Black-Scholes Monte Carlo Pricer

You are building a simple Monte Carlo pricer for a European call option. The underlying stock follows geometric Brownian motion: $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$ with initial price $S_0$, constant volatility $\sigma$, and risk-free rate $r$. The call has strike $K$ and maturity $T$. 1. Derive the risk-neutral distribution of $S_T$. Write an expression for the option price as $e^{-rT} \, E^Q[(S_T - K)^+]$. 2. Describe an algorithm that approximates this expectation via Monte Carlo using $M$ simulation paths. Specify exactly how you generate $S_T$ on each path. Analyze the time complexity in terms of $M$. 3. How would you reduce the variance of your Monte Carlo estimator? Discuss at least two techniques (e.g., antithetic variates, control variates) and explain how you would empirically verify that the pricer is converging correctly.