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.


1 Answer 1


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)

enter image description here

Note, all of this was done before Monte Carlo simulations were a thing in finance. So, it has nothing to do with this.

  • 1
    $\begingroup$ Interesting, thanks ! $\endgroup$
    – TmSmth
    Apr 21, 2020 at 19:29

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