Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 202 / 4 / c

Table of contents
- Solution c

Solution

In the Black-Scholes model,

S_{t} = S_{0} exp (σ W_{t} - \frac{1}{2} σ^{2} t) .

Conditioning on

F_{t}

and using the moment-generating function of the independent Gaussian increment

W_{T} - W_{t}

gives

C_{t} = S_{t} exp (- \frac{1}{8} σ^{2} (T - t)) = c (t, S_{t}) .

c

satisfies the zero-rate Black-Scholes equation, so Itô formula leaves only its stochastic term:

d C_{t} = \partial_{s} c (t, S_{t}) d S_{t} .

Consequently the required delta hedge is

Δ (t, s) = \partial_{s} c (t, s) = \frac{1}{2 s} exp (- \frac{1}{8} σ^{2} (T - t)) .

Solved by gpt-5.6-sol high.

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