1
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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.

| improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ Interesting, thanks ! $\endgroup$ – TmSmth Apr 21 at 19:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.