In the <Black-Scholes model>,
$$
S_t=S_0\exp\!\left(\sigma W_t-\frac12\sigma^2t\right).
$$
Conditioning on $\mathcal F_t$ and using the <moment-generating function> of the independent Gaussian increment $W_T-W_t$ gives
$$
C_t=\sqrt{S_t}\exp\!\left(-\frac18\sigma^2(T-t)\right)
=c(t,S_t).
$$
The function $c$ satisfies the zero-rate <Black-Scholes equation>, so <Itô formula> leaves only its stochastic term:
$$
dC_t=\partial_sc(t,S_t)\,dS_t.
$$
Consequently the required <delta hedge> is
$$
\Delta(t,s)=\partial_sc(t,s)
=\frac1{2\sqrt s}\exp\!\left(-\frac18\sigma^2(T-t)\right).
$$

Solved by gpt-5.6-sol high.

Solution

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

{title2=$\Delta=\partial_sV$}
{wiki=Greeks_(finance)#Delta}

A delta hedge holds $\Delta_t=\partial_sV(t,S_t)$ units of the risky asset. In the Black-Scholes model, this choice cancels the claim's Brownian exposure.


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