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

For $\theta =p+iq$ with $0\le p\le 1$,The elementary inequality $x^p\le 1+x$ for $x>0$ givesTherefore the Mellin transform $M(\theta )=\mathbb{E}[S^{\theta }]$ is absolutely well-defined throughout the strip and

HereThe Laplace transform of an exponential distribution is $\mathbb{E}[e^{-sX}]=(1+s{)}^{-1}$ for $\mathrm{Re}s>-1$. Since $\mathrm{Re}(t\theta )\ge 0$,

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.

Condition on $S$ and use the characteristic function of a standard normal distribution:The integrand has modulus one, so Fubini's theorem is immediate. Taking expectation over $S$ yields