Geometric Brownian Motion and Lognormal Asset Prices

Summary

The discussion clarifies how two forms of a geometric Brownian motion model relate to simulated asset prices. In continuous time, the price process associated with the stochastic differential equation produces lognormally distributed prices, so modeled prices remain above zero. Taking logarithms transforms the process and explains why log prices have a normal distribution, while percentage changes under the alternative representation can be normally distributed and may mathematically fall below a full loss.

The answer distinguishes the continuous model from a discrete numerical approximation. A sufficiently large time step can produce a negative simulated price in that approximation, even though the continuous process does not cross zero; smaller steps make this less likely. The document offers a conceptual explanation rather than simulation results or a detailed derivation. It also does not specify a time-step threshold, model calibration, or how to choose a discretization scheme for a particular application.

Key ideas

• The continuous geometric Brownian price process yields lognormally distributed, positive prices.
• Log prices are normally distributed under the corresponding transformed process.
• A discrete approximation can produce negative prices when its time step is large.
• The two displayed stochastic equations are related through a change of variables.

Tags

# Geometric Brownian Motion: d(S) vs. d(ln(S))


# Geometric Brownian Motion: d(S) vs. d(ln(S))












I am quoting from "Tools for Computational Finance, 5th Edition" [Seydel].

I wonder whether the histogram of simulations of the first (yellow) SDE makes sense... especially given that Seydel (correctly) states that the resulting percentage changes are normally distributed (i.e. can go below -100%).

Shouldn't the histogram much rather correspond to asset prices simulated from the second (red) SDE, as the second SDE would produce log-prices which are normally distributed, i.e. prices which are log-normally distributed?

The related question How to simulate stock prices with a Geometric Brownian Motion? does not quite cover this.

## Answer by Alex C (score 1)

https://quant.stackexchange.com/a/22567

Equation 1.33 results in lognormally distributed prices. The price can't go below zero, because the equation is being integrated over infinitesimally short periods of time. The discrete approximation 1.34a could go negative if $\Delta t$ is big, if it is reasonably small that is unlikely.

Furthermore, although Equations 1.33 and 1.46 look completely different, they are atually intimately related (through a change of variables), as you will soon find out.

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