Simulate the path of a Brownian motion over a year (using your favorite programming language or Excel) by simulating N standard normal random variables zi and calculating Bti = Bti−1 +zi

√t for i = 1,...,N, where

t = 1/N and B0 = 0. (To simulate a standard normal random variable in a cell of an Excel worksheet, use the formula = NORMSINV(RAND()).)

(a) Plot the path—the set of points(ti,Bti

).

(b) Calculate the sum of the (Bti

)2. Confirm that for large N the sum is approximately equal to 1.

(c) Calculate the sum of |Bti

|. Confirm that this sum increases as N increases. Note: The sum converges to ∞ as N → ∞ (because a Brownian motion has infinite total variation), but this may be difficult to see.

(d) Use the simulated Brownian motion to simulate a path of a geometric Brownian motion via the formula (12.23). Plot the path.