Why Geometric Brownian Motion Uses Log Returns Instead of Simple Returns
Summary
The document considers whether Monte Carlo simulation can be applied directly to discrete stock percentage changes rather than to stock prices. The response highlights a key distinction between simple returns and the log returns used in geometric Brownian motion. A price falling from a positive level to zero corresponds to a log return of negative infinity, while its simple return is negative one, so the two representations behave differently at the boundary.
A process built from discrete returns would need to handle the possibility of invalid negative prices, which may require substantial changes to the usual formulas. The text does not provide an alternative process, formal conditions for simulation, or empirical evidence. It offers a conceptual caution about transferring a GBM setup to simple percentage changes without accounting for the different return scale and price constraints.
Key ideas
- Geometric Brownian motion is formulated using log returns rather than discrete percentage returns.
- A move to zero has a finite simple return but an unbounded negative log return.
- A stochastic process based on discrete returns must account for the possibility of negative simulated prices.
- The document cautions that adapting GBM requires changes but does not specify a complete alternative model.
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# Monte Carlo simulations of stock price percentage change rather than stock price
# 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 $i<j$.
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?
## Answer by ZRH (score 1, accepted)
https://quant.stackexchange.com/a/43806
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.