Monte Carlo pricing: which process to simulate

Question

You want to value a European call by Monte Carlo. Should the geometric Brownian motion you simulate describe the option price or the price of the underlying asset? Explain the procedure.

Short answer

Simulate the underlying. Under the risk-neutral measure the stock follows GBM with drift $r$ ; the option price does not follow GBM. Generate many terminal stock prices, compute $\max(S_T-K,0)$ on each path, average, and discount at $e^{-rT}$ .

Adapted from Crack, Heard on the Street, Q2.26, p.28; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Monte Carlo Option Pricing
Applications — Monte Carlo Estimation, Error, and Confidence Intervals

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