Exercise: Stock-price control for vanilla option

When no specialised control variate is available, the underlying itself can be used. Under $\mathbb{Q}$, $\mathbb{E}[e^{-rT}S_T] = S_0$ exactly.

For a vanilla European call with $S_0 = K = 100, T = 1, r = 0.05, \sigma = 0.2$:

Tasks

1. Implement the call MC pricer using $Y = e^{-rT}S_T$ as the control with $\mu_Y = S_0 = 100$.

2. Estimate the variance reduction. Hint: derive $\rho_{XY}$ analytically using the joint distribution of $(S_T, (S_T-K)^+)$.

3. Forward as control. Try also using $Y = S_T - S_0 e^{rT}$ (centred forward), which has known mean 0. Does this give the same variance reduction? Why?
4. Comparison with antithetic. From the previous lesson, antithetic gave variance reduction factor $\sim 0.10$ for the same problem. How does CV compare?