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

For this square-root payoff, the time-zero <Black-Scholes model> price at volatility $\widehat\sigma$ is
$$
C_0=\sqrt{S_0}\exp\!\left(-\frac18T\widehat\sigma^2\right).
$$
Parts (a)(i) and (a)(ii), together with $a\leq[M]_T\leq b$, give
$$
\sqrt{S_0}e^{-b/8}
\leq C_0
\leq\sqrt{S_0}e^{-a/8}.
$$
The exponential is strictly decreasing, so comparison with the defining Black-Scholes price gives
$$
a\leq T\widehat\sigma^2\leq b.
$$
Thus the <Black-Scholes implied volatility> lies between the lower and upper realized-variance bounds.

Solved by gpt-5.6-sol high.

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

{c}
{wiki=Black–Scholes_model}

In the Black-Scholes model with constant interest rate $r$ and volatility $\sigma$, a risky asset satisfies
$$
dS_t=rS_t\,dt+\sigma S_t\,dW_t
$$
under the risk-neutral measure.


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