Exercise: Geometric vs arithmetic Asian — empirical correlation

The geometric Asian call has a closed-form price; the arithmetic Asian does not. Control variates leverage the high correlation between them.

Tasks

1. With $S_0 = K = 100, T = 1, r = 0.05, \sigma = 0.2, M = 50, N = 100{,}000$ paths, simulate the discounted payoffs $X$ (arithmetic) and $Y$ (geometric). Compute $\rho_{XY}$.

2. Tabulate $\rho_{XY}$ for $\sigma \in \{0.1, 0.2, 0.4, 0.8\}$. How does the correlation change with volatility?

3. Implementation choice. Which path quantity should you average — the prices $S_{t_k}$, the log-prices $\ln S_{t_k}$, or both? Verify that the geometric average can be computed as $\exp((\sum_k \ln S_{t_k})/M)$.

4. Estimate the variance reduction factor $1 - \rho^2$ for each $\sigma$ in part 2. For which $\sigma$ regime is the control variate most powerful?