The Method of Common Random Numbers: An Example

Variance reduction is of great interest to the creators of Monte Carlo experiments. For example, investment banks use very complicated Monte Carlo simulations to price esoteric mortgage-backed securities. These simulations often run overnight because many Monte Carlo trials are necessary to obtain (by the central limit theorem) a point estimate of some true population parameter, bounded by a relatively small confidence interval. One way to reduce the number of required Monte Carlo trials is to use a variance reduction technique.

The method of common random numbers is one such technique. It is useful in Monte Carlo experiments generally, including Monte Carlo integration. We illustrate its use with a simple example.

Let and , and and . Mathematica's numerical integration techniques tell us that , but this fact is not necessarily clear from the graphs of these two functions over the unit interval.

We propose two different techniques for estimating .

1. Suppose that we estimate and by and , where and are sequences of independent random numbers from the unit interval. Then .

2. Now consider the following alternative. Let and be positively correlated, but identically distributed uniform random variables. Estimate according to the rule .

The Demonstration output shows that is a modestly better way to estimate than . The histogram on the left features a simulated collection of the quantity . The histogram on the right features a simulated collection of the quantity .

Run the simulation repeatedly to see that in this example, the method of common random numbers always results in a reduction of variance.