Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 211 / 4 / c / Solution

Suppose a numéraire strategy $\eta$ existed and write $N_t={\eta }_{t+1}\cdot P_t={\eta }_t\cdot P_t>0$, using zero consumption. Since ${\eta }_{t+1}$ is ${\mathcal{F}}_t$-measurable and $M_t=(-1{)}^t{Z}_t{P}_t$ is a martingale,ThusorThe left side is nonnegative and the right side nonpositive, so $Z_t{N}_t=0$ almost surely. Strict positivity of $N_t$ implies $Z_t=0$ almost surely for every $t$, contradictingTherefore the market has no numéraire strategy.