 # Convergence hacks.  Variance reduction tools for Monte Carlo Simulation

We look at  ways in which solutions derived from Monte Carlo simulation techniques may be improved. Whether you run Monte Carlo Simulators in Excel or in C, the number of iterations required before we get to our desired solutions represent the cost of running the simulation. Variance reduction techniques are a mechanism for speeding up convergence without increasing the number of iterations required in a simulation.

While using a programming language, programming efficiency allow us to easily run a 100,000 iterations within our simulation, an Excel model is not as lucky.  A large portfolio, a long look back period, exotic options are all drivers that can push requirement for running substantially larger number of iterations to a point where they can no longer be executed within our Excel spreadsheet.

Our two part series presents Quasi Monte Carlo methods an Antithetic Variable technique. Two tools that are simple to implement (in Excel or otherwise) and lead to a significant improvement over conventional Monte Carlo Simulation methods.

We begin with a brief overview of the relationship between the number of trials used and the resultant degree of accuracy in the solution:

Monte Carlo Simulation: Convergence and Variance reduction techniques for option pricing models This is then followed by a closer look at two of the techniques used from improving the accuracy of the model results – a simple increase in the number of trials, Quasi Random Sequences and Anti-Thetic Variable Technique.

The  two methods improve the results without adding unnecessarily to the computational capacity or time. These two variance reduction procedures decrease the standard error between the simulated result and the true value as well as converge to the true result at a faster rate than the method reliant on a simple increase in the number of trials.

The results of these three methods are assessed against each other and a tabular and graphical output shows how the latter two methods are an improvement over the former increase-in-number-of-trials method:

Variance Reduction procedures: Antithetic Variable Technique & Quasi Random sequences 