# (Re) normalisation of random variable in Monte-Carlo simulations

I have a very simple model (CIR) with a very simple discretisation scheme (Euler) and I use it to do Monte-Carlo Simulations. It is working.

Someone insisted that renormalization of my random variables would give better results. I.e. after drawing my normally distributed random variables I should translate them to obtain exactly mean 0 and multiply it to obtain variance 1. I have never heard of this simple technique before.

On a theoretical point of view I am unsure that my new random variables have the normal distribution I wanted. On a practical point of view the change in the result is ridiculous.

Is this technique really working ? Do you have an exemple where this is usefull ? Or a counter exemple where this is very wrong ?

Yes, this technique is called moment matching variance reduction and it may indeed lead to a form of variance reduction. The first and second order moments correspond to the mean and the variance of the distribution. You can extend to higher order moments, which is of course more difficult to implement and creates some extra overhead.

The mean can be adjust by either a linear correction:

$$\tilde{X}_i=X_i -\bar{X} + \mu_X$$

or a multiplicative one: $$\tilde{X}_i=X_i \frac{\mu_X}{\bar{X}}$$

where $\bar{X}$ is the sample mean, $\mu_X$ the exact mean. It depends on the application at hand to decide which of these is more suitable. If the process $X_i$ has mean zero, you of course would not use the multiplicative version. Similarly, if the process is strictly non-negative, then the linear method might cause some of the $X_i$ to become negative, which is again not desirable. So keep this in mind when performing moment matching.

The combined mean and variance moment matching may look like this:

$$\tilde{X}_i = (X_i - \bar{X})\frac{\sigma_X}{s_X} + \mu_X$$

where $s_X$ and $\sigma_X$ are the sample and distribution standard deviation respectively. This transformation actually introduces some bias, which leads to biased estimators of whatever option you are trying to price. But these are, according to the experts, usually very small so it's should be ok. Besides, the bias vanishes with increasing sample size.

You can think of this technique as a special type of Control Variate. A control variate works by using another process $Y_i$ to replace the samples $X_i$ by:

$$\tilde{X}_i=X_i - b (Y_i - \mu_Y)$$

where $b$ is optimal by setting it to $Cov[X,Y]/Var[X]$. The similarity should be clear. Note that by adjusting both the mean and the variance you are essentially using two control variates, so this is definitely something worth considering.

Finally, the downside is that you still need to know what the actual mean and variance of the process are; potentially also covariances if you are dealing with multiple processes. This might require some heavy math crunching (but hopefully someone has done this for you). Higher order moments become even more difficult to obtain.

Another problem is that these transformation typically introduce dependency among the samples. As a result, you can no longer use the usual rules for confidence interval construction, since this is based on independent samples. This is a problem for things like confidence levels of your estimates. You can partially get around this by creating different batches of samples, and apply moment matching to the individual batches. This is again more overhead for perhaps negligible gain.

Finally, Boyle, Broadie and Glasserman proved that it if you have a moment available, then you are usually better off using it as a control variate instead of using it as moment matching (i.e. you can gain more variance reduction this way). The reason is that moment matching is essentially a form of using a control variate, but with a sub-optimal coefficient $b$. Their proof is in the appendix of this paper.