May 07

Monte Carlo simulations. Part 1: Process

by David Harper, CFA, FRM, CIPM

FRM | Risk | Quant |

A Monte Carlo simulation is an experiment repeated thousands or millions of times over. It is a massively parallel experiment, where the results are synthesized into an prediction about the future. For example, we may simulate a portfolio over a million possible future scenarios into order to determine, "if we are really unlucky, say 1-in-100 unlucky, the worst 1% of outcomes, what would be the loss to the portfolio?"

Monte Carlo game of craps

Here is a simple Monte Carlo simulation: we grab two six-sided die and "roll the dice" 100 times in a row. When I do this (in Excel), I get the following histogram:

This Monte Carlo simulation, out of 100 experiments (trials), produced a seven exactly 21 times. You can see that, even with only two die, the distribution hints toward a normal shape. If we were to increase the number of dice per roll (i.e., roll three dice, four dice, five dice, ...), a marvelous thing happens. The Central Limit Theorem (CLT) starts to kick in. The shape of the histogram inevitably morphs into a normal "bell-curved" distribution. The CLT says: The sample mean (or the sample sum, for that matter, in this case) converges toward a normal distribution as the sample size increases. The marvelous thing is that the individual random variable (i.e., a single six-sided die) is not normally distributed. It has a uniform distribution: equal odds for each outcome, 1, 2, 3, 4, 5, or 6. Although the random variables are non-normally distributed, their average (or summation) converges toward normal!

Process dimensions: time and variable

There are two dimension to a random process: time and variable. Our experiment above is a discrete time process: one hundred discrete rolls (1st roll, 2nd roll, 3rd roll, ..., 99th roll, 100th roll). But actual time is continuous. Between one moment and another is some smaller fragment of time, and so on. The time steps in a process model can be either chunky or smooth.

Further, our experiment randomized discrete variables. The outcomes of each trial are drawn from the set {2,3,4,5,6,7,8,9,10,11,12}. But variables, like time, can be discrete or continuous. In summary, both the time steps in the process and the randomized variables can be either discrete or continuous:

Next we will characterize a stock price as a continuous-time, continuous-variable random process.

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