Formula Sheet

Long: S_T − F₀(T); short: F₀(T) − S_T

S_T

underlying price at expiry

F₀(T)

forward price agreed at the start for delivery at T

No premium changes hands at a fair forward price, so profit equals payoff.

Used in: A1 Derivative Payoffs, C1 Strategy Builder

Cash-and-carry profit = F(market) − F₀; reverse cash-and-carry profit = F₀ − F(market)

F(market)

forward price quoted in the market

F₀

no-arbitrage forward price from carry

Per unit at expiry: sell the rich forward and carry the asset, or buy the cheap forward and short the asset.

Used in: A2 Forward and Futures Pricing

F₀ = (S₀ − PV(income) + PV(costs)) × (1 + r)^T

F₀

no-arbitrage forward price

S₀

spot price today

PV(income)

present value of dividends or coupons paid before expiry

PV(costs)

present value of storage and insurance costs

r

risk-free rate, annual compounding

T

years to expiry

The Level I convention and the default in A2 and A4; Level II forward pricing uses it too.

Used in: A2 Forward and Futures Pricing, A4 Put–Call Parity and Replication

F₀ = (S₀ − PV(income) + PV(costs)) × e^(rT)

F₀

no-arbitrage forward price

S₀

spot price today

PV(income), PV(costs)

present values of discrete income and costs

r

risk-free rate, continuously compounded

T

years to expiry

Same carry argument with e^(rT) as the growth factor; the present values use e^(−rt) too.

Used in: A2 Forward and Futures Pricing, A4 Put–Call Parity and Replication

F₀ = (S₀ − PV(income) + PV(costs)) × growth × e^((c − q − y) × T)

growth

(1 + r)^T or e^(rT)

c

storage cost as a continuous yield

q

income yield: a dividend yield or a foreign interest rate

y

convenience yield of holding the physical asset

With no discrete items and continuous compounding this is F₀ = S₀ e^((r + c − q − y)T).

Used in: A2 Forward and Futures Pricing, D5 Commodity Swaps

Vₜ = (Fₜ − F₀) ÷ growth(T − t), Fₜ = (Sₜ − PV(remaining income) + PV(remaining costs)) × growth(T − t)

Vₜ

value to the long at time t (the short holds −Vₜ)

Fₜ

forward price today for the same delivery date

F₀

price agreed in the contract

growth(T − t)

(1 + r)^(T − t) or e^(r(T − t))

Equivalent to Sₜ − PV(remaining income) + PV(remaining costs) − F₀ ÷ growth(T − t).

Used in: A2 Forward and Futures Pricing

b = S − F

S

spot price of the asset being hedged

F

futures or forward price

Spot minus futures, the hedger's convention; it converges to zero at the futures' delivery date.

Used in: A2 Forward and Futures Pricing, C6 Hedging with Futures and Forwards, D5 Commodity Swaps

Gainₜ = ±units × (settlementₜ − settlementₜ₋₁); if balance < maintenance level: call = initial deposit − balance

±

+ for a long position, − for a short

units

contracts × contract size

settlementₜ

settlement price on day t (the entry price on day 1)

initial deposit

initial margin per contract × contracts

maintenance level

maintenance margin per contract × contracts

Gains and losses settle daily; a call restores the account to the initial level, not the maintenance level.

Used in: A6 Futures Margin and Mark-to-Market

FRA = [(1 + L₂ × t₂) ÷ (1 + L₁ × t₁) − 1] ÷ (t₂ − t₁)

L₁, L₂

money-market rates (MRR) to the start and to the end of the FRA period

t₁, t₂

days to start and to end ÷ day-count basis (360 or 365)

Simple money-market interest; the implied forward MRR for the FRA period.

Used in: A3 Forward Rate Agreements

Settlement to the long = Notional × (L − FRA) × period ÷ (1 + L × period)

L

MRR set on the settlement date for the FRA period

FRA

fixed rate agreed

period

length of the FRA period in years

The long pays fixed and receives floating; paid at the start of the period, so the interest difference is discounted at L.

Used in: A3 Forward Rate Agreements

Vₜ = Notional × (FRA new − FRA agreed) × period ÷ (1 + L₂(new) × t₂(new))

FRA new

FRA rate today for the same period

L₂(new)

today's MRR to the end of the period

t₂(new)

years from today to the end of the period

Value to the long; the short holds the negative.

Used in: A3 Forward Rate Agreements

F = S × (1 + i_p × d ÷ B_p) ÷ (1 + i_b × d ÷ B_b)

S, F

spot and forward rates, price currency per unit of base currency

i_p, i_b

price- and base-currency money-market rates

d

days to delivery

B_p, B_b

day-count bases (360 or 365) of each currency

The higher-rate currency trades at a forward discount. Continuous form: F = S e^((r_p − r_b)T), which A2 gives with the base-currency rate as the income yield q.

Used in: D4 FX Forwards and Swaps, A2 Forward and Futures Pricing

Points = (F − S) ÷ pip size; (F − S) ÷ S = (i_p τ_p − i_b τ_b) ÷ (1 + i_b τ_b)

pip size

0.0001 for most pairs, 0.01 for yen pairs

τ_p, τ_b

days ÷ basis in each currency

Positive points: the base currency trades at a forward premium.

Used in: D4 FX Forwards and Swaps

Vₜ = amount × (F_t − F₀) ÷ (1 + i_p × days left ÷ B_p)

amount

base-currency amount bought forward

F_t

forward rate today for the same delivery date

F₀

contract rate

i_p

price-currency rate for the remaining days

In price currency, to the party buying the base currency; the seller holds the negative.

Used in: D4 FX Forwards and Swaps

F₀ = S₀ e^((r − q)T)

S₀

index level today

r

continuously compounded risk-free rate

q

continuous dividend yield of the index

T

years to expiry

C2 values futures this way; a futures position agreed at F₀ is then worth F − F₀ and has delta e^((r − q)T).

Used in: A2 Forward and Futures Pricing, C2 Option Portfolio

QFP = [FV(S₀ + AI₀) − FV(coupons) − AI_T] ÷ CF

S₀

quoted (clean) bond price today

FV(S₀ + AI₀)

full price carried to delivery: × (1 + r)^T or × e^(rT)

FV(coupons)

coupons before delivery, carried to delivery

AI₀, AI_T

accrued interest today and at delivery

CF

conversion factor of the delivered bond

The bond's quoted forward price ÷ CF. A7 accrues by months: AI = (months since coupon ÷ months in period) × coupon per period.

Used in: A7 Bond Futures and Cheapest-to-Deliver

CF ≈ Σ (c ÷ 2) ÷ (1 + y ÷ 2)ᵏ + 1 ÷ (1 + y ÷ 2)ⁿ

c

the bond's annual coupon rate

y

the contract's notional yield (6%)

n

semiannual periods left, maturity rounded down to quarters

Price per 1 of par at the notional yield; an odd quarter is valued 3 months ahead, ÷ √(1 + y ÷ 2), less c ÷ 4.

Used in: A7 Bond Futures and Cheapest-to-Deliver

Invoice = (F × CF + AI_T) × par ÷ 100; cost of delivering = quoted price − F × CF

F

quoted futures price per 100 par

CF

conversion factor

AI_T

accrued interest at delivery per 100 par

The short picks the bond with the lowest cost of delivering: the cheapest to deliver.

Used in: A7 Bond Futures and Cheapest-to-Deliver

Implied repo = (invoice ÷ full price − 1) × 360 ÷ days

invoice

amount received for delivering the bond

full price

price paid for the bond today, accrued interest included

days

days to delivery

The return from buying the bond and delivering it; the bond with the highest implied repo is usually the CTD.

Used in: A7 Bond Futures and Cheapest-to-Deliver

Payoff = n × max(S_T − K, 0); long profit = payoff − n × c₀

n

number of options (payoff is negated for a written call)

S_T

underlying price at expiry

K

strike price

c₀

call premium paid at the start

At expiry only; break-even S_T* = K + c₀. The writer's profit is the negative: premium received less the payoff.

Used in: A1 Derivative Payoffs, C1 Strategy Builder

Payoff = n × max(K − S_T, 0); long profit = payoff − n × p₀

n

number of options (payoff is negated for a written put)

S_T

underlying price at expiry

K

strike price

p₀

put premium paid at the start

At expiry only; break-even S_T* = K − p₀. A long put's largest gain is K − p₀, reached if the underlying falls to zero.

Used in: A1 Derivative Payoffs, C1 Strategy Builder

max(0, S e^(−qT) − K e^(−rT)) ≤ c ≤ S e^(−qT); max(0, K e^(−rT) − S e^(−qT)) ≤ p ≤ K e^(−rT)

c, p

European call and put prices

S e^(−qT)

spot less the present value of the yield

K e^(−rT)

present value of the strike

Level I writes the same bounds with q = 0 and PV(K) = K ÷ (1 + r)^T; B2 checks the strict form (<), since a price on a bound fits no volatility.

Used in: B2 Implied Volatility and Smile

c + PV(K) = p + S − PV(D)

c, p

European call and put, same strike and expiry

PV(K)

K ÷ (1 + r)^T or K × e^(−rT)

PV(D)

present value of dividends paid before expiry

A fiduciary call equals a protective put; a gap between them is an arbitrage.

Used in: A4 Put–Call Parity and Replication

c + K e^(−rT) = p + S e^(−qT)

q

continuous dividend yield; for a currency option, the base-currency (foreign) rate

r

continuously compounded rate of the price (domestic) currency

BSM prices with a yield satisfy this exactly.

Used in: B1 Option Pricing and Greeks, B5 Greeks Visualizer

c − p = PV(F) − PV(K) = PV(F − K)

F

forward price for the options' expiry, (S − PV(D)) × growth

PV(·)

present value at the risk-free rate

Replaces S − PV(D) with PV(F) in put–call parity.

Used in: A4 Put–Call Parity and Replication

c − p = e^(−rT)(F − K)

F

futures price

c, p

European options on the futures, same strike and expiry

The Black-model form of parity.

Used in: B4 Black Model

h = (V⁺ − V⁻) ÷ (S⁺ − S⁻), S⁺ = uS, S⁻ = dS

V⁺, V⁻

option values after an up and a down move

S⁺, S⁻

underlying prices after an up and a down move

u, d

up and down factors

Shares per option that make the hedge riskless over one step; B3 reports it at the first node as delta.

Used in: A5 One-Period Binomial Model, B3 Binomial Tree

B = (V⁻ − h S⁻) ÷ (1 + r)^T; V = hS + B

h

hedge ratio (shares held)

B

bond position: negative means borrowing

r

risk-free rate, annual compounding

A call is replicated by buying shares and borrowing; a put by shorting shares and lending.

Used in: A5 One-Period Binomial Model

π = ((1 + r)^T − d) ÷ (u − d); V = [π V⁺ + (1 − π) V⁻] ÷ (1 + r)^T

π

risk-neutral probability of the up move

u, d

up and down factors

V⁺, V⁻

option payoffs after an up and a down move

No arbitrage requires d < (1 + r)^T < u; gives the same value as replication.

Used in: A5 One-Period Binomial Model

Δt = T ÷ n; u = e^(σ√Δt), d = 1 ÷ u; p = (e^((r − q)Δt) − d) ÷ (u − d)

n

number of steps

σ

volatility of the underlying

r, q

risk-free rate and dividend yield

p

risk-neutral probability of an up move per step

Cox–Ross–Rubinstein; with annual compounding B3 uses (1 + r)^Δt ÷ (1 + q)^Δt in place of e^((r − q)Δt).

Used in: B3 Binomial Tree

V = e^(−rΔt)[p V⁺ + (1 − p) V⁻]; American: V = max(exercise value, e^(−rΔt)[p V⁺ + (1 − p) V⁻])

V⁺, V⁻

values at the two following nodes

exercise value

max(S − K, 0) for a call, max(K − S, 0) for a put

Start from the payoffs max(S uʲ d^(n−j) − K, 0) at expiry and step back to today; under annual compounding B3 discounts each step by (1 + r)^(−Δt).

Used in: B3 Binomial Tree

d₁ = [ln(S ÷ K) + (r − q + σ² ÷ 2)T] ÷ (σ√T); d₂ = d₁ − σ√T

S

spot price

K

strike

r

continuously compounded risk-free rate

q

continuous dividend yield

σ

volatility

T

years to expiry

All rates continuous; N(d₂) is the risk-neutral probability that a call finishes in the money.

Used in: B1 Option Pricing and Greeks, B2 Implied Volatility and Smile, B5 Greeks Visualizer, C1 Strategy Builder, C2 Option Portfolio, C3 Delta and Gamma Hedging, C4 Delta-Hedge Simulator, C5 Scenario Analysis

c = S e^(−qT) N(d₁) − K e^(−rT) N(d₂)

N(·)

standard normal cumulative distribution

e^(−qT), e^(−rT)

yield and rate discount factors

European exercise; a long stock-and-borrowing replication with N(d₁) e^(−qT) shares.

Used in: B1 Option Pricing and Greeks, B2 Implied Volatility and Smile, B5 Greeks Visualizer, C1 Strategy Builder, C2 Option Portfolio, C3 Delta and Gamma Hedging, C4 Delta-Hedge Simulator, C5 Scenario Analysis

p = K e^(−rT) N(−d₂) − S e^(−qT) N(−d₁)

N(−d)

1 − N(d)

K e^(−rT)

present value of the strike

European exercise; consistent with put–call parity with a yield.

Used in: B1 Option Pricing and Greeks, B2 Implied Volatility and Smile, B5 Greeks Visualizer, C1 Strategy Builder, C2 Option Portfolio, C3 Delta and Gamma Hedging, C4 Delta-Hedge Simulator, C5 Scenario Analysis

c = e^(−rT)[F N(d₁) − K N(d₂)], p = e^(−rT)[K N(−d₂) − F N(−d₁)], d₁ = [ln(F ÷ K) + σ²T ÷ 2] ÷ (σ√T), d₂ = d₁ − σ√T

F

futures price

σ

volatility of the futures price

e^(−rT)

discount factor to expiry

BSM with the futures in place of the spot and q = r: the futures needs no financing.

Used in: B4 Black Model

Caplet = N × τ × DF(payment) × [F N(d₁) − K N(d₂)]; floorlet = N × τ × DF(payment) × [K N(−d₂) − F N(−d₁)]

N

notional

τ

accrual period of the caplet in years

F

forward MRR for the period

K

cap or floor rate

T

time to the reset date (in d₁ and d₂)

A cap is the sum of its caplets; cap − floor = N × Σ τᵢ DFᵢ (Fᵢ − K), a payer swap at K.

Used in: B4 Black Model

Payer = N × AF × [F N(d₁) − K N(d₂)]; receiver = N × AF × [K N(−d₂) − F N(−d₁)]; F = (DF_start − DF_end) ÷ AF

AF

annuity factor Σ τᵢ DFᵢ over the underlying swap's payment dates

F

forward swap rate

K

swaption strike rate

T

time to the swaption's expiry (in d₁ and d₂)

Payer − receiver = N × AF × (F − K), a forward-starting payer swap.

Used in: B4 Black Model

Call: Δ = e^(−qT) N(d₁); put: Δ = e^(−qT) [N(d₁) − 1]

Δ

change in option value per 1 change in the underlying

Per option held long; a written option has the opposite sign.

Used in: B1 Option Pricing and Greeks, B5 Greeks Visualizer, C1 Strategy Builder, C2 Option Portfolio, C3 Delta and Gamma Hedging, C4 Delta-Hedge Simulator, C5 Scenario Analysis

Γ = e^(−qT) n(d₁) ÷ (S σ √T)

n(·)

standard normal density

Γ

change in delta per 1 change in the underlying

Same for calls and puts; largest near the money and close to expiry.

Used in: B1 Option Pricing and Greeks, B5 Greeks Visualizer, C2 Option Portfolio, C3 Delta and Gamma Hedging, C5 Scenario Analysis

ν = S e^(−qT) n(d₁) √T, ÷ 100 per percentage point

ν

change in option value per unit change in volatility

Same for calls and puts; the tools show it per 1 percentage point of volatility, except C4's hedging-error rule of thumb, which uses ν per 1.00 of volatility.

Used in: B1 Option Pricing and Greeks, B2 Implied Volatility and Smile, B5 Greeks Visualizer, C2 Option Portfolio, C3 Delta and Gamma Hedging, C4 Delta-Hedge Simulator

Call: Θ = −S e^(−qT) n(d₁) σ ÷ (2√T) − r K e^(−rT) N(d₂) + q S e^(−qT) N(d₁); put: Θ = −S e^(−qT) n(d₁) σ ÷ (2√T) + r K e^(−rT) N(−d₂) − q S e^(−qT) N(−d₁); ÷ 365 per day

Θ

change in option value as time passes, per year

Usually negative for long options; the tools show it per calendar day.

Used in: B1 Option Pricing and Greeks, B5 Greeks Visualizer, C2 Option Portfolio, C5 Scenario Analysis

Call: ρ = K T e^(−rT) N(d₂); put: ρ = −K T e^(−rT) N(−d₂); ÷ 100 per percentage point

ρ

change in option value per unit change in the risk-free rate

Calls gain and puts lose when rates rise; shown per 1 percentage point of the rate.

Used in: B1 Option Pricing and Greeks, B5 Greeks Visualizer, C2 Option Portfolio

Δ = Σ nᵢ Δᵢ, Γ = Σ nᵢ Γᵢ, ν = Σ nᵢ νᵢ

nᵢ

quantity of position i: negative when written or short

Δᵢ, Γᵢ, νᵢ

Greeks per unit; a share has Δ = 1 and Γ = 0

Greeks add across positions; dollar delta is Δ × S.

Used in: C1 Strategy Builder, C2 Option Portfolio, C3 Delta and Gamma Hedging, C5 Scenario Analysis

ΔV ≈ Δ ΔS + ½ Γ ΔS² + ν Δσ

ΔS

change in the underlying price

Δσ

change in volatility (in the units vega is quoted in)

A Taylor estimate; full revaluation V(S + ΔS, σ + Δσ) − V(S, σ) is exact, and the gap grows with the move.

Used in: C2 Option Portfolio, C3 Delta and Gamma Hedging, C5 Scenario Analysis

Find σ such that BSM(S, K, T, r, q, σ) = market price

σ

implied volatility

market price

observed option premium

Exactly one σ fits because price rises with volatility (vega > 0), provided the price lies inside the no-arbitrage bounds.

Used in: B2 Implied Volatility and Smile

Covered call: max profit = K − S₀ + c₀, break-even = S₀ − c₀; protective put: max loss = S₀ + p₀ − K, break-even = S₀ + p₀

S₀

price paid for the underlying

K

strike of the option

c₀

premium received for the written call

p₀

premium paid for the put

One share and one option, held to expiry.

Used in: C1 Strategy Builder

Profit(S_T) = Σ ±n × [payoff per unit(S_T) − premium]

±n

quantity of each leg: + long, − short or written

payoff per unit

max(S_T − K, 0), max(K − S_T, 0), S_T − F or S_T for each leg

premium

price paid per unit for the leg

Profit is piecewise linear in S_T, so break-evens and extremes are found between and at the strikes.

Used in: C1 Strategy Builder

N_S = −(Δ_B + Σ nᵢΔᵢ)

N_S

shares to trade (negative: sell)

Δ_B

delta of the book before hedging

nᵢ, Δᵢ

quantities and deltas of hedging options traded

With no hedging options N_S = −Δ_B. After a move ΔS the hedged delta is about Γ ΔS, so trade −Γ ΔS shares to rebalance.

Used in: C3 Delta and Gamma Hedging, C4 Delta-Hedge Simulator

Written: P&L = e^(−rT) [cash_T + shares_T × S_T − payoff(S_T)] × options; long: P&L = e^(−rT) [cash_T + shares_T × S_T + payoff(S_T)] × options

cash_T

premium (written) less share purchases; each step cash ← cash × e^(rΔt) + shares × S × (e^(qΔt) − 1), so dividends on the shares held are included

shares_T

shares held after the last rebalance: +Δ per option written, −Δ per option bought, Δ at σ_imp

payoff(S_T)

the option's payoff at expiry

Discounted to today. Mean near zero when realised volatility equals implied; its spread falls roughly like 1 ÷ √(rebalances).

Used in: C4 Delta-Hedge Simulator

std ≈ √(π ÷ 4) × ν × σ_real ÷ √N × options; mean (written) ≈ [BSM(σ_imp) − BSM(σ_real)] × options

ν

BSM vega per 1.00 of volatility (not per percentage point)

σ_real, σ_imp

realised volatility of the stock and implied volatility used to price and hedge

N

number of rebalances over the option's life

The volatility-gap mean is exact in expectation for continuous hedging when μ = r; a long hedged option has the opposite sign, so the writer gains when σ_real < σ_imp.

Used in: C4 Delta-Hedge Simulator

n₁ = −Γ_B ÷ Γ₁, then N_S = −(Δ_B + n₁Δ₁)

Γ_B

gamma of the book

Γ₁, Δ₁

gamma and delta of the hedging option

n₁

options to trade

Shares have no gamma, so gamma is fixed with options first and delta with shares last.

Used in: C3 Delta and Gamma Hedging

n₁ = (−Γ_B ν₂ + ν_B Γ₂) ÷ D, n₂ = (−ν_B Γ₁ + Γ_B ν₁) ÷ D, D = Γ₁ν₂ − Γ₂ν₁

Γ_B, ν_B

gamma and vega of the book

Γᵢ, νᵢ

gamma and vega of hedging option i

D

determinant; must not be zero

Solves n₁Γ₁ + n₂Γ₂ = −Γ_B and n₁ν₁ + n₂ν₂ = −ν_B; shares then restore delta neutrality.

Used in: C3 Delta and Gamma Hedging

h* = ρ × σ_S ÷ σ_F; hedge effectiveness = ρ²

ρ

correlation of spot and futures price changes

σ_S, σ_F

standard deviations of spot and futures price changes over the hedge

For a cross hedge, where the futures' underlying differs from the asset.

Used in: C6 Hedging with Futures and Forwards

N* = h* × Q_A ÷ Q_F; tailed: N* = h* × V_A ÷ V_F, V_A = S₁ × Q_A, V_F = F₁ × Q_F

Q_A

units of the asset exposed

Q_F

units of the underlying per contract

V_A

value of the exposure

V_F

value of one futures contract

Tailing adjusts for daily settlement, with h* from one-day percentage changes; round to whole contracts in practice.

Used in: C6 Hedging with Futures and Forwards

Effective price = S₂ + h × (F₁ − F₂), h = N × Q_F ÷ Q_A

S₂, F₂

spot and futures prices when the hedge is closed

F₁

futures price when the hedge is put on

With h = 1 this is F₁ + b₂: the hedge leaves only basis risk.

Used in: C6 Hedging with Futures and Forwards

N = (β* − β) × V_A ÷ (F × multiplier)

β, β*

current and target beta

V_A

portfolio value

F × multiplier

value of one index futures contract

β* = 0 hedges market risk fully; negative N means sell contracts.

Used in: C6 Hedging with Futures and Forwards

Fᵢ = (DFᵢ₋₁ ÷ DFᵢ − 1) ÷ τᵢ

DFᵢ₋₁, DFᵢ

discount factors to the start (reset) and end (payment) of period i; DF₀ = 1

τᵢ

accrual period in years

The FRA strip behind a swap, cap or floor.

Used in: B4 Black Model, D1 Interest Rate Swaps, D2 Currency Swaps

DF₀.₅ = 1 ÷ (1 + c₀.₅ ÷ 2); DF₁ = (1 − c₁ ÷ 2 × DF₀.₅) ÷ (1 + c₁ ÷ 2); S₁ = 1 ÷ DF₁ − 1; f(1,2) = DF₁ ÷ DF₂ − 1; r꜀ = ln(1 + S₁)

cₜ

par yield (coupon rate) for maturity t, paid semiannually

DFₜ

discount factor for t years

S₁

one-year spot rate, annual compounding

f(1,2)

one-year rate one year ahead

r꜀

continuously compounded equivalent rate

Use r꜀ in BSM, Black and trees; Level I forward and parity problems use the annual rate.

Used in: E1 Market Monitor

DFᵢ = (1 + zᵢ)^(−tᵢ)

zᵢ

spot (zero) rate to date i, annual compounding

tᵢ

years to date i

The swap tools and D6's flat-rate option use annual compounding; D5's flat rate uses e^(−r tᵢ).

Used in: D1 Interest Rate Swaps, D2 Currency Swaps, D3 Equity and Total Return Swaps, D5 Commodity Swaps, D6 CDS Pricer

V(payer) = N × Σ (fᵢ − c) × τᵢ × DFᵢ

fᵢ

forward MRR for period i, (DFᵢ₋₁ ÷ DFᵢ − 1) ÷ τᵢ

c

fixed rate of the swap

A swap is a strip of FRAs; with known first fixing f₁ is that MRR.

Used in: D1 Interest Rate Swaps

c = (1 − DFₙ) ÷ Σ τᵢ DFᵢ

c

fixed rate that gives the swap zero value

DFₙ

discount factor to the last payment

Σ τᵢ DFᵢ

annuity A of the fixed leg

On a reset date; between resets D1 uses (floating leg − DFₙ) ÷ A.

Used in: D1 Interest Rate Swaps, D2 Currency Swaps

V(payer) = N × (B_float − B_fixed), B_fixed = c × A + DFₙ, B_float = (1 + MRR × τ₁) × DF₁

c

fixed rate of the swap

A

annuity Σ τᵢ DFᵢ

MRR

floating rate already set for the current period

B_float

1 per 1 of notional on a reset date

Pay fixed = long a floating-rate bond, short a fixed-rate bond; the receiver holds the negative.

Used in: D1 Interest Rate Swaps

V = B(dom) − S × B(for), B = N × [c × Σ(τ × DF) + DFₙ] for a fixed leg

B(dom), B(for)

values of the domestic and foreign legs, each in its own currency

S

spot rate, domestic per unit of foreign

N

notional of each leg

Value to the party receiving domestic; notionals are exchanged at the end, and a floating leg is worth N on a reset date.

Used in: D2 Currency Swaps

V(receive equity) = N × S_t ÷ S_last − other leg; fixed leg = N × (c × Σ τᵢ DFᵢ + DFₙ)

S_t

index level today

S_last

index level at the last settlement

other leg

fixed, floating or another equity leg, valued like a bond

A total-return leg adds dividends gone ex: N × (S_t + D × DF) ÷ S_last.

Used in: D3 Equity and Total Return Swaps

Net to the equity receiver = N × (S_T ÷ S_last − 1) − N × c × τ₁

S_T

index level on the settlement date

c

fixed rate (MRR + spread for a floating leg)

τ₁

settlement period in years

A negative equity return is paid by the equity receiver on top of the fixed leg.

Used in: D3 Equity and Total Return Swaps

K* = Σ Qᵢ × DFᵢ × Fᵢ ÷ Σ Qᵢ × DFᵢ

K*

fixed price per unit that gives the swap zero value

Qᵢ

quantity settled in period i

Fᵢ

forward or futures price for settlement date i

DFᵢ

discount factor to settlement date i

A Q × DF-weighted average of the forwards: below the near forward in backwardation, above it in contango.

Used in: D5 Commodity Swaps

V(pay fixed) = Σ Qᵢ × DFᵢ × (Fᵢ − K) = (K* − K) × Σ Qᵢ × DFᵢ; net to the fixed payer = Q × (P − K)

K

fixed price in the swap

P

realised average floating price for the period

Σ Qᵢ × DFᵢ

value change for a parallel 1.00 move in the forward curve

The fixed payer receives floating; the fixed receiver holds the negative.

Used in: D5 Commodity Swaps

POSₜ = POSₜ₋₁ × (1 − h) = (1 − h)^t; PODₜ = POSₜ₋₁ × h; ELₜ = Notional × (1 − R) × PODₜ

h

hazard rate: probability of default in a year, given survival to its start

POSₜ

probability of surviving to year t

PODₜ

probability of defaulting in year t

R

recovery rate; 1 − R is the loss given default

ELₜ

expected loss in year t, undiscounted

A constant annual hazard rate, as in D6.

Used in: D6 CDS Pricer

PV(protection) = Σ (1 − R) × PODₜ × DFₜ; RPV01 = Σ POSₜ × DFₜ

DFₜ

risk-free discount factor to year t

RPV01

PV of 1 per year of premium paid while the name survives

Per 1 of notional; premiums at year end with no accrued premium on default.

Used in: D6 CDS Pricer

s = PV(protection) ÷ RPV01 = h × (1 − R) ÷ (1 − h) ≈ h × (1 − R)

s

spread that makes the CDS worth zero at inception

h

constant annual hazard rate

R

recovery rate

Exact in D6's convention (annual premiums at year end, constant hazard rate); the credit-triangle shortcut h × (1 − R) is low by the factor 1 − h (1.20% against 1.2245% at h = 2%, R = 40%).

Used in: D6 CDS Pricer

Upfront = PV(protection) − c × RPV01 ≈ (s − c) × RPV01; price per 100 = 100 − upfront (%)

c

standard coupon of the contract (e.g. 1% or 5%)

s

par spread

Paid by the protection buyer when s > c, received when s < c. A 1 bp spread move changes the value by about RPV01 × Notional × 0.0001.

Used in: D6 CDS Pricer