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

For initial capital zero and a predictable strategy $H$, let $C_t^{0,H}$ denote consumption after the time-$t$ portfolio payoff and before choosing the next holdings. An investment-consumption arbitrage haswith strictly positive consumption at some date with positive probability. A terminal-consumption arbitrage is a finite-horizon such strategy whose consumption is zero before its terminal date $T$, while $C_T^{0,H}\ge 0$ almost surely and $\mathbb{P}(C_T^{0,H}>0)>0$.

A numéraire strategy $\eta$ has zero consumption and strictly positive wealthat every date. Given an investment-consumption arbitrage $H$, retain its holdings and invest each nonnegative consumption $C_s^{0,H}$ in the numéraire. Withthe self-financing identity for $\eta$ gives zero intermediate consumption for $K$. At a deterministic $T$ after a date at which positive consumption occurs with positive probability, liquidating givesIt is strictly positive with positive probability. Hence $K$ is a terminal-consumption arbitrage.

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.