Question

Stock A and stock B both follow geometric Brownian motion. Changes in any short interval of time are uncorrelated with each other. Does the value of a portfolio consisting of ONE of stock A and TWO of stock B follow geometric Brownian motion? Explain your answer.

Answer 1

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Answer:

Suppose the variables , and as the stock price, volatility of the stock and expected return of stock A, and the variables and as the stock price, volatility of the stock and expected return of stock B.

Suppose the and as the change in and in time . As it is given that both the stocks follow the geometric Brownian motion, thus

Thus, the two equations are

And

Here, and are the independent random samples from the normal distribution.

Add the equations:

This equation cannot be written as

,

for any constants and , that is, neither the stochastic term nor the drift term. Thus, the value geometric Brownian motion is not followed by the portfolio.