Antithetic variates

The antithetic variates technique consists, for every sample path obtained, in taking its antithetic path — that is given a path ${\displaystyle \{\varepsilon _{1},\dots ,\varepsilon _{M}\}}$ to also take ${\displaystyle \{-\varepsilon _{1},\dots ,-\varepsilon _{M}\}}$. The advantage of this technique is twofold: it reduces the number of normal samples to be taken to generate N paths, and it reduces the variance of the sample paths, improving the precision.

Suppose that we would like to estimate

${\displaystyle \theta =\mathrm {E} (h(X))=\mathrm {E} (Y)\,}$

For that we have generated two samples

${\displaystyle Y_{1}{\text{ and }}Y_{2}\,}$

An unbiased estimate of ${\displaystyle {\theta }}$ is given by

${\displaystyle {\hat {\theta }}={\frac {{\hat {\theta }}_{1}+{\hat {\theta }}_{2}}{2}}.}$

And

${\displaystyle {\text{Var}}({\hat {\theta }})={\frac {{\text{Var}}(Y_{1})+{\text{Var}}(Y_{2})+2{\text{Cov}}(Y_{1},Y_{2})}{4}}}$

so variance is reduced if ${\displaystyle {\text{Cov}}(Y_{1},Y_{2})}$ is negative.

If the law of the variable X follows a uniform distribution along [0, 1], the first sample will be ${\displaystyle u_{1},\ldots ,u_{n}}$, where, for any given i, ${\displaystyle u_{i}}$ is obtained from U(0, 1). The second sample is built from ${\displaystyle u'_{1},\ldots ,u'_{n}}$, where, for any given i: ${\displaystyle u'_{i}=1-u_{i}}$. If the set ${\displaystyle u_{i}}$ is uniform along [0, 1], so are ${\displaystyle u'_{i}}$. Furthermore, covariance is negative, allowing for initial variance reduction.

We would like to estimate

${\displaystyle I=\int _{0}^{1}{\frac {1}{1+x}}\,\mathrm {d} x.}$

The exact result is ${\displaystyle I=\ln 2\approx 0.69314718}$. This integral can be seen as the expected value of ${\displaystyle f(U)}$, where

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

and U follows a uniform distribution [0, 1].

The following table compares the classical Monte Carlo estimate (sample size: 2n, where n = 1500) to the antithetic variates estimate (sample size: n, completed with the transformed sample 1 − u_i):

	Estimate	Standard deviation
Classical Estimate	0.69365	0.00255
Antithetic Variates	0.69399	0.00063

The use of the antithetic variates method to estimate the result shows an important variance reduction.