Why Log Returns Solve the Geometric Brownian Motion SDE
Summary
The document discusses why the Black-Scholes stock-price model uses a stochastic differential equation with drift and volatility proportional to the stock price, and why applying Itô’s lemma to the logarithm yields an explicit solution. The explanation places the transformation in the history of financial modeling rather than treating it as a trick developed for simulation.
The response notes that Brownian motion was a simple continuous-time price model but allows normally distributed prices, including negative values. Samuelson had already studied geometric Brownian motion, which instead gives normally distributed log returns and positive stock prices. Itô’s lemma was used to connect the proportional price process to its explicit solution. The account says this work predates Monte Carlo simulation in finance, so simulation was not the motivation. The document offers historical context and a conceptual rationale, but does not derive the solution step by step or discuss the model’s empirical limitations.
Key ideas
- Applying Itô’s lemma to the logarithm of price solves the geometric Brownian motion equation.
- A Brownian motion model for price can assign positive probability to negative prices.
- Geometric Brownian motion models log returns as normally distributed and keeps prices positive.
- Samuelson studied geometric Brownian motion before Black-Scholes.
- The historical use of the log transformation predates financial Monte Carlo simulation.
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# Idea of using logarithm for solving SDE in Black-Scholes model
# Idea of using logarithm for solving SDE in Black-Scholes model
In the Black-Scholes model they consider that the stock follows this stochastic differential equation: $$ dS = \mu S dt + \sigma S\ dW $$
I was wondering, was it common at the time they work on this to use $\log S$ with Itô lemma to solve this kind of equation, or do they discover it ?
Or do they use it because they assume from the beginning it was lognormal, then with applying it, we would obtain :
$$ \begin{aligned} S_T = S_0 * \exp^{\left(\mu - \frac{\sigma^2}{2}\right)dt + \sigma dW} \end{aligned} $$
I'm a bit confused about this because it seems obivous to apply $\log S$ when they give this equation for example to simulate the paths with Monte-Carlo.
## Answer by Kevin (score 6, accepted)
https://quant.stackexchange.com/a/53500
Black and Scholes (1973) were not the first ones to use the geometric Brownian motion as a model for stock prices. For example, Samuelson did it before them.
It all started with a Brownian motion as simplest time continuous stock price model. However, then the stock price is normally distributed and can be negative. Not a great property! So, Samuelson exponentiated the model and studied a geometric Brownian motion, in which the log returns are normally distributed. But, of course, the researchers at the time knew Ito's Lemma very well and how to get from $\mathrm{d}S_t=\mu S_t \mathrm{d}t+\sigma S_t\mathrm{d}W_t$ to an explicit solution for $S_t$.
Here is a part of Paul Samuelson's seminal 1965 paper (Rational theory of warrant pricing)
Note, all of this was done before Monte Carlo simulations were a thing in finance. So, it has nothing to do with this.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.