IFoA SP6 flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Put-call parity

For European options on a non-dividend stock: c + Ke^{-rT} = p + S0. Any deviation allows arbitrage by trading the option, stock and bond to lock in a riskless profit.

Risk-neutral measure

A probability measure under which the discounted asset price is a martingale. Derivative prices equal the discounted expected payoff under this measure, not under real-world probabilities.

Black-Scholes d1

d1 = [ln(S/K) + (r - q + 0.5σ²)T]/(σ√T). It appears in both call and put formulas and in the delta and gamma expressions for European options.

Black-Scholes d2

d2 = d1 - σ√T. N(d2) is the risk-neutral probability the call finishes in the money; Ke^{-rT}N(d2) is the present value of the strike paid on exercise.

Delta of a call

For a European call on a dividend-paying stock, delta = e^{-qT}N(d1). It ranges between 0 and e^{-qT} and gives the number of shares needed to hedge one option.

Gamma

Gamma = e^{-qT}n(d1)/(Sσ√T) for a European option. It measures the change in delta per unit change in the underlying and is largest for at-the-money near-dated options.

Vega

Vega = Se^{-qT}n(d1)√T. It measures sensitivity of the option price to a one-unit change in volatility and is largest for at-the-money long-dated options.

Implied volatility

The volatility input that, when substituted into a pricing model, reproduces the observed market option price. It varies across strikes and maturities, producing the smile or skew.

Binomial CRR parameters

u = e^{σ√Δt}, d = 1/u, p = (e^{(r-q)Δt} - d)/(u - d). These ensure the tree matches the mean and variance of the lognormal process in the limit.

Early exercise of American call

An American call on a non-dividend-paying stock is never optimally exercised before expiry; its value equals the European call. With dividends, early exercise can be optimal just before ex-dividend.

Forward price with income

F0 = (S0 - I)e^{rT} when the underlying pays known income with present value I. For a continuous dividend yield q, F0 = S0e^{(r-q)T}.

Value of a forward

For a long forward struck at K, value = (F0 - K)e^{-rT}, where F0 is the current forward price for the same maturity. At inception the value is zero.

Futures convexity adjustment

Futures prices differ from forward prices because of daily settlement. When the underlying is positively correlated with interest rates, the futures price exceeds the forward price.

Black model

Prices options on futures: c = e^{-rT}[F0N(d1) - KN(d2)] with d1 = [ln(F0/K) + 0.5σ²T]/(σ√T) and d2 = d1 - σ√T. Used for caps, floors and swaptions.

Garman-Kohlhagen model

Prices European currency options by treating the foreign interest rate rf as a continuous dividend yield: call = Se^{-rf T}N(d1) - Ke^{-rT}N(d2).

Interest rate swap value

Value to the fixed-rate payer equals the PV of the floating leg minus the PV of the fixed leg. At inception the fixed rate is the par swap rate and value is zero.

Forward rate agreement

An FRA fixes an interest rate for a future period. Its value is the notional times the rate difference times the accrual factor, discounted to the valuation date.

Credit default swap spread

The CDS spread approximately equals the loss given default (1 - recovery) times the hazard rate. The premium leg is paid while the reference entity survives.

Barrier option

An option that is activated (knock-in) or extinguished (knock-out) when the underlying crosses a specified barrier. Knock-outs are cheaper than vanillas but carry discontinuous delta at the barrier.

Asian option

An option whose payoff depends on the average underlying price over the option life. Averaging reduces effective volatility, so Asian options are cheaper than comparable European options.

Variance swap

A contract paying the difference between realised variance and a fixed strike variance. The fair strike equals the risk-neutral expected realised variance and can be replicated with a portfolio of options.

Delta hedging P&L

For a delta-hedged option over a small interval, P&L ≈ 0.5Γ(ΔS)² - 0.5σ²S²ΓΔt. Gains from realised moves are offset by time decay of gamma, giving the gamma-theta trade-off.

Longstaff-Schwartz method

A least-squares Monte Carlo technique for American options. At each exercise date, regress continuation values on basis functions of the state to decide whether to exercise early.

Finite difference method

Solves the Black-Scholes PDE numerically on a grid of asset prices and times. It handles early exercise and barrier conditions well but scales poorly to high-dimensional problems.

Heston stochastic volatility

A model where variance follows a mean-reverting square-root process correlated with the asset. It generates volatility smiles and skews that pure Black-Scholes cannot produce.

Merton jump diffusion

Extends Black-Scholes by adding Poisson jumps to the asset price. It produces fat tails and short-dated skew, improving calibration to observed option smiles.

Option price bounds

For European options: c ≥ max(0, Se^{-qT} - Ke^{-rT}) and p ≥ max(0, Ke^{-rT} - Se^{-qT}). Violations create arbitrage opportunities using the underlying and a risk-free bond.

Monte Carlo standard error

The standard error of a Monte Carlo price estimate is proportional to 1/√n, where n is the number of simulated paths. Antithetic variates and control variates reduce variance.

Swaption

An option to enter an interest rate swap at a future date. Priced with the Black model on the forward swap rate or with a short-rate or LIBOR market model.

Caps and floors

A cap is a series of caplets paying the excess of a floating rate over a strike; a floor pays the shortfall below a strike. Each caplet is priced with the Black model.

Volatility surface

The mapping of implied volatility across strikes and maturities. It is not flat in practice; smiles and skews reflect fat tails, leverage effects and supply-demand imbalances.

Self-financing portfolio

A portfolio whose value changes only through asset price movements, with no external cash injections or withdrawals. No-arbitrage pricing relies on constructing such a replicating portfolio.