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]

Underlying principle

Let the parameter of interest be $\mu$ , 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 $\mu$ 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{aligned}{\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{aligned}}

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 $\rho _{mt}$ 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 cariates	0.69295	0.00060

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

Notes

^ Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering (Stochastic Modelling and Applied Probability) (1 ed.). New York: Springer., p. 185.

References

Ross, Sheldon M. Simulation 3rd edition ISBN 978-0125980531

Averill M. Law & W. David Kelton, Simulation Modeling and Analysis, 3rd edition, 2000, ISBN 0-07-116537-1

S. P. Meyn. Control Techniques for Complex Networks, Cambridge University Press, 2007. ISBN 9780521884419. Online: https://netfiles.uiuc.edu/meyn/www/spm_files/CTCN/CTCN.html

[1] Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering (Stochastic Modelling and Applied Probability) (1 ed.). New York: Springer., p. 185.

[1]

Underlying principle

Example

See also

Notes

References