Exercise: Optimal Esscher tilt for an OTM call

Price the OTM call $S_0 = 100, K = 150, T = 1, r = 0.05, \sigma = 0.2$ using importance sampling. The Black-Scholes price is \approx \0.6225$.

Tasks

1. Compute $z^* = (\ln(K/S_0) - (r - \sigma^2/2)T)/(\sigma\sqrt{T})$. This is the threshold for $Z$ such that $S_T > K$.

2. With $N = 10^5$ samples, compute the IS estimate using $\mu = z^*$ (centre on the strike).

3. Optimum search. Vary $\mu \in [z^*, z^* + 2]$ and find the empirical optimum. Compare with the prediction $\mu^* = z^* + \sigma\sqrt{T}$ (the optimum Esscher for a call, accounting for the linear payoff growth).
4. Effective sample size. Compute $N_{\text{eff}} = (\sum w_i)^2/\sum w_i^2$ at $\mu = z^*$ and $\mu = z^* + 2$. Which gives higher ESS?