Exercise: Risk-Neutral Pricing of a European Call Under GBM

Prerequisites: Geometric Brownian Motion, Log-Normal Distribution, Risk-Neutral Measure

Problem

Under the risk-neutral measure $\mathbb{Q}$, a stock follows $dS_t = rS_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}$ with $r = 0.04$, $\sigma = 0.30$, $S_0 = 100$. Consider a European call with strike $K = 105$ and maturity $T = 0.5$ (six months).

The fair price is $C_0 = e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T - K)^+]$.

1. Price the option by Monte Carlo using $N = 1{,}000{,}000$ risk-neutral GBM paths. Report the estimate and a 95% confidence interval.
2. Compute the Black-Scholes closed-form price $C_0 = S_0\Phi(d_1) - Ke^{-rT}\Phi(d_2)$ and compare to the Monte Carlo estimate.
3. Estimate the exercise probability $\mathbb{Q}(S_T > K)$ from the same simulation, and compare to the closed-form $\Phi(d_2)$.
4. The "real-world" drift under $\mathbb{P}$ for this stock is $\mu = 0.10$. Would the option price change if we used $\mu = 0.10$ instead of $r = 0.04$ in the simulation? If yes, by how much (numerically)? If no, explain why not.

Hint

For the confidence interval, use $\hat C \pm 1.96 \cdot \hat\sigma_C / \sqrt{N}$ where $\hat\sigma_C$ is the sample standard deviation of the discounted payoffs.