Learn about Geometric Brownian Motion and download a spreadsheet
Stock prices are often modeled as the sum of
- the deterministic drift, or growth, rate
- and a random number with a mean of 0 and a variance that is proportional to dt
This is known as Geometric Brownian Motion, and is commonly model to define stock price paths. It is defined by the following stochastic differential equation.
St is the stock price at time t, dt is the time step, μ is the drift, σ is the volatility, Wt is a Weiner process, and ε is a normal distribution with a mean of zero and standard deviation of one .
Substituting Equation 2 into Equation 1 gives
Hence dSt is the sum of a general trend, and a term that represents uncertainty.
Simulate Geometric Brownian Motion in Excel
Converting Equation 3 into finite difference form gives
Bear in mind that ε is a normal distribution with a mean of zero and standard deviation of one. This can be represented in Excel by NORM.INV(RAND(),0,1).
The spreadsheet linked to at the bottom of this post implements Geometric Brownian Motion in Excel using Equation 4.