About this question

Martingale Representation Theorem and Option Pricing

Hard · Stochastic Calculus · Quant Trader interview question · martingale-representation, stochastic-calculus, option-pricing, hedging

Suppose the price of a stock, $S_t$, follows a geometric Brownian motion under the risk-neutral measure $Q$: $dS_t = r S_t dt + \sigma S_t dW_t$, where $r$ is the risk-free rate and $\sigma$ is the volatility. Consider a European call option on this stock with payoff $C_T = max(S_T - K, 0)$ at maturity $T$, where $K$ is the strike price. According to the Martingale Representation Theorem, the discounted option price, $e^{-r(T-t)}C_T$, can be represented as $C_t + \int_t^T H_s dW_s$ for some adap