## Monday, November 25, 2013

### Monte Carlo Simulation, Explained

By Tim Dulaney, PhD, FRM and Tim Husson, PhD

Valuing products with exotic derivatives can be difficult since these products typically have complex payoff formulas.  One of the most flexible methods for valuing such products is called Monte Carlo simulation.  At SLCG, we use Monte Carlo simulation in a lot of our work, so we thought it would be helpful to explain a bit about it and show how it can be used to estimate the future returns of an asset.

The basic idea behind Monte Carlo simulation is to determine the statistical properties (e.g., mean and standard deviation) of the underlying security's returns, then to randomly pick returns that have those characteristics.  If you do this many times, you will generate many 'paths' of the underlying asset.  Then you can apply the payoff formula of the product to each path and average the result to determine the value of the product.

This might sound a little complicated.  We've created a simple tool that lets you simulate possible asset values based on log-normally distributed returns (which are commonly assumed for financial assets). Change any of the sliders to see their effect on the projected returns.

 Number of Years: Number of Simulations: Mean Return: Volatility:

For a given average (mean) return and volatility, Monte Carlo simulation let us draw many possible price paths and average the results. With enough simulations, the resulting returns are distributed log-normally.  These returns could be those of the S&P 500, the spot price of a commodity, a foreign exchange rate, or any other asset (though the log-normal assumption may not always be appropriate).

The graphs above plot many possible price paths (grey lines) for the asset and the average of all paths (black line). You can change the number of years to simulate, the number of simulations, the mean annual return, and the volatility using the sliders. We also plot the distribution of final returns in the lower panel, as well as the theoretical log-normal distribution implied by the chosen mean return and volatility.

This type of simulation is used to value options embedded in structured products and many other financial products that depend on the future value of an asset.  The flexibility of  the Monte Carlo framework allows practitioners to include the effect of, for example, mortality risk when valuing annuities, which is very difficult to value using any other method.  As products become ever more complex, Monte Carlo simulation is likely to become more and more important to both practitioners and investors.