Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 211 / 3

Choose any strictly positive pricing kernel $Z\in \mathcal{Z}$. The assumed componentwise expectations and the linearity of expectation giveThe integrand is nonnegative, so its expectation is also nonnegative and hence zero. A nonnegative random variable with zero expectation vanishes almost surely. Since $Z>0$ almost surely, this forces $H\cdot P=0$ almost surely. Both conclusions follow:The strict positivity of the pricing kernel is essential: a kernel allowed to vanish could miss a positive payoff on its zero set. No normalization $\mathbb{E}Z=1$ is assumed in this abstract market; that normalization follows only if a unit-priced unit-payoff cash asset is included.

For fixed $\gamma$, differentiation under the integral sign is justified by bounded $P$ and $X$. At the unconstrained minimizer $H_{\gamma }$, the first-order condition isDefine the strictly positive, bounded pricing kernelThe first-order condition says $\mathbb{E}[Z_{\gamma }P]=p$, so $Z_{\gamma }\in \mathcal{Z}$. When taking the requested partial derivative, hold $H$ fixed and only afterwards set $H=H_{\gamma }$. Writing $A_{\gamma }=H_{\gamma }\cdot p-x$, we obtainThe assumed dual bound applies to this particular pricing kernel. ThusNo derivative of the map $\gamma \mapsto H_{\gamma }$ is needed; confusing a partial derivative with a derivative along the minimizing path would add an unnecessary hypothesis.

Let $M={\sup }_{\gamma >1}{F}_{\gamma }(H_{\gamma })<\infty$. Choose ${\gamma }_j\to \infty$. Boundedness of the minimizers and the Bolzano-Weierstrass theorem yield a subsequence, still indexed by $j$, with $H_{{\gamma }_j}\to H^{\ast }\in {\mathbb{R}}^n$. Each $H_{{\gamma }_j}$ is deterministic, hence so is $H^{\ast }$. The first exponential term givesand passing to the limit gives $H^{\ast }\cdot p\le x$.

To obtain the superhedging inequality, suppose instead that $\mathbb{P}(X>H^{\ast }\cdot P)>0$. There is then an $\epsilon >0$ such that the event $E=\{X-H^{\ast }\cdot P\ge \epsilon \}$ has positive probability. Let $\Vert P\Vert \le B$ almost surely. For all sufficiently large $j$,Consequently $X-H_{{\gamma }_j}\cdot P\ge \epsilon /2$ on $E$, givinga contradiction. The limiting portfolio meets both constraints:Bounded $P$ is used precisely to turn convergence of deterministic holdings into a uniform bound on the payoff error.

Let $h$ be the stock holding and let $c$ be the initial cost, so the cash holding is $b=c-10h$. Terminal portfolio wealth at stock price $s$ is $c+h(s-10)$. Dominating the European call option payoff at the three possible prices givesAdding the two endpoint inequalities yields $2c\ge 1$. Equality is attained by $h=1/2$ and $c=1/2$, which imply $b=-9/2$. The corresponding terminal wealth is $0,1/2,1$ at $s=9,10,11$, respectively, compared with the required payoff $0,0,1$. The cheapest super-replication strategy isThe middle-state excess shows why this is superhedging rather than exact claim replication. The endpoint bound proves global minimality, without relying on the physical state probabilities.