May 08

Monte Carlo simulations. Part 2: GBM

by David Harper, CFA, FRM, CIPM

FRM | Risk | Quant |

MCS is lots of trials

Previously we understood a Monte Carlo simulation to be a massively parallel trial (millions of simultaneous experiments). In order to conduct trials, we need a specific idea about the behavior of the asset. Really, we just need to specify how a random variable marches through time. Time seems to be actually continuous. But we can model time discretely or continuously. In regard to the random variable, it also can be discrete or continuous. Example of discrete random variables include dice, stock price quotes expressed in one-eighth increments (e.g., $4 1/8), binomial and Poisson distributions.

We assume stock follows GBM

For purposes of modeling stock price returns, the most common approach is to assume Geometric Brownian Motion (GBM). Technically, that's a continuous-time continuous-variable process, but it is easier to model in discrete time:

 

GBM says that stock returns follow a process based on two pieces: drift and shock. Drift is the the expected return (over the time interval). Shock is the random component: volatility times (x) a standard normal variable (epsilon) times (x) the square root of the time interval:

 

In regard to the shock, to multiply volatility by a standard normal variable is to simply randomize the volatility; e.g., it ensures that about 2/3rds of the time, the shock will fall within one standard deviation. Then the familiar square-root-rule: volatility scales with the square root of time because variance scales linearly with time.

This GBM model is the assumption about the stochastic (random) process we can use to conduct many trials in the Monte Carlo simulation. Next post we will actually do that...but first let's put GBM into process perspective

GBM is a Markov process

A Markov process (i.e., that exhibits the Markov property) does not care about history. Under Markov, tomorrow's stock price is a function of today's stock price but not yesterday's. (Note: this is consistent with all forms of market efficiency, including weak market efficiency which says the price already impounds all past prices.)

The Generalized Wiener process is a Markov Process. And further, GBM is a particular type of Wiener process:

Comments

  1. Vasillis Vontikakis 23 May 2008

    The formula that you describe above for GBM is not GBM. The above formula is the simple linear stochastic differential equation the solution of above slsd is the geometric brownian motion which is a different formula and it can be obtained by applying the Ito rule to slsd

Leave a Comment