 Solution

ID: past-exam-of-the-martingales-and-financial-mathematics-course-of-the-university-of-oxford/2011/b/5/d/solution

V_{t} = A (t) S_{t}^{3}

, where

A^{'} (t) = - (2 r + 3 σ^{2}) A (t)

. Since

d S_{t} / S_{t} = μ d t + σ d W_{3} (t)

, the drift and diffusion are

\frac{d V _{t}}{V _{t}} = (- (2 r + 3 σ^{2}) + 3 μ + 3 σ^{2}) d t + 3 σ d W_{3} (t) .

Thus

\overline{μ} = 3 μ - 2 r, \overline{σ} = 3 σ .

Under the risk-neutral measure its drift becomes

3 r - 2 r = r

, as required for a traded, non-dividend-paying replicating asset.

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