IFoA SP6 cheat sheet

Forward price with known income: F0 = (S0 - I)e^{rT}; with known yield q: F0 = S0e^{(r-q)T}. Value of a long forward = (F0_current - K)e^{-rT}.

Black-Scholes: d1 = [ln(S/K) + (r - q + 0.5σ²)T]/(σ√T), d2 = d1 - σ√T; call = Se^{-qT}N(d1) - Ke^{-rT}N(d2); put via put-call parity c + Ke^{-rT} = p + Se^{-qT}.

Greeks: delta_call = e^{-qT}N(d1), delta_put = -e^{-qT}N(-d1); gamma = e^{-qT}n(d1)/(Sσ√T); vega = Se^{-qT}n(d1)√T; theta and rho follow from partial differentiation of the BS formula.

Put-call parity holds for European options on non-dividend stock: c + Ke^{-rT} = p + S0. Violations imply arbitrage via a replicating portfolio of stock, bond and option.

Binomial CRR parameterisation: u = e^{σ√Δt}, d = 1/u, p = (e^{(r-q)Δt} - d)/(u - d); price by backward induction, discounting expected payoffs at the risk-free rate.

Risk-neutral valuation: any derivative price equals e^{-rT}E^Q[payoff]; Q is the measure under which the discounted asset price is a martingale, not the real-world measure.

Interest rate swaps: value = PV of fixed leg minus PV of floating leg; the fixed rate at inception makes the swap value zero (par swap rate). FRAs are single-period equivalents.

Futures vs forwards: futures are marked to market daily, so their price differs from forwards when interest rates correlate with the underlying; convexity adjustment applies.

Option pricing bounds: c ≥ max(0, Se^{-qT} - Ke^{-rT}); p ≥ max(0, Ke^{-rT} - Se^{-qT}); American calls on non-dividend stock are never optimally exercised early.

Implied volatility is the σ that reproduces the observed market price; the volatility smile/skew reflects non-lognormal tails and can be modelled with local or stochastic volatility.

Delta hedging error arises from discrete rebalancing and gamma; the P&L of a delta-hedged position over a small interval ≈ 0.5 Γ (ΔS)² - 0.5 σ² S² Γ Δt (theta/gamma trade-off).

Monte Carlo pricing: simulate risk-neutral paths, discount the mean payoff; use antithetic or control variates to reduce variance; standard error ∝ 1/√n.

American options: use a binomial/trinomial tree or least-squares Monte Carlo (Longstaff-Schwartz); early exercise value is compared to continuation value at each node.

Black model for futures options: c = e^{-rT}[F0N(d1) - KN(d2)], d1 = [ln(F0/K) + 0.5σ²T]/(σ√T), d2 = d1 - σ√T.

Currency options (Garman-Kohlhagen): treat foreign rate rf as a continuous dividend yield; call = Se^{-rf T}N(d1) - Ke^{-rT}N(d2).

Credit default swaps: spread ≈ (1 - R) × hazard rate; value a CDS as protection leg minus premium leg, both discounted and survival-weighted.

Exotic structures: barrier options are cheaper than vanillas (knock-out) and carry discontinuity in delta at the barrier; Asian options average the underlying and reduce volatility.

Numerical methods trade-off: trees handle early exercise but scale poorly with dimensions; finite differences handle Greeks well; Monte Carlo handles high dimensions but struggles with early exercise.

No-arbitrage replication: a derivative is priced by constructing a self-financing portfolio of the underlying and a risk-free bond that replicates its payoff.

Continuous dividend yield q: replace S0 by S0e^{-qT} in the BS formula and discount the strike at r; equivalently use drift r - q under Q.

Martingale condition: under Q, dS/S = (r - q)dt + σdW; the discounted, dividend-reinvested asset price is a martingale.

Variance swap fair strike equals the risk-neutral expected realised variance; replication uses a static portfolio of options across strikes (the log contract).

Forward rate agreement: value = notional × (R_K - R_M) × τ × discount factor; settle in arrears, discount from settlement to valuation date.

Bond option pricing: use the Black model with the forward bond price and the bond's yield volatility, or a short-rate model (Vasicek, Hull-White) for the full term structure.

Swap spread and basis: the swap rate reflects the fixed-rate government yield plus a spread compensating for credit and liquidity; basis swaps exchange floating indices.

Gap risk and jump models: pure diffusion BS cannot generate the observed smile; add jumps (Merton) or stochastic volatility (Heston) to capture skew and fat tails.

Model risk controls: calibrate to liquid vanillas, backtest hedge performance, stress the volatility surface, and document parameter stability and limitations.