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

By the bound in part (a), Fubini's theorem applies. Conditional on $S$,where the Characteristic function of the Cauchy distribution was used. Since $\mathbb{E}S=1$,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.