 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ii/paper-3/27i/c/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

A power option need not have bounded payoff, but all real powers of a positive lognormal stock have finite moments, so risk-neutral valuation still applies. The Gaussian moment-generating identity gives

E [e^{nσ B_{T}}] = e^{n^{2} σ^{2} T /2}

. Therefore

V (0, S_{0}) = S_{0}^{n} exp ([(n - 1) r + \frac{1}{2} n (n - 1) σ^{2}] T) .

For

n = 1

this reduces to

S_{0}

, and for

n = 0

to the time-zero value

e^{- r T}

of one unit paid at expiry.

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