MATLAB Language Financial Applications Univariate Geometric Brownian Motion
Example
The dynamics of the Geometric Brownian Motion (GBM) are described by the following stochastic differential equation (SDE):
I can use the exact solution to the SDE
to generate paths that follow a GBM.
Given daily parameters for a year-long simulation
mu = 0.08/250;
sigma = 0.25/sqrt(250);
dt = 1/250;
npaths = 100;
nsteps = 250;
S0 = 23.2;
we can get the Brownian Motion (BM) W starting at 0 and use it to obtain the GBM starting at S0
% BM
epsilon = randn(nsteps, npaths);
W = [zeros(1,npaths); sqrt(dt)*cumsum(epsilon)];
% GBM
t = (0:nsteps)'*dt;
Y = bsxfun(@plus, (mu-0.5*sigma.^2)*t, sigma*W);
Y = S0*exp(Y);
Which produces the paths
plot(Y)
