Assume S is a geometric Brownian motion: dS S = dt + dB for constants and

Assume S is a geometric Brownian motion:

dS S = μdt +σ dB for constants μ and σ and a Brownian motion B.

(a) Show that vart

St+1 St



= e 2μ

*

e

σ 2

−1

+

.

Hint: Compare Exercise 1.7.

(b) Use the result of the previous part, the formula (12.22), and the approximation ex ≈ 1+ x to derive approximate formulas for vart(St+1/St) and Et[St+1/St].