Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 211 / 5 / b / Solution

Set $z=x_0+iy$. Since $|M(z)|\le \mathbb{E}{e}^{x_0\xi }$ and $|z(z-1){|}^{-1}$ is integrable over $y\in \mathbb{R}$, the proposed contour integral is absolutely convergent. The same bound justifies the Fubini's theorem when substituting $M(z)=\mathbb{E}{e}^{z\xi }$:The last equality uses the contour identity provided in the original PDF. ConsequentlyThis is contour inversion for call prices. The supplied TeX corrupts the permitted identity into a reciprocal and drops the imaginary units from its endpoints; the PDF has $(e^a-1{)}^{+}$ and the vertical contour used above.