Calculating VaR using Monte Carlo Simulation

Monte Carlo Simulations correspond to an algorithm that generates random numbers that are used to compute a formula that does not have a closed (analytical) form – this means that we need to proceed to some trial and error in picking up random numbers/events and assess what the formula yields to approximate the solution. Drawing random numbers over a large number of times (a few hundred to a few million depending on the problem at stake) will give a good indication of what the output of the formula should be.

It is believed actually that the name of this method stems from the fact that the uncle of one of the researchers (the Polish mathematician Stanislaw Ulam) who popularized this algorithm used to gamble in the Monte Carlo casino and/or that the randomness involved in this recurring methodology can be compared to the game of roulette.

We will now look at how Monte Carlo simulation can be applied to Value at Risk.

As we know, Monte Carlo Simulations correspond to an algorithm that generates random numbers that are used to compute a formula that does not have a closed (analytical) form – this means that we need to proceed to some trial and error in picking up random numbers/events and assess what the formula yields to approximate the solution. Drawing random numbers over a large number of times (a few hundred to a few million depending on the problem at stake) will give a good indication of what the output of the formula should be.

In the section, we will present the algorithm, and apply it to compute the VaR for a sample stock.

Computing VaR with Monte Carlo Simulations is very similar to Historical Simulations. The main difference lies in the first step of the algorithm – instead of using the historical data for the price (or returns) of the asset and assuming that this return (or price) can re-occur in the next time interval, we generate a random number that will be used to estimate the return (or price) of the asset at the end of the analysis horizon.

Step 1 – Determine the time horizon t for our analysis and divide it equally into small time periods, i.e. dt = t/n).

For illustration, we will compute a monthly VaR consisting of twenty-two trading days. Therefore n = 22 days and $\delta t$ = 1 day. In order to calculate daily VaR, one may divide each day per the number of minutes or seconds comprised in one day – the more, the merrier.