# Simulation of geometric Brownian motion

## Math 339, Spring 2007

### Version 5.6.07

- A B C D E
1 Normal random variable BM Geometric BM mu sigma
2 - (starting value) (starting value) (value of mu) (value of sigma)
3 (see below) =B2+A3 =C2 + C2*(\$E\$2+\$F\$2*A3) - -
4 (copy and paste to cells below) (copy and paste to cells below) (copy and paste to cells below) - -

### Notes

• This spreadsheet gives a simulation of both Brownian motion (in cloumn B) and geometric Brownian motion (in column C; this is the standard model for stock prices).
• The stochastic differential equation for geometric Brownian motion is dS/S = (mu)dt + (sigma)W_t. Enter the starting value for the Brownian motion in cell B2, and the starting value for geometric Brownian motion in cell C2. Put the value of mu in cell D2 and the value of sigma in cell E2.
• Column A contains random numbers drawn from standard normal distribution. In cell A3 put the following:
= RAND() + RAND() + RAND() + RAND() + RAND() + RAND() + RAND() + RAND() + RAND() + RAND() + RAND() + RAND() - 6. The function RAND() gives a uniformly distributed random number between 0 and 1. There are 12 "RAND()'s" in this expression. When these are added and six is subtracted the result is a pretty good approximation of a normal random variable.

Version 3.11.07