cost}_{\text{cash}} \approx (r + s)\,N - r_f\,N + \tau_{\text{tx}}\,N

The swap costs the financing rate plus spread; buying the shares costs the client’s own funding and any transaction tax the swap avoids.

rfr_f
the client’s own funding rate
τtx\tau_{\text{tx}}
stamp duty or FTT on a purchase

Margin and risk

Leverage
L=NIM\displaystyle L = \frac{N}{IM}

The exposure per unit of initial margin.

Worked example 100,000,000 / 20,000,000 = 5 times

Variation margin
VMt=Vt−Vt−1\displaystyle VM_t = V_t - V_{t-1}

The change in the swap’s value, paid in cash every day.

Worked example a 5% fall in the basket: USD 5,000,000 called from the client

Initial margin
IM=N max⁡ ⁣(stress, mmin⁡)+add-ons\displaystyle IM = N\,\max\!\left(\text{stress},\ m_{\min}\right) + \text{add-ons}

The prime broker sizes the margin on a stress move of the basket, with add-ons for concentration and liquidity.

mmin⁡m_{\min}
floor rate

Worked example 20% for a diversified mid-cap basket

Withholding under Section 871(m)
W=30%×Divi\displaystyle W = 30\%\times \text{Div}_i

Dividend equivalents paid to a non-US person are taxed as US dividends, at 30% or the treaty rate.

Worked example the June reset’s dividends of USD 424,800: USD 127,440 withheld

Counterparty

Gap loss of the bank
L=max⁡ ⁣(N ∣Δ∣close-out−IM, 0)\displaystyle L = \max\!\left(N\,|\Delta|_{\text{close-out}} - IM,\ 0\right)

The bank loses only if the basket moves against the client by more than the margin before it can sell the hedge.

∣Δ∣close-out|\Delta|_{\text{close-out}}
move over the close-out period

Worked example a 25% gap: 25,000,000 − 20,000,000 = USD 5,000,000

Close-out period
h=Qu ADV\displaystyle h = \frac{Q}{u\,\text{ADV}}

The days needed to sell the hedge without moving the price too far; concentrated positions need more margin.

QQ
hedge shares
uu
participation rate
Exposure with margin
EE(t)=E[max⁡(−Vt−IM, 0)]\displaystyle EE(t) = \mathbb{E}\left[\max(-V_t - IM,\ 0)\right]

The expected exposure after the margin, the input to CVA and to the counterparty limits.

US worked example: USD 100,000,000 notional, SOFR + 0.60%, Act/360, monthly resets from October 29, 2026; the basket path, the dividends and the SOFR fixings are illustrative.
Equities›ETF

Formulas

NAV and premium

Net asset value per share
NAV=∑jqj Pj+cash−liabilitiesSout\displaystyle NAV = \frac{\sum_j q_j\,P_j + \text{cash} - \text{liabilities}}{S_{\text{out}}}

The fund’s holdings at the closing prices, net of accrued expenses, per share outstanding.

qj, Pjq_j,\ P_j
holding and closing price of security j
SoutS_{\text{out}}
shares outstanding

Worked example USD 100.00 at the close of October 28

Indicative NAV
iNAVt=∑jqj Pj,t+cashSout\displaystyle iNAV_t = \frac{\sum_j q_j\,P_{j,t} + \text{cash}}{S_{\text{out}}}

The same sum on live prices, published during the day.

Worked example USD 100.00 when the ETF trades at 100.02

Premium or discount
π=PETF−NAVNAV\displaystyle \pi = \frac{P_{\text{ETF}} - NAV}{NAV}

How far the market price is from the value of the holdings.

Worked example (100.02 − 100.00) / 100.00 = +2.0 bp; the client’s price, 100.015, is +1.5 bp

Expense accrual
NAVt=NAVt−1 (1+Rt)−NAVt−1 ER365\displaystyle NAV_t = NAV_{t-1}\,(1 + R_t) - NAV_{t-1}\,\frac{ER}{365}

The expense ratio comes out of the NAV a little every day.

ERER
expense ratio
RtR_t
return of the holdings

Worked example 0.09% a year on USD 200,000,000: about USD 493 a day

Tracking difference
TD=RETF−Rindex\displaystyle TD = R_{\text{ETF}} - R_{\text{index}}

The ETF’s return against its index over a period: about minus the expenses, plus lending income, less withholding taxes the index does not bear.

Creation and redemption

Creation unit
1 CU=basket+C\displaystyle 1\ \text{CU} = \text{basket} + C

A block of new shares issued against the basket of the portfolio composition file and a cash component.

CC
cash component

Worked example 40 units of 50,000 shares for 2,000,000 shares

Cash component
C=U⋅NAV−∑jbj Pj\displaystyle C = U \cdot NAV - \sum_j b_j\,P_j

The difference between the value of a unit at NAV and the basket: accrued dividends, cash in the fund and rounding.

UU
shares per unit
bjb_j
shares of security j in the basket

Worked example USD 0.12 a share of accrued dividends: USD 6,000 a unit, USD 240,000 for 40 units

When to create or redeem
PETF>NAV (1+ccr)⇒createPETF<NAV (1−crd)⇒redeem\displaystyle P_{\text{ETF}} > NAV\,(1 + c_{\text{cr}}) \Rightarrow \text{create} \qquad P_{\text{ETF}} < NAV\,(1 - c_{\text{rd}}) \Rightarrow \text{redeem}

The AP acts when the gap to NAV pays for trading the basket and the fees.

ccr, crdc_{\text{cr}},\ c_{\text{rd}}
costs of creating and redeeming, as a fraction

Worked example +2 bp premium against about 1 bp of basket cost: create

AP margin
Π=q (Pclient−NAV−i)−f\displaystyle \Pi = q\,(P_{\text{client}} - NAV - i) - f

What the bank keeps: the price it sells at less the NAV and the impact of buying the basket, less the creation fee.

ii
impact per share
ff
creation fee

Worked example 2,000,000 × (100.015 − 100.00 − 0.01) − 500 = USD 9,500

Client settlement
A=q Pclient\displaystyle A = q\,P_{\text{client}}

The client pays on T+1 against the ETF shares.

Worked example 2,000,000 × 100.015 = USD 200,030,000

Distribution
Dist=q d\displaystyle \text{Dist} = q\,d

The fund pays out the income it received, less its expenses, to the holders of record.

dd
distribution per share

Worked example 2,000,000 × 0.30 = USD 600,000

Costs to the investor

Total cost of holding
TCO=ER⋅h+12(sbuy+ssell)+πbuy−πsell\displaystyle TCO = ER \cdot h + \tfrac{1}{2}\left(s_{\text{buy}} + s_{\text{sell}}\right) + \pi_{\text{buy}} - \pi_{\text{sell}}

Expenses over the holding period, half the spread on each trade and the premium paid on entry less the one received on exit.

hh
holding period, years
ss
bid-offer spread, as a fraction

Worked example one year: 0.09% + spreads + 1.5 bp of premium paid at entry

Primary market cost
ccr=fq NAV+iNAV\displaystyle c_{\text{cr}} = \frac{f}{q\,NAV} + \frac{i}{NAV}

What a creation costs the AP per unit of value: the fee and the impact of buying the basket.

Worked example 500 / 200,000,000 + 0.01 / 100.00 = 1.03 bp

Risk

Tracking error
TE=σ ⁣(RETF−Rindex)252\displaystyle TE = \sigma\!\left(R_{\text{ETF}} - R_{\text{index}}\right)\sqrt{252}

The volatility of the daily return difference, annualized.

Redemption arbitrage
Πrd=q (NAV−PETF−i)−f\displaystyle \Pi_{\text{rd}} = q\,(NAV - P_{\text{ETF}} - i) - f

An AP buys shares at a discount, redeems them and sells the basket.

Worked example on 2,000,000 shares at 99.97: 2,000,000 × (100.00 − 99.97 − 0.01) − 500 = USD 39,500

Liquidity of the ETF
capacity≈min⁡ju ADVjwj\displaystyle \text{capacity} \approx \min_j \frac{u\,\text{ADV}_j}{w_j}

An ETF can absorb as much as its basket can trade: the least liquid name, scaled by its weight, sets the limit.

wjw_j
weight of security j
uu
participation rate

US worked example: 2,000,000 shares at USD 100.015 against a NAV of USD 100.00, 40 creation units of 50,000 shares; the cash component, the expense ratio (0.09%) and the distribution are illustrative.
Equities›Convertible Bond

Formulas

Valuation

Conversion ratio and parity
CR=1000CPparity=CR×S1000×100\displaystyle CR = \frac{1000}{CP} \qquad \text{parity} = \frac{CR \times S}{1000} \times 100

The number of shares per bond and their value as a percentage of face.

CPCP
conversion price
SS
share price

Worked example CR = 20 shares; parity = 20 × 42.00 / 10 = 84.00

Premium over parity
premium=Pparity−1\displaystyle \text{premium} = \frac{P}{\text{parity}} - 1

How much more than the shares the bond costs: the price of the floor and of the time value.

PP
bond price

Worked example 102.00 / 84.00 − 1 = 21.4%

Bond floor
B=∑k=1nc/2(1+y/2)k−1+f+100(1+y/2)n−1+f−AI\displaystyle B = \sum_{k=1}^{n}\frac{c/2}{(1 + y/2)^{k-1+f}} + \frac{100}{(1 + y/2)^{n-1+f}} - AI

The straight bond on the issuer’s credit: what the convertible is worth if the option is worthless.

yy
Treasury yield plus the credit spread
ff
fraction of the current period left
AIAI
accrued interest

Worked example y = 3.70% + 3.11% = 6.81%: B = 78.27

Conversion option
O=CR10  e−rT[F N(d1)−CP N(d2)]F=S e(r−b)T\displaystyle O = \frac{CR}{10}\;e^{-rT}\left[F\,N(d_1) - CP\,N(d_2)\right] \qquad F = S\,e^{(r - b)T}

The holder’s right to the shares, as Black-Scholes calls struck at the conversion price, with the borrow cost in place of a dividend yield.

bb
borrow cost
TT
years to maturity

Worked example 2 calls of 11.74 = 23.48 per 100 of face

Convertible value
V=B+O\displaystyle V = B + O

A first estimate, before the issuer call and the other terms.

Worked example 78.27 + 23.48 = 101.75; at 102.00 the fund pays a quarter point of margin, about USD 50,000

Implied volatility
σimp: B+O(σimp)=P\displaystyle \sigma_{\text{imp}}:\ B + O(\sigma_{\text{imp}}) = P

The volatility the market price implies, compared with listed volatility to judge whether the bond is cheap.

Worked example 102.00 implies 34.4% against 32% on listed options

Risk and the hedge

Delta (hedge ratio)
Δ=e−bT N(d1)shares short=Δ×CR×face1000\displaystyle \Delta = e^{-bT}\,N(d_1) \qquad \text{shares short} = \Delta \times CR \times \frac{\text{face}}{1000}

The fraction of the conversion shares that hedges the bond against small moves in the stock.

Worked example 62.2%: 249,000 of the 400,000 shares

Gamma
Γ=e−bT φ(d1)S σT\displaystyle \Gamma = \frac{e^{-bT}\,\varphi(d_1)}{S\,\sigma\sqrt{T}}

How fast the hedge ratio changes as the stock moves.

Worked example about 1,949 shares of delta for a 1% move

Vega
ν=CR face1000  S e−bT φ(d1)T\displaystyle \nu = CR\,\frac{\text{face}}{1000}\;S\,e^{-bT}\,\varphi(d_1)\sqrt{T}

The change in value for one point of volatility.

Worked example about USD 136,000 per vol point on USD 20,000,000

Credit spread sensitivity
CS01=B(y−0.5 bp)−B(y+0.5 bp)\displaystyle \text{CS01} = B(y - 0.5\,\text{bp}) - B(y + 0.5\,\text{bp})

The change in the bond floor for one basis point of credit spread.

Worked example about USD 7,100 per bp on USD 20,000,000

Carry of the hedged position
carry=face⋅c+shares⋅S (rrebate)−financing of the bond\displaystyle \text{carry} = \text{face}\cdot c + \text{shares}\cdot S\,(r_{\text{rebate}}) - \text{financing of the bond}

The coupon and the interest on the short sale proceeds, less the cost of financing the bond at the prime broker.

rrebater_{\text{rebate}}
short rebate rate, the rate less the borrow fee

Worked example coupon USD 300,000 a year; rebate 249,000 × 42.00 × 3.20% = USD 334,656

Cash flows

Settlement amount
face×P100+AI\displaystyle \text{face} \times \frac{P}{100} + AI

What the fund pays on T+1.

Worked example 20,000,000 × 102.00% + 36,667 = USD 20,436,667

Accrued interest (30/360)
AI=face×c×days360\displaystyle AI = \text{face} \times c \times \frac{\text{days}}{360}

The coupon earned by the seller since the last payment date.

Worked example 44 days from September 15: USD 36,667

Coupon
face×c2\displaystyle \text{face} \times \frac{c}{2}

Paid every March 15 and September 15.

Worked example USD 150,000

At conversion or maturity

Value at maturity
max⁡(face, CR×ST)=face+CR max⁡(ST−CP, 0)\displaystyle \max(\text{face},\ CR \times S_T) = \text{face} + CR\,\max(S_T - CP,\ 0)

Par, or the shares if they are worth more: a bond plus calls struck at the conversion price.

Worked example stock 60.00: 400,000 shares worth USD 24,000,000 against USD 20,000,000 at par

Net share settlement
cash=faceshares=CR face1000 ST−faceST\displaystyle \text{cash} = \text{face} \qquad \text{shares} = \frac{CR\,\frac{\text{face}}{1000}\,S_T - \text{face}}{S_T}

Par in cash and only the excess in shares: less dilution for the issuer.

Worked example USD 20,000,000 in cash and 66,667 shares

Issuer soft call
callable if St≥130% CP on 20 of 30 trading days\displaystyle \text{callable if } S_t \ge 130\%\,CP \text{ on 20 of 30 trading days}

Once the stock is well above the conversion price the issuer calls, and the holders convert.

Worked example trigger USD 65.00 from year 3

Data of the worked example, illustrative: stock USD 42.00, conversion price USD 50.00 (20 shares per USD 1,000), 1.50% semiannual coupon to September 15, 2031, settlement October 29, 2026, 5-year Treasury 3.70%, credit spread 311 bp, implied volatility 34%, borrow cost 0.50%, no dividend.
Equities›Structured Note

Formulas

Payoffs

Contingent coupon
N c4 1{Xi≥70%}\displaystyle N\,\frac{c}{4}\,\mathbf{1}\{X_i \ge 70\%\}

Paid on each observation where the index is at or above the coupon barrier.

XiX_i
index on observation i, as a % of initial
cc
coupon a year

Worked example 8.10% / 4 = 2.025%: USD 506,250

Autocall
N(1+c4) if Xi≥100%, i≥2\displaystyle N\left(1 + \frac{c}{4}\right) \text{ if } X_i \ge 100\%,\ i \ge 2

The note ends early at par plus the coupon.

Worked example observation 2 at 103%: USD 25,506,250

Redemption at maturity
N(1+c4) if XT≥70%N XT if XT<70%\displaystyle N\left(1 + \frac{c}{4}\right) \text{ if } X_T \ge 70\% \qquad N\,X_T \text{ if } X_T < 70\%

If never called: par and the coupon above the barrier, the index below it.

Worked example at 60%: USD 15,000,000

Capital-protected note
N[1+p max⁡(XT−1, 0)]\displaystyle N\left[1 + p\,\max(X_T - 1,\ 0)\right]

Par at maturity plus a share of the index’s rise.

pp
participation

Worked example p = 56%

Valuation and the estimated value

Index path
Xti=Xti−1exp⁡ ⁣[(r−q−12σ2)Δt+σΔt Zi]\displaystyle X_{t_i} = X_{t_{i-1}}\exp\!\left[\left(r - q - \tfrac{1}{2}\sigma^2\right)\Delta t + \sigma\sqrt{\Delta t}\,Z_i\right]

Black-Scholes dynamics simulated from one observation date to the next.

ZiZ_i
independent standard normals
Estimated value
EV=E[∑iCFi e−(r+sf) ti]\displaystyle EV = \mathbb{E}\left[\sum_i CF_i\,e^{-(r + s_f)\,t_i}\right]

The average of the note’s payments over the paths, discounted at the internal funding rate.

sfs_f
internal funding spread, 70 bp

Worked example 96.23 per 100

Its parts
EV=PV(par)+PV(coupons)−PV(loss)\displaystyle EV = PV(\text{par}) + PV(\text{coupons}) - PV(\text{loss})

The principal repaid, the coupons, and the down-and-in put the investor sold.

Worked example 94.53 + 7.60 − 5.89 = 96.23

Issue price
100=EV+fee+margin\displaystyle 100 = EV + \text{fee} + \text{margin}

The gap between the price and the estimated value pays the distributor and the issuer.

Worked example 96.2 + 2.50 + 1.30 ≈ 100

Participation
p=EV−100 DfDf C erTDf=e−(r+sf)T\displaystyle p = \frac{EV - 100\,D_f}{D_f\,C\,e^{rT}} \qquad D_f = e^{-(r + s_f)T}

The share of a call the budget buys once the protected principal is set aside.

CC
at-the-money call, % of notional

Worked example (96.2 − 87.58) / (0.8758 × 15.59 × e^(rT)) = 56%

Probabilities
P(call before maturity)P(XT<70% and never called)\displaystyle P(\text{call before maturity}) \qquad P(X_T < 70\% \text{ and never called})

Read from the same simulation.

Worked example 77% and 14%

Risk

Vega of the note
∂EV∂σ\displaystyle \frac{\partial EV}{\partial \sigma}

The investor is short volatility: the note loses value when volatility rises.

Worked example about −USD 128,000 per vol point on USD 25,000,000

Jump at the barrier
N(1+c4−70%)\displaystyle N\left(1 + \frac{c}{4} - 70\%\right)

What changes hands for an index at 70% or just below on the final observation.

Worked example USD 8,006,250

Jump at the autocall level
N(1+c4)−Vnot called\displaystyle N\left(1 + \frac{c}{4}\right) - V_{\text{not called}}

The value of being called against continuing: largest just before an observation.

Flat at-the-money volatility
EV(σATM)−EV(σ70%)\displaystyle EV(\sigma_{\text{ATM}}) - EV(\sigma_{70\%})

The skew is the price of the investor’s sold downside: at 19% the same note would be worth more than its issue price.

Worked example 100.2 − 96.2

Dates and amounts

Coupon amount
N×c4\displaystyle N \times \frac{c}{4}

Paid three business days after the observation.

Worked example USD 506,250 on February 2, 2027

Call amount
N×(1+c4)\displaystyle N \times \left(1 + \frac{c}{4}\right)

Par and the coupon on the call payment date.

Worked example USD 25,506,250 on May 3, 2027

Data of the worked example, illustrative: S&P 500 initial level 5,000.00, dividend yield 1.40%, USD rate 3.70%, internal funding SOFR + 70 bp; the note is valued by Monte Carlo under Black-Scholes dynamics (50,000 paths) at a flat 26.4% volatility, the level of the 3-year 70% strike on the skew (19% at the money).
Commodities›Energy Swap

Formulas

Monthly settlement

Monthly average
Am=1nm∑d=1nmPd\displaystyle A_m = \frac{1}{n_m}\sum_{d=1}^{n_m} P_d

The floating price of month m: the arithmetic average of the daily settlement prices of the front-month futures over the pricing days of the calendar month.

PdP_d
first-nearby settlement price on pricing day d
nmn_m
pricing days in month m

Worked example January 2027: USD 68.50 (illustrative)

Settlement
Xm=Q (K−Am)\displaystyle X_m = Q\,(K - A_m)

What the fixed receiver gets for the month; a negative amount is paid.

QQ
barrels a month
KK
fixed price

Worked example 500,000 × (72.20 − 68.50) = USD 1,850,000 to the producer

Payment date
Tmpay=month endm+5 business days\displaystyle T^{\text{pay}}_m = \text{month end}_m + 5\ \text{business days}

The settlement is known once the last price of the month is published and is paid a few days later.

Worked example Jan 31, 2027, a Sunday, plus 5 business days: Feb 5, 2027

Hedged revenue
Q Pmloc+Q (K−Am)=Q (K+bm)bm=Pmloc−Am\displaystyle Q\,P^{\text{loc}}_m + Q\,(K - A_m) = Q\,(K + b_m) \qquad b_m = P^{\text{loc}}_m - A_m

The producer sells its oil at the local price and settles the swap on the benchmark: the basis between the two is what stays unhedged.

PmlocP^{\text{loc}}_m
price received for the physical barrels
bmb_m
location and quality basis

Worked example a local price USD 1.10 over the benchmark: USD 73.30 a barrel

The year
∑m=112Xm=Q(12 K−∑m=112Am)\displaystyle \sum_{m=1}^{12} X_m = Q\left(12\,K - \sum_{m=1}^{12} A_m\right)

The net of the twelve settlements is the fixed price against the average of the year’s averages.

Worked example 500,000 × (12 × 72.20 − 12 × 71.3833) = +USD 4,900,000 on the illustrative path

Valuation

Forward of a monthly average
Fm=1nm∑d=1nmF c(d)(0)\displaystyle F_m = \frac{1}{n_m}\sum_{d=1}^{n_m} F^{\,c(d)}(0)

The expected average is the average of the futures prices of the contracts that will be front month on each pricing day.

c(d)c(d)
the contract that is first nearby on day d
Month partly priced
E[Am]=knm Aˉk+nm−knm Fm\displaystyle \mathbb{E}[A_m] = \frac{k}{n_m}\,\bar{A}_{k} + \frac{n_m - k}{n_m}\,F_m

Inside the month the fixings already published are known; only the remaining days are forwards.

kk
pricing days already fixed
Aˉk\bar{A}_k
their average
Swap value
Vrec=∑m=112Q (K−Fm) P(0,Tmpay)\displaystyle V_{\text{rec}} = \sum_{m=1}^{12} Q\,(K - F_m)\,P(0,T^{\text{pay}}_m)

The value to the fixed receiver: each month’s fixed price against the forward of its average, discounted from the payment date.

P(0,T)P(0,T)
SOFR discount factor

Worked example strip 73.50 to 71.30: −USD 1,211,682 to the producer at the trade, the dealer’s margin of about 20 cents

Fair fixed price
K∗=∑mFm P(0,Tmpay)∑mP(0,Tmpay)\displaystyle K^{*} = \frac{\sum_m F_m\,P(0,T^{\text{pay}}_m)}{\sum_m P(0,T^{\text{pay}}_m)}

The fixed price that makes the swap worth zero: the discount-weighted average of the strip.

Worked example K* = 72.41 against the 72.20 dealt

Risk and hedge

Delta by month
Δm=∂Vrec∂Fm=−Q P(0,Tmpay)\displaystyle \Delta_m = \frac{\partial V_{\text{rec}}}{\partial F_m} = -Q\,P(0,T^{\text{pay}}_m)

The fixed receiver is short the commodity: it loses Q dollars a month for each dollar on the forward.

Worked example month 1: −494,686 barrels discounted

Futures hedge
Nm=Q1,000\displaystyle N_m = \frac{Q}{1{,}000}

The dealer, long the floating leg, sells one WTI contract of 1,000 barrels for each 1,000 barrels a month.

Worked example 500 contracts a month, 6,000 in all

Variation margin on the hedge
VMt=−1,000∑mNm ΔFm,t\displaystyle VM_t = -1{,}000\sum_m N_m\,\Delta F_{m,t}

The futures pay and collect margin every day while the swap settles once a month.

Worked example USD 1 on the whole strip: USD 6,000,000

Buy-back through the month
Nm(k)=Nm nm−knm\displaystyle N_m(k) = N_m\,\frac{n_m - k}{n_m}

As each day’s price fixes part of the average, the dealer buys back the matching futures, so the hedge of the month is gone by its last pricing day.

kk
pricing days fixed

Worked example 500 contracts over 20 pricing days: 25 a day

Collars and three-ways

Swap
X=Q (K−A)\displaystyle X = Q\,(K - A)

The settlement of a month against its average: linear both ways.

Worked example at 68.50: +USD 1,850,000

Costless collar
X=Q[max⁡(Kp−A, 0)−max⁡(A−Kc, 0)]\displaystyle X = Q\left[\max(K_p - A,\,0) - \max(A - K_c,\,0)\right]

A bought put and a sold call on the monthly average: a floor and a cap, the call’s premium paying for the put.

Kp, KcK_p,\ K_c
put and call strikes

Worked example 60 / 82.50 on 150,000 barrels: at 55.00, 5.00 × 150,000 = USD 750,000

Three-way
X=Q[max⁡(K2−A, 0)−max⁡(K1−A, 0)−max⁡(A−Kc, 0)]\displaystyle X = Q\left[\max(K_2 - A,\,0) - \max(K_1 - A,\,0) - \max(A - K_c,\,0)\right]

A collar with a sold put below the floor: the protection covers the band between the two puts only.

K1<K2K_1 < K_2
sold and bought put strikes

Worked example 45 / 62 / 81: at 55.00 pays 7.00, USD 1,050,000; at 40.00 only 17.00

Option on a monthly average
σ2TA=σ2(t1+t2−t13)C=P(0,T)[Fm N(d1)−K N(d2)]\displaystyle \sigma^2 T_A = \sigma^2\left(t_1 + \frac{t_2 - t_1}{3}\right) \qquad C = P(0,T)\left[F_m\,N(d_1) - K\,N(d_2)\right]

An Asian option on a month’s average priced with Black-76 on the forward of the average, its variance reduced for the averaging (a common approximation).

t1, t2t_1,\ t_2
start and end of the pricing month, years from today
TAT_A
the equivalent time for the average

Worked example flat 32%, strip 70.00: 60 put 2.69, 82.50 call 2.96, 45 put 0.32, 62 put 3.32, 81 call 3.28 a barrel

Value to the producer
Vcollar=P(Kp)−C(Kc)V3-way=P(K2)−P(K1)−C(Kc)\displaystyle V_{\text{collar}} = P(K_p) - C(K_c) \qquad V_{\text{3-way}} = P(K_2) - P(K_1) - C(K_c)

Dealt at zero premium, the structures are worth slightly less than nothing to the producer: the bank’s margin.

Worked example collar −0.27, three-way −0.27 a barrel

Credit and funding

Exposure
EE(t)=E[max⁡(Vtbank, 0)]\displaystyle EE(t) = \mathbb{E}\left[\max(V^{\text{bank}}_t,\,0)\right]

The dealer is exposed to the producer when prices rise and the producer owes the remaining settlements.

CVA
CVA≈(1−R)∑k=1KEE(tk)  PD(tk−1,tk)  P(0,tk)\displaystyle \text{CVA} \approx (1-R)\sum_{k=1}^{K} EE(t_k)\;PD(t_{k-1},t_k)\;P(0,t_k)

The expected loss from the producer’s default, often reduced by the security of the reserve-based loan.

RR
recovery rate
PDPD
probability of default between two dates
Funding of the hedge margin
FVA≈∑ksF E[Mk]τk P(0,tk)\displaystyle \text{FVA} \approx \sum_k s_F\,\mathbb{E}\left[M_k\right]\tau_k\,P(0,t_k)

The dealer funds the margin posted on its futures while the uncollateralized swap pays nothing until each month’s settlement.

sFs_F
funding spread
MkM_k
cumulative margin posted at t_k
τk\tau_k
length of period k

Market data of the worked example, illustrative: the WTI strip for calendar 2027 from USD 73.50 to 71.30 (average 72.40), SOFR flat at 3.90%; the producer program’s options on a flat 70.00 strip at 32% volatility.
Commodities›Energy Future

Formulas

Contract and margin

Notional
N×q×F0\displaystyle N \times q \times F_0

The value of the position at the fill price.

NN
contracts
qq
contract size, 42,000 gallons
F0F_0
fill price

Worked example 1,000 × 42,000 × 2.4500 = USD 102,900,000

Tick value
q×δ\displaystyle q \times \delta

The value of the minimum price move of one contract.

δ\delta
tick, USD 0.0001 a gallon

Worked example 42,000 × 0.0001 = USD 4.20

Variation margin
VMt=N q (Ft−Ft−1)\displaystyle VM_t = N\,q\,(F_t - F_{t-1})

Each day the change in the settlement price is paid in cash, received by the long when the price rises.

FtF_t
settlement price of day t

Worked example Nov 2: 1,000 × 42,000 × (2.4710 − 2.4290) = USD 1,764,000

Cumulative variation margin
∑s=1tVMs=N q (Ft−F0)\displaystyle \sum_{s=1}^{t} VM_s = N\,q\,(F_t - F_0)

The margins add up to the gain or loss since the fill: the futures has no other value to settle.

Worked example after five days: 1,000 × 42,000 × (2.4850 − 2.4500) = USD 1,470,000

Initial margin
IM=N×m\displaystyle IM = N \times m

Collateral against one or two days of adverse moves, set by the clearing house’s model and returned at the close.

mm
margin per contract

Worked example 1,000 × 9,000 = USD 9,000,000

The curve: carry and convenience

Cost of carry
F(T)=S e(r+u−y) T\displaystyle F(T) = S\,e^{(r + u - y)\,T}

A storable commodity’s futures price is spot carried at the financing rate and the storage cost, less the convenience of holding the physical.

SS
spot price
rr
financing rate
uu
storage cost, as a yield
yy
convenience yield
Implied convenience yield
y=r+u−1T2−T1ln⁡F(T2)F(T1)\displaystyle y = r + u - \frac{1}{T_2 - T_1}\ln\frac{F(T_2)}{F(T_1)}

Read from two contract months, the convenience yield is what makes the curve slope as it does.

Worked example December 2.4650, January 2.4500: y = 4.90% + 3.00% + 7.32% = 15.22% a year

Calendar spread
spread=F(T2)−F(T1)\displaystyle \text{spread} = F(T_2) - F(T_1)

Positive in contango, negative in backwardation; the price of rolling a position one month.

Worked example January − December = −0.0150 a gallon

Contango or backwardation
F(T2)>F(T1)  ⟺  y<r+u\displaystyle F(T_2) > F(T_1) \iff y < r + u

The curve rises when the convenience of holding product is worth less than the cost of carrying it.

The roll

Roll yield
ρ=F(T1)−F(T2)F(T2)\displaystyle \rho = \frac{F(T_1) - F(T_2)}{F(T_2)}

Selling the expiring month and buying the next earns the spread in backwardation and pays it in contango.

Worked example sell December at 2.4650, buy January at 2.4500: +0.61% for the month

Return of a rolled long
R≈ΔSS+ρ+rcollateral\displaystyle R \approx \frac{\Delta S}{S} + \rho + r_{\text{collateral}}

An index fund’s return is the spot move, the roll yield and the interest on the cash held as collateral.

Roll schedule (index)
wk=0.2on business days 5 to 9\displaystyle w_k = 0.2 \quad \text{on business days } 5 \text{ to } 9

A large commodity index moves a fifth of its position each day from the 5th to the 9th business day of the month.

Hedge result

Futures P&L at the close-out
Π=N q (FT−F0)\displaystyle \Pi = N\,q\,(F_T - F_0)

What the long has received in variation margin when it sells at F_T.

Worked example at 2.6000: USD 6,300,000

Hedged fuel cost
C=STjet−(FT−F0)=F0+bTbT=STjet−FT\displaystyle C = S^{\text{jet}}_T - (F_T - F_0) = F_0 + b_T \qquad b_T = S^{\text{jet}}_T - F_T

The airline pays the jet price and keeps the futures gain: what remains is the fill price plus the basis at the close.

bTb_T
jet fuel price less the ULSD future at the close-out
Minimum variance hedge ratio
h∗=ρ σΔSσΔFN∗=h∗ Qq\displaystyle h^{*} = \rho\,\frac{\sigma_{\Delta S}}{\sigma_{\Delta F}} \qquad N^{*} = h^{*}\,\frac{Q}{q}

The proxy hedge sizes the futures to the correlation and the relative volatility of jet fuel and ULSD.

QQ
gallons of jet fuel hedged
ρ\rho
correlation of the price changes

Risk and margin

Delta
Δ=N q\displaystyle \Delta = N\,q

Gallons equivalent of the position, by month and in total.

Worked example 42,000,000 gallons

Margin at risk
call≈N q ∣ΔF∣+ΔIM\displaystyle \text{call} \approx N\,q\,|\Delta F| + \Delta IM

A stressed move calls the variation margin and, as volatility rises, a higher initial margin the same day.

Worked example a 10 cent fall: USD 4,200,000 plus any increase in the initial margin

Use of the position limit
u=∣Nspot∣Lspot\displaystyle u = \frac{|N_{\text{spot}}|}{L_{\text{spot}}}

The share of the spot-month limit used as the contract nears expiry; exemptions lift it for bona fide hedges.

LspotL_{\text{spot}}
spot-month limit

Worked example, illustrative: 1,000 NYMEX ULSD contracts of 42,000 gallons; settlement prices of the first days as in the cash flows; the curve December 2.4650, January 2.4500, February 2.4350; funding at 4.90%, storage at 3% a year.
Commodities›Precious Metals

Formulas

Spot and settlement

Consideration
C=Q×P\displaystyle C = Q \times P

The dollars paid on the value date for the ounces bought.

QQ
troy ounces
PP
price per ounce

Worked example 10,000 × 2,651.00 = USD 26,510,000

Bar premium
CP=Q (P+pbar)\displaystyle C_P = Q\,(P + p_{\text{bar}})

Allocated bars carry a premium for fabrication, location and logistics.

pbarp_{\text{bar}}
premium per ounce

Worked example 10,000 × (2,651.00 + 1.50) = USD 26,525,000

Troy ounce
1 oz t=31.1035 g1 kg=32.1507 oz t\displaystyle 1\ \text{oz t} = 31.1035\ \text{g} \qquad 1\ \text{kg} = 32.1507\ \text{oz t}

The unit of every precious metals price.

Fine weight of a bar
Qfine=Wgross×ϕ\displaystyle Q_{\text{fine}} = W_{\text{gross}} \times \phi

A bar is credited for its fine gold: the gross weight times the fineness assayed by the refiner.

ϕ\phi
fineness, at least 0.995 for good delivery

Worked example 400.000 oz gross at 0.9995 = 399.800 fine oz

Forwards, lease and EFP

Forward price
F=S(1+(r−l) d360)\displaystyle F = S\left(1 + (r - l)\,\frac{d}{360}\right)

Gold is carried like a currency: the dollar rate against the lease rate earned on the metal.

rr
USD rate
ll
lease rate
dd
days to the forward date

Worked example 92 days: 2,650.25 × (1 + 1.753% × 92/360) = 2,662.12

Gold forward rate
GOFO=r−l\displaystyle \text{GOFO} = r - l

The swap rate at which dealers lend gold against dollars; positive when the dollar rate is above the lease rate.

Worked example 3.90% − 2.147% = 1.753%

EFP
EFP=FCOMEX−SLondon\displaystyle \text{EFP} = F^{\text{COMEX}} - S^{\text{London}}

The futures price over London spot: the carry to the futures delivery plus any premium for metal in New York.

Worked example +8.00 on December

Lease rate implied by the EFP
l=r−EFPS 360d\displaystyle l = r - \frac{\text{EFP}}{S}\,\frac{360}{d}

With no premium for location, the EFP reveals the lease rate the market charges for metal.

Worked example 3.90% − (8 / 2,650.25) × 360/62 = 2.147%

Lease fee
L=Q S0 l d360\displaystyle L = Q\,S_0\,l\,\frac{d}{360}

A holder of unallocated metal earns the lease rate on the value of the ounces lent, paid in dollars or in metal.

Worked example 10,000 × 2,651.00 × 1.90% × 92/360 = USD 128,721

Risk and hedge

Delta
Δ=Q\displaystyle \Delta = Q

The position in ounces; USD per dollar on the price.

Worked example 10,000 oz: USD 10,000 per dollar an ounce

Futures hedge
N=Q100\displaystyle N = \frac{Q}{100}

COMEX gold futures are 100 oz.

Worked example 10,000 / 100 = 100 contracts

Lease rate sensitivity
∂F∂l=−S d360\displaystyle \frac{\partial F}{\partial l} = -S\,\frac{d}{360}

A rise in the lease rate lowers the forward and the value of a long forward position.

Worked example 1% on 92 days: −USD 6.77 an ounce

Unallocated against allocated

Exposure of an unallocated holder
E=Q×S\displaystyle E = Q \times S

Unallocated metal is an unsecured claim on the bank for the whole value of the ounces.

Worked example USD 26,502,500 at mid

Storage of allocated metal
Cs=Q S c d365\displaystyle C_s = Q\,S\,c\,\frac{d}{365}

Allocated bars pay storage and insurance, quoted as a rate on value or per bar.

cc
storage rate a year

Worked example 0.12% a year: USD 31,812

Required stable funding
RSF=85%×Vgold\displaystyle RSF = 85\% \times V_{\text{gold}}

Under the NSFR a bank funds 85% of the gold it holds with stable funding, the cost behind smaller unallocated books.

Worked example USD 22,533,500 for the ounces bought

Market data of the worked example, illustrative: spot 2,650.00 / 2,650.50, EFP +8 on the December contract (carry to Dec 31, 2026), USD rate 3.90%, Act/360 for leases and forwards.
Commodities›Base Metals

Formulas

Prompt dates and settlement

Three-month prompt
T3M=t+3 calendar months\displaystyle T_{3M} = t + 3\ \text{calendar months}

The benchmark date is three calendar months after the trade date, moved to a business day; cash is two business days after the trade.

Worked example Oct 28, 2026: cash Oct 30, 2026, three months Jan 28, 2027

Contract value
V0=n×25×P0\displaystyle V_0 = n \times 25 \times P_0

Lots of 25 tonnes at the trade price.

nn
lots
P0P_0
price a tonne

Worked example 200 × 25 × 9,258 = USD 46,290,000

Close-out settled on the prompt
X=(Pclose−P0)×Qclosedpaid on Tprompt\displaystyle X = (P_{\text{close}} - P_0)\times Q_{\text{closed}} \quad \text{paid on } T_{\text{prompt}}

An opposite trade for the same date fixes the result; the cash moves on the prompt date.

Worked example (9,410 − 9,258) × 4,000 = USD 608,000

Delivery
pays P0×Qopen against Qopen/25 warrants\displaystyle \text{pays } P_0 \times Q_{\text{open}} \text{ against } Q_{\text{open}}/25 \text{ warrants}

A position open on its prompt date settles by warrants against the contract price, delivery versus payment through LME Clear.

Worked example 9,258 × 1,000 = USD 9,258,000 for 40 warrants

The carry

Cash to three months
C3M=F3M−Scash\displaystyle C_{3M} = F_{3M} - S_{\text{cash}}

Positive in contango (three months above cash), negative in backwardation.

Worked example 9,252 − 9,180 = +72

Full carry
Cmax⁡=S r d360+u d\displaystyle C^{\max} = S\,r\,\frac{d}{360} + u\,d

The most the contango can reach: financing the metal and paying rent until the later date, or a merchant buys cash, sells three months and stores the metal.

uu
rent a tonne a day
dd
days between the dates

Worked example 9,180 × 4.90% × 90/360 + 0.60 × 90 = USD 166

Share of full carry
C3MCmax⁡\displaystyle \frac{C_{3M}}{C^{\max}}

A contango well inside full carry, or a backwardation, signals metal in demand near the front.

Worked example 72 / 166 = 43%

Implied convenience yield
y=rc+uS−1Tln⁡F3MScash\displaystyle y = r_c + \frac{u}{S} - \frac{1}{T}\ln\frac{F_{3M}}{S_{\text{cash}}}

The return to holding metal that the curve implies, with every rate continuous.

rcr_c
financing rate, continuous

Worked example 4.16% a year

Lending and borrowing
borrow: buy T1, sell T2lend: sell T1, buy T2\displaystyle \text{borrow: buy } T_1,\ \text{sell } T_2 \qquad \text{lend: sell } T_1,\ \text{buy } T_2

A carry moves a position from one prompt to another at the spread: the cable maker could roll its prompt by borrowing.

Margin and risk

Delta by prompt date
ΔT=∑i: Ti=TQi\displaystyle \Delta_{T} = \sum_{i:\,T_i = T} Q_i

Tonnes long or short on each prompt date; the outright sums them across dates.

Worked example +5,000 t on Jan 28, 2027

Contingent variation margin
CVMt=Q (Ft(T)−P0)\displaystyle CVM_t = Q\,(F_t(T) - P_0)

The mark to the prompt date’s price is collateralized each day and settled only on the prompt.

Worked example at 9,050: 5,000 × (9,050 − 9,258) = −USD 1,040,000

Value of a USD 100 move
Q×100\displaystyle Q \times 100

The scale of the margin calls.

Worked example 5,000 × 100 = USD 500,000

Initial margin
IM=Q×m\displaystyle IM = Q \times m

Set on the portfolio by LME Clear, passed on with the bank’s own terms.

mm
margin a tonne

Worked example 5,000 × 650 = USD 3,250,000

Average price swaps

Monthly average price
Pˉm=1nm∑dPdoff\displaystyle \bar{P}_m = \frac{1}{n_m}\sum_{d} P^{\text{off}}_d

Physical contracts and swaps settle on the average of the LME official cash price over the month.

PdoffP^{\text{off}}_d
official cash price on day d
Swap settlement
Xm=Qm (Pˉm−K)\displaystyle X_m = Q_m\,(\bar{P}_m - K)

The consumer receives the average and pays the fixed price, fixing its cost for the month.

Provisional pricing
final invoice=Q (PˉQP−Pˉprov)\displaystyle \text{final invoice} = Q\,\left(\bar{P}_{QP} - \bar{P}_{\text{prov}}\right)

A miner’s concentrates are invoiced on a provisional price and settled on the average of a later quotational period, the exposure the swaps hedge.

QPQP
quotational period

Market data of the worked example, illustrative: cash USD 9,180, three months USD 9,250 / 9,254; financing at 4.90% (SOFR plus 1.00%), warehouse rent USD 0.60 a tonne a day; 90 days from the cash date (Oct 30, 2026) to the three-month prompt.
Commodities›Agricultural Future

Formulas

Contract and margin

Notional
N×q×F0\displaystyle N \times q \times F_0

The value of the position at the fill.

qq
bushels a contract

Worked example 500 × 5,000 × 4.50 = USD 11,250,000

Value of a cent
N×q×0.01\displaystyle N \times q \times 0.01

Grain prices move in quarter cents; a cent a bushel is USD 50 a contract.

Worked example 500 × 5,000 × 0.01 = USD 25,000

Variation margin
VMt=N q (Ft−Ft−1)\displaystyle VM_t = N\,q\,(F_t - F_{t-1})

The change in the settlement price, paid in cash the next morning.

Worked example week to Nov 6: 500 × 5,000 × (4.6100 − 4.4600) = USD 375,000

Initial margin
IM=N×mhedge\displaystyle IM = N \times m_{\text{hedge}}

Commercial hedgers post the hedge rate, set at the maintenance level.

Worked example 500 × 1,500 = USD 750,000

The basis

Basis
bt=Stloc−Ft\displaystyle b_t = S^{\text{loc}}_t - F_t

The local cash price less the futures price: negative where grain must be shipped to the delivery points.

Worked example 4.42 − 4.72 = −0.30

Hedged purchase price
C=STloc−(FT−F0)=F0+bT\displaystyle C = S^{\text{loc}}_T - (F_T - F_0) = F_0 + b_T

A long hedge locks the futures price; the cost moves only with the basis at the close.

Worked example 4.42 − (4.72 − 4.50) = 4.50 − 0.30 = USD 4.20

Change in basis
Δb=bT−b0\displaystyle \Delta b = b_T - b_0

A long hedger gains when the basis weakens (falls) and loses when it strengthens; the short hedger the reverse.

Calendar spread and carry
F(T2)−F(T1)≤(s+S r d360)\displaystyle F(T_2) - F(T_1) \le \left(s + S\,r\,\tfrac{d}{360}\right)

A deferred month can sit above the nearby by at most the storage and financing until it: the full carry.

ss
storage a bushel for the period

Delivery

Invoice
I=q×(Fsettle+ploc)\displaystyle I = q \times \left(F_{\text{settle}} + p_{\text{loc}}\right)

The long pays the settlement price of the notice day, adjusted for the delivery location and grade.

plocp_{\text{loc}}
location premium or discount
Storage on a certificate
charges=q×sd×days\displaystyle \text{charges} = q \times s_d \times \text{days}

The holder of a shipping certificate pays the elevator’s storage rate from the last payment date.

sds_d
storage a bushel a day
Delivery dates
FND=last business day of (m−1)LTD=business day before the 15th\displaystyle \text{FND} = \text{last business day of } (m-1) \qquad \text{LTD} = \text{business day before the } 15^{\text{th}}

The first notice day and the last trading day of a CBOT grain contract for month m.

Worked example December 2026: first notice Nov 30, 2026, last trade Dec 14, 2026, last delivery Dec 16, 2026

Risk and limits

Delta
Δ=N q\displaystyle \Delta = N\,q

Bushels equivalent of the position.

Worked example 2,500,000 bushels

Move to the daily limit
VMmax⁡=N q L\displaystyle VM^{\max} = N\,q\,L

The largest variation margin of a day when the market closes at the limit.

LL
daily limit, USD a bushel
Futures gain at the close-out
Π=N q (FT−F0)\displaystyle \Pi = N\,q\,(F_T - F_0)

The sum of the variation margins paid to the long.

Worked example 2,500,000 × (4.72 − 4.50) = USD 550,000

Use of the position limit
u=∣N∣Lall\displaystyle u = \frac{|N|}{L_{\text{all}}}

The share of the all-months limit used, before any hedge exemption.

LallL_{\text{all}}
all-months limit

Worked example, illustrative: 500 CBOT December corn at USD 4.50; the settlements of the first days as in the cash flows; local basis 30 cents under December.
Commodities›Emission Allowance

Formulas

Settlement and compliance

Consideration
C=Q×P\displaystyle C = Q \times P

Paid by wire against the registry transfer, with no delivery versus payment.

Worked example 1,000,000 × 30.50 = USD 30,500,000

Annual surrender (California)
Sy=30%×Ey−1\displaystyle S_y = 30\% \times E_{y-1}

Each November 1 an emitter surrenders allowances for 30% of the previous year’s verified emissions.

Ey−1E_{y-1}
covered emissions of the prior year

Worked example 30% × 3,500,000 = 1,050,000 allowances by November 1, 2026

Final surrender
Sfinal=∑y∈periodEy−∑Sy\displaystyle S_{\text{final}} = \sum_{y \in \text{period}} E_y - \sum S_y

At the end of the three-year period the balance of the period’s emissions is surrendered.

Offset limit
O≤6%×obligation\displaystyle O \le 6\% \times \text{obligation}

Offsets may cover up to 6% of a California obligation from 2026, no more than half from projects outside the state.

Late surrender
owed=4×shortfall\displaystyle \text{owed} = 4 \times \text{shortfall}

California’s penalty obligation for allowances not surrendered on time; the EU charges EUR 100 a tonne, indexed, plus the allowances.

Forward price and carry

Cost of carry
F=S(1+r d360)\displaystyle F = S\left(1 + r\,\frac{d}{360}\right)

An allowance pays nothing and costs nothing to store, so its forward is spot financed to the delivery date.

rr
funding rate
dd
days from the spot date

Worked example 30.50 × (1 + 4.90% × 60/360) = USD 30.75

Curve across Decembers
F(T2)=F(T1) er (T2−T1)\displaystyle F(T_2) = F(T_1)\,e^{r\,(T_2 - T_1)}

Each December is the previous one carried another year.

Worked example 76.00 → 78.23 → 80.53 at 2.90%

Implied carry
rimpl=1T2−T1ln⁡F(T2)F(T1)\displaystyle r_{\text{impl}} = \frac{1}{T_2 - T_1}\ln\frac{F(T_2)}{F(T_1)}

The rate the curve implies; above the holders’ funding rate, buying the near date and selling the far one earns the difference.

Worked example a December 2027 at 79.50 against 76.00 would imply 4.51%, above the 2.90% funding: buy December 2026, sell December 2027

Bounds from banking
S≤F(T)≤S erT\displaystyle S \le F(T) \le S\,e^{rT}

With no convenience yield, a forward cannot sit above full carry; banking keeps spot and futures linked.

Compliance economics

Abate or buy
abate if MAC(q)<P\displaystyle \text{abate if } MAC(q) < P

An emitter reduces emissions while the marginal cost of a tonne avoided is below the allowance price.

MACMAC
marginal abatement cost
Fund price ceiling (Alberta)
c=min⁡(Pcredit, Pfund)\displaystyle c = \min(P_{\text{credit}},\, P_{\text{fund}})

Credits and offsets are worth buying only below the fund price.

Worked example (95 − 55) × 400,000 = CAD 16,000,000 saved

Clean spark spread
CSS=Ppower−h Pgas−e PCO2\displaystyle CSS = P_{\text{power}} - h\,P_{\text{gas}} - e\,P_{\text{CO}_2}

A gas plant’s margin per megawatt hour after fuel and allowances; the generator fixes all three together.

hh
heat rate
ee
tonnes of CO2 per megawatt hour
Free allocation and the CBAM
FA=B×HAL×CBAMy\displaystyle FA = B \times HAL \times CBAM_y

Free allowances follow a product benchmark and the historical activity, cut each year by the CBAM factor for the covered sectors.

BB
product benchmark
HALHAL
historical activity level
CBAMyCBAM_y
factor, 97.5% in 2026 falling to 0 in 2034

Risk and hedge

Delta
Δ=Q\displaystyle \Delta = Q

Tonnes long or short by delivery date.

Worked example the bank: short 2,000,000 for December, long 2,000 futures

Futures hedge
N=Q1,000\displaystyle N = \frac{Q}{1{,}000}

Allowance futures are 1,000 tonnes.

Worked example 2,000,000 / 1,000 = 2,000 contracts

Margin against the forward
VMt=1,000 N (Ft−Ft−1)\displaystyle VM_t = 1{,}000\,N\,(F_t - F_{t-1})

The futures margin moves daily; the forward with the emitter settles once on delivery, so the bank funds the gap.

Worked example USD 1 a tonne: USD 2,000,000

Exposure to the emitter
Et=Q max⁡(K−Ft, 0)\displaystyle E_t = Q\,\max(K - F_t,\,0)

The bank loses if the emitter defaults when the price has fallen below the forward price it agreed.

Worked examples of the four regimes, illustrative; forwards carried at the overnight rate plus 1.00% (4.90% in USD, 2.90% in EUR); ICE delivery dates from the exchange rules.
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