Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 29K (Stochastic Financial Models)

An equivalent martingale measure is a probability measure $\mathbb{Q}$ on the terminal sigma-algebra such that $\mathbb{Q}$ and the physical measure $\mathbb{P}$ are mutually absolutely continuous and every asset price discounted by the chosen numéraire is a $\mathbb{Q}$-martingale with respect to the market filtration.

The fundamental theorem of asset pricing for a finite discrete-time market states that the market is free of arbitrage if and only if it admits an equivalent martingale measure. Subject to the usual finite-market hypotheses, the arbitrage-free market is complete if and only if that measure is unique.

Define a new probability measure by the strictly positive Radon-Nikodym derivativeThe moment-generating function of a normal distribution shows that this density has expectation one, so $\mathbb{Q}$ is a probability measure equivalent to $\mathbb{P}$. Moreover,Thus the discounted risky asset is a one-period $\mathbb{Q}$-martingale. The fundamental theorem of asset pricing implies that there is no arbitrage.

For each period setand defineIndependence of the $Z_i$ shows that this strictly positive density has expectation one. Under $\mathbb{Q}$, the conditional mean of each gross return isConsequentlyso the discounted price process is a $\mathbb{Q}$-martingale. This equivalent martingale measure rules out arbitrage.