Variations of Geometric Brownian Motion

Overview

Variations on GBM

Geometric Brownian Motion is a rather idealized process, which is unlikely to be realized in the real world. Many processes, such as asset prices, can be modeled as gbm's, however, they deviate from a pure GBM in rather predictable ways.

The first issue is that a pure GBM has a constant volatility, that is, {% \sigma %} is constant. Asset prices have been known to have varying volatility.

Another common issue is that the distribution of returns deviates from normality, especially in the tails. Asset prices exhibit fat tails, that is, for a given volatility, the return distribution shows more probability in the extreme wings of the tails. One way this can be accomodated is through a varying volatility (or {% \sigma %}). Other ways to potentially fix this issues is through jumps.

The last issue is that some processes can attain a zero value. In particular, an asset price, such as a stock, could in theory go to zero, whereas a GBM is always positive. This can by easily accomodated by assuming that when the GBM goes below some threshold, it is assumed to be zero.