 Solution

ID: past-exam-of-the-mathematical-models-of-financial-derivatives-course-of-the-university-of-oxford/2021/1/a/ii/solution

For

V \in C^{1, 2}

d V = (V_{t} + μ s V_{s} + \frac{σ ^{2}}{2} V_{ss}) d t + σ V_{s} d W .

ϕ = V_{s}

and bond value

ψ B = V - s V_{s}

. Equating drifts cancels the physical drift

μ

and gives

V_{t} + rs V_{s} + \frac{σ ^{2}}{2} V_{ss} - r V = 0, (t, s) \in [0, T) \times R, V (T, s) = F (s) .

This is the pricing equation for a linear asset with additive noise. Its spatial domain is the whole real line, because the asset can be negative. In particular the diffusion term is

σ^{2} V_{ss} /2

, not

σ^{2} s^{2} V_{ss} /2

. The derivation is delta hedging and does not impose a boundary condition at

s = 0

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