Control variates

Control variates

The control variates method is a variance reduction technique used in Monte Carlo methods. It exploits information about the errors in estimates of known quantities to reduce the error of an estimate of an unknown quantity.[1]

Contents

Underlying principle

Let the parameter of interest be μ, and assume we have a statistic m such that \mathbb{E}\left[m\right]=\mu. Suppose we calculate another statistic t such that \mathbb{E}\left[t\right]=\tau is a known value. Then

m^\star = m + c\left(t-\tau\right) \,

is also an unbiased estimator for μ for any choice of the coefficient c. The variance of the resulting estimator m^{\star} is

\textrm{Var}\left(m^{\star}\right)=\textrm{Var}\left(m\right) + c^2\,\textrm{Var}\left(t\right) + 2c\,\textrm{Cov}\left(m,t\right);

It can be shown that choosing the optimal coefficient

c^\star = - \frac{\textrm{Cov}\left(m,t\right)}{\textrm{Var}\left(t\right)};

minimizes the variance of m^{\star}, and that with this choice,

\begin{align}
\textrm{Var}\left(m^{\star}\right) & =\textrm{Var}\left(m\right) - \frac{\left[\textrm{Cov}\left(m,t\right)\right]^2}{\textrm{Var}\left(t\right)} \\
& = \left(1-\rho_{m,t}^2\right)\textrm{Var}\left(m\right);
\end{align}

where

\rho_{m,t}=\textrm{Corr}\left(m,t\right); \,

hence, the term variance reduction. The greater the value of \vert\rho_{mt}\vert, the greater the variance reduction achieved.

In the case that \textrm{Cov}\left(m,t\right), \textrm{Var}\left(t\right), and/or ρmt are unknown, they can be estimated across the Monte Carlo replicates. This is equivalent to solving a certain least squares system; therefore this technique is also known as regression sampling.

Example

We would like to estimate

I = \int_0^1 \frac{1}{1+x} \, \mathrm{d}x.

The exact result is I=\ln 2 \approx 0.69314718. Using Monte Carlo integration, this integral can be seen as the expected value of f(U), where

f(x) = \frac{1}{1+x}

and U follows a uniform distribution [0, 1]. Using a sample of size n denote the points in the sample as u_1, \cdots, u_n. Then the estimate is given by

I \approx \frac{1}{n} \sum_i f(u_i);

If we introduce T=\int_0^1 1+x \, \mathrm{d}x. as a control variate with a known expected value \textrm{E}\left[T\left(U\right)\right]=\frac{3}{2}

Using n = 1500 realizations and an estimated optimal coefficient  c^\star \approx 0.4773 we obtain the following results

Estimate Variance
Classical estimate 0.69475 0.01947
Control variates 0.69295 0.00060

The variance was significantly reduced after using the control variates technique.

See also

Notes

  1. ^ Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. New York: Springer. ISBN 0387004513 (p. 185)

References


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