Exercise: Multiple control variates

For an arithmetic Asian call, we have two natural controls:

• $Y_1$ = geometric Asian (closed form).
• $Y_2$ = $e^{-rT}S_T$ (terminal stock, mean $S_0$).

Combine both: $\hat\theta = \bar X - \beta_1(\bar Y_1 - \mu_1) - \beta_2(\bar Y_2 - \mu_2)$.

The optimal $\beta = (\beta_1, \beta_2)$ minimising variance is the OLS regression of $X$ on $(Y_1, Y_2)$.

Tasks

1. Implement two-control variate pricing for the arithmetic Asian. Use the same parameters as before.

2. Compare the variance reduction with single-control (using just $Y_1$).

3. Diminishing returns. When does adding $Y_2$ help, and when is the geometric Asian alone sufficient?
4. Vector formula. Derive the optimal $\beta$ vector: it's the slope of the OLS regression of $X$ on $(Y_1, Y_2)$. State this in matrix form.