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

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.