Option Pricing and Black-Scholes Framework
This domain tests risk-neutral valuation, put-call parity, the Black-Scholes formula and the binomial model.
Core pricing relationships
- A European call pays max(S_T - K, 0) at maturity. A European put pays max(K - S_T, 0). For non-dividend-paying shares, put-call parity is C - P = S_0 - K e^{-rT}.
- Under the risk-neutral measure, the call value is e^{-rT} E^Q[max(S_T - K, 0)]. In the Black-Scholes model, C = S_0 N(d1) - K e^{-rT} N(d2), where d1 = [ln(S_0/K) + (r + 0.5 sigma^2)T]/(sigma sqrt(T)) and d2 = d1 - sigma sqrt(T).
- The Greeks measure sensitivities: delta, gamma, vega, theta and rho. Implied volatility is the volatility that reproduces a market price.
- The one-period binomial model uses risk-neutral probability p = (e^{rT} - d)/(u - d) and discounts expected payoffs.
Worked example
A one-period binomial model has S_0 = 100, u = 1.2, d = 0.8, r = 5% and T = 1. A call has K = 100, so payoffs are 20 and 0. The risk-neutral probability is p = (e^{0.05} - 0.8)/(1.2 - 0.8) = 0.62818. The call value is e^{-0.05} x 0.62818 x 20 = 11.95. The real-world probability of an up move is not used; only the risk-neutral probability matters for pricing.
Traps that cost marks
Use continuous compounding consistently in Black-Scholes. Do not use the real-world expected return as the drift in risk-neutral pricing. Put-call parity must be adjusted for dividends. American options may be exercised early, so a plain European formula may not apply. Do not mix up d1 and d2.