Derivatives formula sheet.

Where $S_T$ = underlying price at maturity, $F_0(T)$ = forward price agreed at inception; multiply by the contract size.

A forward is a zero-sum firm commitment: the long gains exactly what the short loses. Because nothing is paid at inception, the payoff is also the profit, unlike an option where the premium must be deducted.

Exam tip The long gains when the underlying ends above the agreed forward price, not above today's spot price.

Lesson: Forward contracts and firm commitments →All Forward Commitment and Contingent Claim Features and Instruments formulas

Where $\Pi$ = profit at expiration, $S_T$ = underlying price at expiration, $X$ = exercise price, $c_0$ = premium paid.

The buyer's loss is capped at the premium while the upside is unlimited. Breakeven is $S_T = X + c_0$. The short call's profit is the exact negative: maximum gain $c_0$, unlimited loss.

Exam tip Payoff ignores the premium, profit deducts it; read whether the question asks for payoff (value at expiration) or profit.

Lesson: Options and call payoffs →All Forward Commitment and Contingent Claim Features and Instruments formulas

Where $S_t$ = underlying price at time $t$, $X$ = exercise price, $c_t$ = market price of the call at $t$.

Intrinsic value measures moneyness: a call is in the money when $S_t > X$ (a put when $S_t < X$), at the money when they are equal. Everything else in the premium is time value, which decays to zero at expiration.

Exam tip An out-of-the-money or at-the-money option has zero intrinsic value, so its whole price is time value; intrinsic value is never negative.

Lesson: Options and call payoffs →All Forward Commitment and Contingent Claim Features and Instruments formulas

Where $\Pi$ = profit at expiration, $X$ = exercise price, $S_T$ = underlying price at expiration, $p_0$ = premium paid.

The put buyer profits when the underlying falls below $X - p_0$, the breakeven. Maximum gain is $X - p_0$ (underlying falls to zero) and maximum loss is the premium. The put seller's profit is the negative: gain capped at $p_0$, loss up to $X - p_0$.

Exam tip Unlike a long call, a long put's gain is capped because the underlying cannot fall below zero.

Lesson: Put payoffs and the four option positions →All Forward Commitment and Contingent Claim Features and Instruments formulas

Where $S_0$ = spot price, $PV_0(I)$ = present value of income (dividends, coupons) before $T$, $PV_0(C)$ = present value of carrying costs (storage, insurance), $r$ = risk-free rate, $T$ = years.

The no-arbitrage forward price is the cost of buying the underlying today and carrying it to $T$. Income lowers the forward price because the forward holder does not receive it; costs raise it because the forward holder avoids them. With neither, $F_0(T) = S_0(1+r)^T$.

Exam tip Subtract income and add costs, each discounted from its own payment date; only cash flows that occur before delivery count.

Lesson: The cost of carry: income and costs of owning the underlying →All Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives formulas

Where $S_0$ = spot price, $r$ = risk-free rate, $c$ = storage cost rate, $y$ = convenience yield (or income rate such as an index dividend yield), all continuously compounded; $T$ = years.

The same carry logic in continuous form, used for equity indexes, commodities and currencies. Costs add to the exponent, benefits subtract. A large convenience yield can push the forward below spot (backwardation).

Exam tip Convenience yield is a benefit of holding the physical commodity, so it reduces the forward price just like a dividend yield.

Lesson: Commodities, convenience yield and carry by asset class →All Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives formulas

Where $S_{0,f/d}$, $F_{0,f/d}(T)$ = spot and forward rates in units of $f$ (price currency) per unit of $d$ (base currency); $r_f$, $r_d$ = their continuously compounded risk-free rates; $T$ = years.

Carry for a currency is the interest rate differential. If the price currency's rate exceeds the base currency's, the forward rate is above spot, so the base currency trades at a forward premium.

Exam tip The price-currency (numerator) rate comes first in the exponent; the currency with the higher interest rate trades at a forward discount.

Lesson: FX forwards: carry as an interest rate differential →All Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives formulas

Where $S_t$ = spot price at $t$, $PV_t(I)$, $PV_t(C)$ = value at $t$ of income and costs still to come before $T$, $F_0(T)$ = forward price agreed at inception, $T - t$ = years remaining.

A forward is worth zero at inception, then its value moves with the underlying. Mid-life, the long's value is the carry-adjusted spot price minus the present value of the price it must pay. The short's value is the negative; at expiration it reduces to $S_T - F_0(T)$.

Exam tip Discount the original forward price only over the remaining $T - t$ years, and ignore income or costs already paid before $t$.

Lesson: Forwards on assets with costs and benefits →All Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities formulas

Where $z_A$, $z_B$ = zero (spot) rates to times $A$ and $B$, $IFR_{A,B-A}$ = implied forward rate for the period of length $B - A$ starting at $A$; rates per period.

Investing to $A$ and rolling over at the forward rate must give the same result as investing straight to $B$. The implied forward rate is the no-arbitrage fixed rate of an FRA for that period and the expected floating rate used to price swaps.

Exam tip Solve as $IFR = \left[(1+z_B)^B/(1+z_A)^A\right]^{1/(B-A)} - 1$; forgetting the root over the forward period's length is the usual slip.

Lesson: Implied forward rates →All Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities formulas

Where $f_t(T)$ = settlement price on day $t$, $f_0(T)$ = price at which the long entered, Initial and Maintenance = margin levels per contract, Contract size = units of underlying per contract.

Futures are marked to market daily, so losses come out of the margin account each day. A long gets a margin call once cumulative losses exceed the cushion between initial and maintenance margin. For a short, the trigger price is above $f_0(T)$ by the same amount.

Exam tip After a margin call the account must be topped back up to the initial margin, not just to the maintenance margin.

Lesson: Mark-to-market: futures versus forwards →All Pricing and Valuation of Futures Contracts formulas

Where $BPV$ = gain or loss per contract for a 1 bp change in the rate, Notional = contract size, Period = fraction of a year of the underlying rate (0.25 for three months).

Interest rate futures are quoted as $100 - (100 \times MRR)$, so the price falls when rates rise. The gain or loss is $BPV$ times the basis point change and is linear in rates, the same per basis point in either direction.

Exam tip A borrower worried about rising rates sells (goes short) the futures, because a rate rise lowers the futures price.

Lesson: Short-term interest rate futures versus FRAs →All Pricing and Valuation of Futures Contracts formulas

Where $F_0(T)$ = forward price, $f_0(T)$ = futures price, $\operatorname{corr}(f, r)$ = correlation between futures prices and interest rates.

Daily settlement only matters if rates move with the futures price. With positive correlation, gains are reinvested at higher rates and losses financed at lower ones, so the long prefers futures and $f_0(T) > F_0(T)$; with negative correlation $F_0(T) > f_0(T)$.

Exam tip This is a classic conceptual question: positive correlation makes the futures price higher, negative makes the forward price higher.

Lesson: Why forward and futures prices can differ →All Pricing and Valuation of Futures Contracts formulas

Where $MRR_{B-A}$ = market reference rate set at $A$ for the period to $B$, $IFR_{A,B-A}$ = FRA fixed rate, Period = accrual fraction (e.g. days/360).

The pay-fixed (long) FRA side receives when the reference rate sets above the FRA rate. The interest difference relates to a period ending at $B$, so when the FRA settles at $A$ it is discounted over that period at the reference rate that just set.

Exam tip Discount at the newly set MRR, not the FRA rate; the receive-fixed side gets the same amount with the opposite sign.

Lesson: Convexity bias and the effect of central clearing →All Pricing and Valuation of Futures Contracts formulas

Where $s_N$ = fixed swap rate for $N$ periods, $DF_i = 1/(1+z_i)^i$ = discount factor for payment date $i$, $DF_N$ = last discount factor.

The swap rate is the fixed rate that makes the present value of fixed payments equal that of the expected floating payments, so the swap is worth zero at inception. It equals the coupon of a fixed-rate bond priced at par off the same zero curve.

Exam tip Use one discount factor per payment date; with semiannual periods the result is a per-period rate, so annualise it by multiplying by 2.

Lesson: Pricing a swap: the par swap rate →All Pricing and Valuation of Interest Rates and Other Swaps formulas

Where $MRR$ = reference rate set at the start of the period, $s_N$ = fixed swap rate, Period = accrual fraction (0.5 semiannual, 0.25 quarterly).

Each swap payment date is like a forward: only the net amount changes hands. A positive result is paid to the fixed-rate payer; a negative one is paid by it. The fixed-rate receiver's settlement is $(s_N - MRR) \times$ Notional $\times$ Period.

Exam tip Use the MRR that was set at the beginning of the period (set in advance, paid in arrears), and remember to multiply the annual rate gap by the period fraction.

Lesson: Swap price versus swap value →All Pricing and Valuation of Interest Rates and Other Swaps formulas

Where $s_{\text{new}}$ = today's par swap rate for the remaining maturity, $s_N$ = original fixed rate, $DF_i^{\text{new}}$ = today's discount factors for the remaining payment dates.

An existing swap is valued as the present value of the gap between today's market swap rate and the locked-in rate. When rates rise, the fixed-rate payer gains because it keeps paying the old, lower rate; the fixed-rate receiver's value is the negative.

Exam tip Sum discount factors only over the remaining payment dates, and check the sign: the fixed payer gains when $s_{\text{new}} > s_N$.

Lesson: How swap values change: time, rates and credit exposure →All Pricing and Valuation of Interest Rates and Other Swaps formulas

Where $c_t$ = European call price at $t$, $S_t$ = underlying price, $X$ = exercise price, $r$ = risk-free rate, $T - t$ = years to expiration.

The lower bound is the call's exercise value, the spot price minus the present value of the exercise price. No one pays more than the underlying itself for the right to buy it. A price outside these bounds offers an arbitrage.

Exam tip Discount the exercise price in the lower bound; using $S_t - X$ understates it, and the bound is zero for an out-of-the-money call.

Lesson: Arbitrage: forwards versus options, and option price bounds →All Pricing and Valuation of Options formulas

Where $p_t$ = European put price at $t$, $X$ = exercise price, $S_t$ = underlying price, $r$ = risk-free rate, $T - t$ = years to expiration.

The lower bound is the put's exercise value, the present value of the exercise price minus the spot price. The put can never be worth more than the exercise price, its payoff when the underlying falls to zero.

Exam tip Call and put bounds mirror each other: the call is capped by the underlying price, the put by the exercise price.

Lesson: Arbitrage: forwards versus options, and option price bounds →All Pricing and Valuation of Options formulas

Where $S_0$ = underlying price, $p_0$, $c_0$ = European put and call prices with the same $X$ and $T$, $X(1+r)^{-T}$ = price of a risk-free zero paying $X$ at $T$.

A protective put (share plus put) and a fiduciary call (call plus bond) both pay $\max(S_T, X)$, so they must cost the same. Rearrange it to price one option from the other or to build synthetic positions, e.g. a synthetic put is a long call, long bond and short share.

Exam tip It holds only for European options on the same underlying with the same exercise price and expiration; a sign error when rearranging is the usual mistake.

Lesson: Fiduciary call, protective put and put–call parity →All Option Replication Using Put–Call Parity formulas

Where $F_0(T)$ = forward price for delivery at $T$, $p_0$, $c_0$ = European put and call prices with the same $X$ and $T$, $r$ = risk-free rate.

A long forward plus a bond paying $F_0(T)$ replicates the underlying, so the present value of the forward price replaces $S_0$ in put-call parity. It follows that $c_0 - p_0$ equals the present value of $F_0(T) - X$.

Exam tip If the exercise price equals the forward price, the call and the put must have the same price.

Lesson: Put–call forward parity →All Option Replication Using Put–Call Parity formulas

Where $\pi$ = risk-neutral probability of an up move, $R^u = S_1^u/S_0$, $R^d = S_1^d/S_0$ = gross up and down returns, $r$ = risk-free rate for the period.

The probability that makes the underlying's expected return equal the risk-free rate. It comes from no-arbitrage, not from investors' beliefs, so the real-world probabilities and the expected return of the underlying play no role in the option value.

Exam tip If the period is $T$ years with an annual rate, use $(1+r)^T$ in place of $1 + r$; check that $\pi S_1^u + (1-\pi)S_1^d = S_0(1+r)$.

Lesson: Risk neutrality and risk-neutral probabilities →All Valuing a Derivative Using a One-Period Binomial Model formulas

Where $c_1^u$, $c_1^d$, $p_1^u$, $p_1^d$ = option payoffs after an up or down move, $\pi$ = risk-neutral probability, $r$ = risk-free rate, $T$ = years.

The option is worth its risk-neutral expected payoff discounted at the risk-free rate. The answer equals the cost of the riskless hedge built with the hedge ratio, and higher volatility (wider up and down moves) raises the value of both calls and puts.

Exam tip Compute the payoffs at the end of the period first, then weight them by $\pi$ and discount at the risk-free rate, never at an expected or risk-adjusted return.

Lesson: Risk neutrality and risk-neutral probabilities →All Valuing a Derivative Using a One-Period Binomial Model formulas