Geometric Brownian Motion and the Distribution of Price Returns

Summary

The document raises a question about how geometric Brownian motion relates to normally distributed percentage returns. It distinguishes simple returns, defined as the change in price relative to the starting price, from log returns, defined as the logarithm of the price ratio. The questioner notices that a normally distributed simple return could fall below minus one, making the log transformation undefined, and asks whether that conflicts with GBM.

The included answer explains that GBM assumes normally distributed log returns, not normally distributed simple returns. Exponentiating a normal log return gives a positive gross return and a price ratio; subtracting one yields a simple return bounded below by minus one, with a transformed, non-normal distribution. A second answer points out that simple returns do not add across periods, while log returns do. The exchange is conceptual rather than empirical, and its closing caveat notes that actual returns can exhibit fat tails, asymmetry, and changing variance, so normality is an imperfect market assumption.

Key ideas

• GBM models normally distributed log returns rather than normally distributed simple returns.
• A normally distributed log return implies a positive price ratio and a simple return bounded below by minus one.
• Simple returns and log returns have different distributions because the conversion is nonlinear.
• Log returns add across successive periods, while simple returns do not.
• Real return data may depart from normality through fat tails, asymmetry, and changing variance.

Tags

# Probability Distribution at each Simulation Period using Geometric Brownian Motion


# Probability Distribution at each Simulation Period using Geometric Brownian Motion












I am using the equation $S_t = S_0e^{(\mu-\frac{\sigma^2}{2})t+\sigma\epsilon\sqrt{t}} $ to simulate a financial metric at each $t$, where $t=1$ and $T=5$. Stated in plain English, I am trying to simulate the financial metric at the end of every year, for $5$ years. Additionally, for this exercise it is important that I know the path of the metric as well. When looking at the distributions of this financial metric at each $t$, after a high number of trials, they start from something that resembles a lognormal distribution at $t=1$ and then decay to 0 with each successive $t$. Can anyone explain the mathematics behind this or why GBM behaves this way?

Note: Volatility is around 20% and I am using annual volatility.

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