Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 211 / 1 / c / Solution

For any holdings $h$, the pricing kernel identity implies $V_0=\mathbb{E}[R{V}_1]$. If $V_1\ge 0$ almost surely, then $V_0\ge 0$, so an arbitrage cannot have $V_0<0$. If also $V_0=0$, then the nonnegative integrable random variable $R{V}_1$ has zero expectation and hence is zero almost surely. Since $R>0$ almost surely, $V_1=0$ almost surely, ruling out a positive terminal gain. Therefore a pricing kernel rules out arbitrage.