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

For a fixed realization $\xi =a$ and $\mathrm{Re}z>1$, direct integration givesTo interchange the integral and expectation, write $x=\mathrm{Re}z>1$ and estimateThe Fubini's theorem now proves the Mellin transform of call prices identityThe required exponential moment is $\mathbb{E}{e}^{x\xi }<\infty$ at the chosen real part; the assertion for every $x>1$ presupposes these exponential moments for every such $x$.

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.

Assume the usual positive initial stock price and strike, so the logarithm is defined, and interpret $U$ as a classical solution of the displayed partial differential equation. For a fixed complex $z$, put $F_t(z)=S_t^{z}U(t,{\sigma }_t,z)$, defining $S_t^z=\exp (z\log S_t)$. The Itô formula for the two Brownian motions with correlation coefficient $\rho$ givesThe partial differential equation cancels the entire drift. By the permission to treat the resulting local martingales as true martingales, and the terminal condition, $F_t(z)=\mathbb{E}[S_T^z\mid {\mathcal{F}}_t]$.

The bounded spot volatility ensures $\mathbb{E}{S}_T^{x_0}<\infty$: the Itô formula for a real power and localization bound its moment by $S_0^{x_0}\exp (x_0(x_0-1)\Vert \sigma {\Vert }_{\infty }^{2}T/2)$. Rewrite the proposed integrand asNow $|F_t(x_0+iy)|\le \mathbb{E}[S_T^{x_0}\mid {\mathcal{F}}_t]$, an integrable bound independent of $y$. Conditional Fubini's theorem and part (b) therefore implyThis route justifies the contour exchange without assuming bounds on $U$ uniform in all complex $z$. Cash is constant, $S$ is a true martingale under the stated allowance, and the displayed $C$ is a true martingale. The original measure is thus an equivalent martingale measure for all three assets. The fundamental theorem of asset pricing gives the market has no arbitrage under the usual admissible trading convention.