 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/iii/paper-39/6/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Put

h = S_{0} - μ / R

. The initial capital of the one-period quadratic hedge is

X_{0}^{*} = ϕ^{*} + (π^{*})^{T} S_{0} = \frac{m}{R} + h^{T} V^{- 1} c .

ρ^{*} = \frac{1}{R} + h^{T} V^{- 1} (S_{1} - μ) .

It is square integrable, and

E [(S_{1} - μ) ξ_{1}] = c

, so

E [ρ^{*} ξ_{1}] = \frac{m}{R} + h^{T} V^{- 1} c = X_{0}^{*} .

The weight depends only on the asset prices and their first two moments, not on the claim. It can be negative, so this identity alone does not supply an equivalent martingale measure or a positive state-price density.

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