By Kardi Teknomo, PhD.

stochastic process

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What is Geometric Brownian Motion?

An exponential Brownian motion is also called Geometric Brownian motion, or GBM. This process is suggested by Black, Scholes and Merton. The actual model of GBM is a stochastic differential equation (SDE) of this form

Or

Suppose  is price, then this model say that the proportional change in the price in the interval of

Similar to AGM SDE, this GBM SDE model also has two parameters:

  1. Drift,
  2. Volatility,

The meaning of drift parameter is a trend or growth rate. If the drift is positive, the trend is going up over time. If the drift is negative, the trend is going down. The meaning of volatility is a variation or the spread of distribution. The value of volatility is always positive (or zero) because it is actually related to standard deviation of the distribution. Note that the drift parameter of GBM is not the same as the drift parameter of ABM. Similarly, the volatility of GBM is not the same value as the volatility of ABM. In fact, the distribution are not the same.

To simulate the GBM, we need to find the solution of the stochastic differential equation above. The solution can be found by the Ito integration to be:

Where

If we input all, we have

This model can be simplified further by setting

Then we have

If we take natural logarithm on both side, we have a linear model:

Notice that GBM also contains Brownian motion (or Wiener Process). This model now has three inputs that is 2 parameters and one initial value:

  1. Drift,
  2. Volatility,
  3. Initial value of the GBM, .

Unlike AGM that is normally distributed, GBM is log-normally distributed with mean and variance .

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These tutorial is copyrighted .

Preferable reference for this tutorial is

Teknomo, Kardi. (2017) Stochastic Process Tutorial .
http://people.revoledu.com/kardi/tutorial/StochasticProcess/