# Monte Carlo simulations of stock price percentage change rather than stock price

Say we have a stock price time series $$S_k$$. We can do monte carlo simulations on the stock price to make predictions about future prices (e.g. through Geometric Brownian Motion SDE's).

Does it make sense to do the same sort of monte carlo simulations on the stock price percentage change? So, if we transform our data $$S_k$$ as such

$$p = 100 \times \frac{S_j - S_i}{S_i}$$

for some time indices $$i.

The transformed quantity $$p$$ is the percentage change of the stock price for some time period $$i < t < j$$. The only difference is that the percentage changes can be negative, whereas stock prices are always positive. Furthermore, in some cases, the quantity $$p$$ will behave like white noise.

Is it valid to do monte carlo simulations on stock price percentage change? If so, what conditions do we have to impose and what changes need to be made to the analysis? If not, why not?

• sure you can, but it can complicate things later, for example with dividends. Why do you want to do this rather than just starting with the spot? – will Jan 29 '19 at 21:17

Agree with will that this approach will complicate things, mostly for the fact that GBM SDEs rely on log returns, and not discrete returns. To go from some finite underlying price level $$S$$ to $$0$$ means a log return of $$-\infty$$, whereas the equivalent discrete return is $$-1$$. To ensure a discrete return - based stochastic process, where $$S$$ can never take a negative value would likely mean cumbersome tinkering with the formulae.