Calculating the Price Density of a Geometric Brownian Motion Model
Summary
The article presents a formula for the probability density of an asset’s future price under geometric Brownian motion (GBM), along with an R function that evaluates the density at a given price. Inputs include the starting price, per-step expected return, volatility, and elapsed steps. A worked illustration assumes a starting value of $100, zero expected return, and 30% annualized volatility, converted to a daily scale, to plot the distribution after 30 trading steps.
The method makes it possible to inspect how a chosen set of GBM assumptions maps into a range of possible prices. Its conclusions are entirely conditional on those assumptions: the example does not test whether returns are lognormal, estimate parameters from data, or compare the model with observed outcomes. The density is therefore a model-based description rather than a forecast guarantee, and the article offers no trading rule or evidence of strategy performance.
Key ideas
- GBM implies a lognormal distribution for positive asset prices under its standard assumptions.
- The price density depends on starting price, drift, volatility, and elapsed time.
- Annualized volatility must be converted to the time step used by the model.
- A plotted density shows outcomes implied by assumptions but does not validate those assumptions.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.