Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 / 3

A claim $\xi$ is replicated if some holdings $h\in {\mathbb{R}}^n$ satisfy $h\cdot P_1=\xi$ almost surely; its replication cost is $h\cdot P_0$. The market is complete when every claim in the specified integrability class can be replicated.

Every indicator random variable must lie in the linear span of the $n$ terminal asset prices. If the sigma-algebra had $n+1$ disjoint events of positive probability, their indicators would be linearly independent, yet completeness would place all of them in an at-most-$n$-dimensional space. Hence, modulo null events, the sample space has at most $n$ positive-probability atoms.

For every replicable claim $\xi =h\cdot P_1$,Completeness makes every bounded claim replicable. Taking increasing bounded truncations of $\mathrm{sgn}(X-Y)$, or using the finite-atom conclusion of part b directly, gives $\mathbb{E}|X-Y|=0$. Thus $X=Y$ almost surely.

The proposed variable satisfiesNo arbitrage and the fundamental theorem of asset pricing provide a strictly positive one-period deflator $Y$ with $\mathbb{E}(Y{P}_1)=P_0$. Part c makes such a deflator unique in a complete market, so $Z=Y>0$ almost surely.