Control variates

Remarks

Clause 1 holds because $\mathbb E(C - \mu) = 0$, and clause 2 is the expansion of the variance of a difference. Clause 3 minimises the quadratic in $\beta$; substituting $\beta^*$ gives $\operatorname{Var} Y - \operatorname{Cov}(Y, C)^2 / \operatorname{Var} C = (1 - \rho^2)\operatorname{Var} Y$. Example: to estimate $\mathbb E e^U = e - 1$ for $U$ uniform on $(0, 1)$, use $C = U$ with $\mu = \frac12$. Here $\operatorname{Cov}(e^U, U) = 1 - \frac{e - 1}{2} \approx 0.1409$ and $\operatorname{Var} U = \frac{1}{12}$, so $\beta^* \approx 1.690$ and $\operatorname{Var} Z_{\beta^*} \approx 0.0039$, against $\operatorname{Var} e^U \approx 0.2420$. In practice $\beta^*$ is estimated from the same samples, by least squares regression of $Y$ on $C$; this introduces a bias of order $1/n$, which is usually negligible. If $C$ is uncorrelated with $Y$, then $\rho = 0$ and no gain is possible.

Going deeper

For a student

Control variates is a variance reduction technique that turns side knowledge, the exact mean of $C$, into accuracy. In option pricing, the price of a similar option with a closed-form value is a classic control. With several controls $C_1, \ldots, C_k$ the optimal coefficients solve a linear least squares problem, and the variance becomes $(1 - R^2)\operatorname{Var} Y$, with $R^2$ the squared multiple correlation. The multilevel Monte Carlo method can be seen as a recursive use of control variates, with coarse approximations as controls for fine ones.