Why Geometric Brownian Motion Produces Lognormal Prices

Summary

The note explains the link between geometric Brownian motion and asset price distributions in the Black–Scholes framework. When a price follows an SDE whose drift and volatility scale with the price, solving the process yields a lognormally distributed price at a given time. Equivalently, modeling log returns as normal and exponentiating them keeps the resulting price positive; Itô’s lemma connects these descriptions. The binomial model is also mentioned as a route to the same limiting distribution as its steps increase.

The model is attractive for its simplicity, Markov property, and tractable derivative pricing. The note cautions that observed stock returns show skewness and fat tails, so normal-return assumptions can be too restrictive. Jump and stochastic-volatility models are cited as ways to represent richer dynamics. No empirical study or quantitative comparison is provided, and the discussion does not establish that asset prices generally follow this distribution; it describes a simplifying modeling assumption and its limitations, especially for Black–Scholes use.

Key ideas

• Geometric Brownian motion implies lognormally distributed prices at each time.
• Normal log returns become positive prices when exponentiated.
• Itô’s lemma links the price process to the return formulation.
• Black–Scholes gains tractability from the simple model, but real returns can be skewed and heavy-tailed.
• Jump and stochastic-volatility models can represent dynamics beyond the basic assumption.

Tags

# Asset return distribution


# Asset return distribution












What is the basis for assumption that asset prices follow a log normal distribution? Then how is it transformed to say that asset return follows a normal distribution? How this relationship between normal and log normal distribution is derived and when to use one vs the other, especially w.r.t. Black Scholes Models?

## Answer by Kevin (score 3)

https://quant.stackexchange.com/a/49079

In the Black Scholes (1973) model, the stock price is assumed to follow a geometric Brownian motion $\mathrm{d}S_t=S_t\mu \mathrm{d}t + S_t \sigma \mathrm{d}W_t$. If you solve the SDE, $(S_t)$ is log-normally distributed for every $t$.

Alternative, you can model the returns by a normal distribution and then take the exponential function to obtain the stock price (for positivity). You can see this transformation from Itô's Lemma.

A geometric Brownian motion is firstly very simple and allows for closed-form solutions of the prices of many derivates. It also satisfies naturally desirable properties such as the Markov property. But also recall the Binomial model in which the stock price converges to a log normal distribution as the amount of steps increases.

Nonetheless, there is plenty of empirical evidence that stock returns are in fact not normally distributed and this assumption is to strong. Models incorporating jumps and stochastic volatility aim to improve capturing real life dynamics. In real life, asset returns are skewed and have fat tails. Keep in mind, that the Black Scholes model is one of the first and simplest models.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.