Geometric Brownian Motion with Nonuniform Time Grid

seed

1

h coefficient

h

1

time steps

n

50

GBM paths

m

100

time to expiry (years)

T

3

starting asset price

S

0

40

strike price

X

45

risk-free rate

r

0.07

dividend yield

δ

0.

volatility

σ

0.3

This Demonstration simulates geometric Brownian motion (GBM) paths with a nonuniform time grid. A GBM is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. In computational finance, GBM is used to model stock prices in the Black–Scholes model and is the most widely used model of stock price behavior.

As an example, this Demonstration generates random GBM paths and uses Monte Carlo simulation to estimate the price of a European put option, with starting stock price

S

0

, strike price

X

, maturity time

T

, stock price volatility

σ

, risk-free interest rate

r

, and stock dividend yield

δ

. The simulation output is compared against the analytic Black–Scholes output, calculated with Mathematica's built-in function FinancialDerivative.

The

h

coefficient (

0<h≤1

) generates nonuniform time steps with decreasing length as we approach maturity, according to the model:

dt

i

=h·

dt

i-1

. If

h=1

, the time grid becomes uniform.