Control variates: Difference between revisions

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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.<ref>Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering (Stochastic Modelling and Applied Probability) (1 ed.). New York: Springer., p. 185.</ref>
 
==Underlying Principleprinciple==
Let the [[Parameter#Statistics_and_econometrics|parameter]] of interest be <math>\mu</math>, and assume we have a statistic <math>m</math> such that <math>\mathbb{E}\left[m\right]=\mu</math>. Suppose we calculate another statistic <math>t</math> such that <math>\mathbb{E}\left[t\right]=\tau</math> is a known value. Then
 
:<math>m^{\star} = m + c\left(t-\tau\right) \, </math>
 
is also [[bias of an estimator|an unbiased estimator]] for <math>\mu</math> for any choice of the coefficient <math>c</math>.
The [[variance]] of the resulting estimator <math>m^{\star}</math> is
 
:<math>\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);</math>;
 
It can be shown that choosing the optimal coefficient
 
:<math>c^{\star} = - \frac{\textrm{Cov}\left(m,t\right)}{\textrm{Var}\left(t\right)}; </math> ;
 
minimizes the variance of <math>m^{\star}</math>, and that with this choice,
:<math>\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} </math>;
 
where
:<math>\rho_{m,t}=\textrm{Corr}\left(m,t\right); \, </math>;
 
hence, the term [[variance reduction]]. The greater the value of <math>\vert\rho_{mt}\vert</math>, the greater the variance reduction achieved.
| align="right" | '''Variance'''
|-
| ''Classical Estimateestimate''
| align="right" | 0.69475
| align="right" | 0.01947
|-
| ''Control Variatescariates ''
| align="right" | 0.69295
| align="right" | 0.00060