Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 211 / 1 / d / Solution

Let $Y$ be a martingale deflator for the original arbitrage-free market. Part b shows thatis a local martingale. Thus $Y$ also deflates the gains of the added asset whose price and dividend are $({\pi }^H,{\xi }^H)$. It already deflates the original $n$ assets, so it is a martingale deflator for the enlarged market. The fundamental theorem of asset pricing implies that the enlarged market has no arbitrage. This expresses the fact that adding a dynamically replicated asset cannot create an arbitrage.