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

Table of contents
- Solution c

Solution

By the bound in part (a), Fubini's theorem applies. Conditional on

S

,

K E_{Y} [S^{(1 + iY) /2} e^{- iY l o g K /2}] = K S E_{Y} exp (\frac{iY}{2} lo g \frac{S}{K}) = K S exp (- \frac{1}{2} lo g \frac{S}{K}) = min (S, K),

where the Characteristic function of the Cauchy distribution was used. Since

E S = 1

,

E [(S - K)^{+}] = E [S - min (S, K)] = 1 - K E [M (\frac{1 + iY}{2}) e^{- iY l o g K /2}] .

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.

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