Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 3 / 29K (Stochastic Financial Models)

Let $V_t$ denote wealth measured in units of the strictly positive numéraire. A predictable self-financing portfolio is an arbitrage ifEquivalently, one may permit $V_0\le 0$ provided one of the two gains is strict.

The finite-horizon discrete-time fundamental theorem of asset pricing states that the market has no arbitrage if and only if there exists a probability measure $\mathbb{Q}$ equivalent to $\mathbb{P}$ under which every integrable discounted asset price $X^i$ is a martingale. Such a $\mathbb{Q}$ is an equivalent martingale measure.

Condition (i) says that each discounted risky price $X^i$ is an integrable $\mathbb{Q}$-martingale. For a bounded predictable risky holding process $\theta$, self-financing gives the discounted wealth equation in discrete timeEach increment is integrable andHence $V$ is a $\mathbb{Q}$-martingale, proving (i) $\Rightarrow$ (ii).

If $V$ is a $\mathbb{Q}$-martingale, thenbecause ${\mathcal{F}}_0$ is $\mathbb{P}$-trivial and equivalence makes it $\mathbb{Q}$-trivial as well. Thus (ii) $\Rightarrow$ (iii).

Assume (iii). Fix $t\in \{1,\dots ,T\}$, a coordinate $i$, and an event $A\in {\mathcal{F}}_{t-1}$. Choose the bounded predictable risky strategyand choose the numéraire holding to make the strategy self-financing with $V_0=0$. Its terminal discounted value isCondition (iii) yieldsfor every $A\in {\mathcal{F}}_{t-1}$. The defining property of conditional expectation impliesEvery coordinate is therefore a $\mathbb{Q}$-martingale, so (iii) $\Rightarrow$ (i). This proves the martingale characterization by bounded self-financing strategies.

Suppose an arbitrage existed. Its discounted terminal wealth would satisfy $V_T\ge 0$ $\mathbb{P}$-almost surely and $\mathbb{P}(V_T>0)>0$. Equivalence of $\mathbb{Q}$ and $\mathbb{P}$ gives the same two statements under $\mathbb{Q}$, soBut part (c), after the usual admissibility or localization needed for integrability, givesa contradiction. Hence the existence of the stated martingale measure rules out arbitrage.