Suppose dS/S = dt + dB for constants and and a Brownian motion B. Let r be a constant. Consider a wealth

Suppose dS/S = μdt + σ dB for constants μ and σ and a Brownian motion B. Let r be a constant. Consider a wealth process W as defined in Section 12.2:

dW W = (1 −π )r dt + π

dS S , where π is a constant.

(a) By observing that W is a geometric Brownian motion, derive an explicit formula for Wt.

(b) For a constant ρ and dates s < t, calculate Es[W1−ρ

t ]. Hint: write W1−ρ

t = e(1−ρ)logWt .

(c) Consider an investor who chooses a portfolio process to maximize E

 1 1−ρ

W1−ρ

T



.

Show that if a constant portfolio πt = π is optimal, then the optimal portfolio is

π = μ− r

ρσ2 .