Exercise: Levy approximation accuracy

The Levy approximation prices an arithmetic Asian by approximating $\bar S$ as log-normal, matching the first two moments of the true distribution.

Tasks

1. Implement the Levy approximation for an arithmetic Asian call.

2. Compare against MC for a range of parameters: $S_0 = K = 100, T = 1, r = 0.05$ and $\sigma \in \{0.1, 0.2, 0.4, 0.8\}$.

3. Where is Levy accurate? Tabulate the relative error against MC. For which $\sigma$ regime is Levy most/least accurate?
4. Theoretical explanation. The arithmetic average of log-normals is not log-normal. The two-moment match captures the mean and variance but ignores higher moments (skewness, kurtosis). Argue why higher-moment errors matter more at high $\sigma$.