# 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.

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.

• Interesting, thanks ! – TmSmth Apr 21 '20 at 19:29