Financial derivatives: pricing, hedging and risk
Financial derivatives: pricing, hedging and risk
SP6 tests whether you can apply no-arbitrage reasoning to value and hedge derivatives, not merely quote formulas. Expect forwards, futures, options and swaps, with questions that connect model prices to tradable prices through replication, discounting and explicit cash-flow timelines.
Core rules
- A derivative is priced from its underlying and a risk-free money-market account. Replication and no-arbitrage must give the same price. A zero-cost strategy with no downside is an arbitrage.
- Forwards: F0 = S0 e^{(r-q)T} for dividend yield q. A forward has zero value at inception; later, a long forward is worth the discounted difference F_t - F0.
- Futures differ from forwards through daily settlement and margin; the basis is spot minus futures, and convergence at maturity is a pricing check, not a guarantee of profit.
- European put-call parity: c + K e^{-rT} = p + S0 e^{-qT}. It is model-free, so check it before Black-Scholes.
- Black-Scholes assumes constant volatility, continuous trading, no arbitrage and frictionless markets. Delta-hedging removes first-order spot risk, not volatility or jump risk.
- Swaps are portfolios of forwards or bonds. Value an interest rate swap as fixed-leg PV minus floating-leg PV; under single-curve assumptions the floating leg resets to par at each reset date.
Worked example
A non-dividend stock is 100, r=5%, T=1, K=100. A European call costs 10. By parity, p = c + K e^{-rT} - S0 = 10 + 95.1229 - 100 = 5.1229. If the put trades at 4.50, buy the put, sell the call, buy the stock, and borrow 95.1229. Net cash today is +0.6229. At maturity the payoff is zero in both states, so the profit is riskless; the fair put is 5.1229.
Traps
Using simple rather than continuous discounting; forgetting q; treating parity as model-dependent; valuing a forward at inception as non-zero; and confusing delta with probability. Always state the replication and the cash-flow timing.