Black–Scholes Assumes Normal Log Returns and a Lognormal Price

Summary

The document resolves an apparent conflict between two descriptions of the Black–Scholes model: that stock prices follow geometric Brownian motion with drift, and that the model assumes a normal distribution. These statements are consistent. Under geometric Brownian motion, the continuously compounded returns over a fixed interval are normally distributed, while the stock price itself has a lognormal distribution.

The answer points to the geometric Brownian motion solution as support for this equivalence. It also mentions the limitation that the model’s normal-return assumption does not represent extreme market moves well, such as crashes. The discussion is a concise conceptual clarification rather than a derivation or empirical test, and it does not address alternative models or how the assumption affects particular option valuations.

Key ideas

• Geometric Brownian motion with drift implies normally distributed log returns over a fixed interval.
• The corresponding stock price distribution is lognormal, rather than normal.
• The normality assumption can underrepresent extreme market movements such as crashes.

Tags

# What is the distribution assumption of the black scholes model


# What is the distribution assumption of the black scholes model












As per wikipedia the Black Scholes assumption is:

(`random walk) The instantaneous log returns of the stock price is an infinitesimal random walk with drift; more precisely, it is a geometric Brownian motion`

But later on, under section, under this section to the right, there is picture and it says:

```
    The normality assumption of the Black–Scholes
 model does not capture extreme movements such as stock market crashes.
```

So does it assume a normal distribution or a GBM with drift?

## Answer by SmallChess (score 1, accepted)

https://quant.stackexchange.com/a/18199

In the Black-Scholes framework, we assume the log returns are normally distributed. This is equal to saying the underlying is log-normally distributed. If you look at Geometric Brownian Motion on wikipedia, you'll see this:

```
The above solution  S_t  (for any value of t) is a **log-normally distributed** random variable
```

The wikipedia is correct.

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