Suppose dS/S = dt + dB for constants and and a Brownian motion B.
Question:
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 .
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