Exercise: Why Asians are cheaper than vanillas

Tasks

1. Variance argument. For Black-Scholes: $\text{Var}(S_T) > \text{Var}(\bar S)$ because averaging reduces variance. State this rigorously and explain why it implies the Asian is cheaper.
2. Numerical comparison. With $S_0 = K = 100, T = 1, r = 0.05, \sigma = 0.2$:
   • Compute the vanilla European call price.
   • Compute the geometric Asian call price (continuous limit).
   • Compute the arithmetic Asian call price (Levy approximation or MC).
   • Tabulate.
3. Sensitivity to $\sigma$. Vary $\sigma \in \{0.1, 0.2, 0.4\}$ and tabulate vanilla, geometric Asian, arithmetic Asian. Compute the "Asian discount" $(C_{\text{vanilla}} - C_{\text{Asian}})/C_{\text{vanilla}}$.
4. Sensitivity to monitoring frequency. For arithmetic Asian, vary $M \in \{12, 252, 10000\}$ (monthly, daily, near-continuous). Use MC for each.