Exercise: Geometric Asian closed-form derivation

For continuously monitored geometric Asian: $\bar S_g = \exp\!\big(\frac{1}{T}\int_0^T \ln S_t \, dt\big)$.

Tasks

1. Derive that under Black-Scholes, $\ln \bar S_g$ is normally distributed. Compute its mean and variance.

2. Show that $\bar S_g$ is log-normal with effective volatility $\sigma_g = \sigma/\sqrt{3}$ over period $T$.

3. Apply the Black-Scholes formula with the adjusted parameters to derive the closed-form price for a geometric Asian call.

4. Numerical verification. Implement and verify against a Monte Carlo with $N = 100{,}000$ paths and $M = 1000$ time steps. The MC and analytic should agree to within MC noise.