By the bound in part (a), <Fubini's theorem> applies. Conditional on $S$,
$$
\begin{aligned}
&\sqrt K\,\mathbb E_Y\left[
S^{(1+iY)/2}e^{-iY\log K/2}
\right]\\
&\qquad=\sqrt{KS}\,
\mathbb E_Y\exp\left(\frac{iY}{2}\log\frac SK\right)\\
&\qquad=\sqrt{KS}\,
\exp\left(-\frac12\left|\log\frac SK\right|\right)
=\min(S,K),
\end{aligned}
$$
where the <Characteristic function of the Cauchy distribution> was used. Since $\mathbb ES=1$,
$$
\mathbb E[(S-K)^+]
=\mathbb E[S-\min(S,K)]
=\boxed{1-\sqrt K\,
\mathbb E\left[M\left(\frac{1+iY}{2}\right)
e^{-iY\log K/2}\right].}
$$
The formula expresses a <European call option> value through complex moments of $S$. In an affine stochastic-volatility model such as the <Heston model>, those moments are available from an explicit transform, so call prices reduce to a one-dimensional Fourier expectation or integral.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 211 / 2 / c / Solution

{c}

For $X\sim\operatorname{Cauchy}(x_0,\gamma)$,
$$
\mathbb E[e^{itX}]=e^{ix_0t-\gamma|t|}.
$$
In particular, the <Standard Cauchy distribution> has characteristic function $e^{-|t|}$.

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 Incoming links: Characteristic function of the Cauchy distribution