# Geometric Brownian Motion: d(S) vs. d(ln(S))

I am quoting from "Tools for Computational Finance, 5th Edition" [Seydel].

I wonder whether the histogram of simulations of the first (yellow) SDE makes sense... especially given that Seydel (correctly) states that the resulting percentage changes are normally distributed (i.e. can go below -100%).

Shouldn't the histogram much rather correspond to asset prices simulated from the second (red) SDE, as the second SDE would produce log-prices which are normally distributed, i.e. prices which are log-normally distributed? The related question How to simulate stock prices with a Geometric Brownian Motion? does not quite cover this.

• OP here, I understand the (continuous version of) 1.33 and the SDE 1.46 are related through the change in variable - however, aren't they two completely different equations resulting in completely different distributions of S? – SuperUser01 Jan 2 '16 at 23:11
• They are the same thing. You'll need to apply Ito's Lemma to go from one to the other. Have you done Ito's Lemma? – SmallChess Jan 3 '16 at 2:12
• Make sure you see the difference between $ds=\mu dt + \sigma dW$, which results in a normal distribution and $ds=\mu S dt + \sigma S dW$ which results in the lognormal. – Alex C Jan 3 '16 at 16:03

Equation 1.33 results in lognormally distributed prices. The price can't go below zero, because the equation is being integrated over infinitesimally short periods of time. The discrete approximation 1.34a could go negative if $\Delta t$ is big, if it is reasonably small that is unlikely.