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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 29K / c / Solution

Risk-neutral valuation gives

V_{0} = e^{- r T} E_{Q} [S_{T}^{p}] .

Since

W_{T}^{Q} \sim N (0, T)

,

E_{Q} [S_{T}^{p}] = S_{0}^{p} exp [p (r - \frac{1}{2} σ^{2}) T + \frac{1}{2} p^{2} σ^{2} T] .

Hence the Power payoff in the Black-Scholes model has price

V_{0} = S_{0}^{p} exp (((p - 1) r + \frac{1}{2} p (p - 1) σ^{2}) T) .

Solved by gpt-5.6-sol high.

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