Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 29K (Stochastic Financial Models)

Let $\varphi =({\varphi }_n^0,{\varphi }_n^1,\dots ,{\varphi }_n^d)$ be a predictable self-financing portfolio, and let $V_n^{\varphi }={\varphi }_n\cdot S_n$ be its value. An arbitrage is such a strategy with

The model is a complete market when every contingent claim in the specified class, usually every ${\mathcal{F}}_T$-measurable payoff satisfying the required integrability condition, is the terminal value of some admissible self-financing portfolio.

Set $Y_n=\log Z_n$ andDefine a probability measure ${\mathbb{P}}^{\ast }$ by the positive densityThe density has expectation one and is strictly positive, so ${\mathbb{P}}^{\ast }$ is equivalent to $\mathbb{P}$. By exponential tilting, the variables $Y_k$ remain independent under ${\mathbb{P}}^{\ast }$ and have distribution $N({\mu }_{\ast },{\sigma }^2)$. Thus $Z_k$ has a log-normal distribution andFor the discounted stock ${\tilde{S}}_n=S_n/S_n^0$,Hence ${\mathbb{P}}^{\ast }$ is an equivalent martingale measure.

There is no arbitrage. Indeed, part (iii) supplies an equivalent martingale measure, so the fundamental theorem of asset pricing implies absence of arbitrage. Directly, the discounted terminal value of a zero-cost admissible self-financing strategy is a ${\mathbb{P}}^{\ast }$-martingale with mean zero; it cannot be nonnegative almost surely and positive with positive probability.

Suppose the claim $H=(S_1{)}^2$, payable at time $1$, were replicable. Holdings chosen at time zero are constants, so the time-$1$ value of any self-financing portfolio has the affine formBut $S_1=S_0{Z}_1$ has a continuous positive density on $(0,\infty )$. No affine function of $S_1$ can equal the quadratic function $S_1^2$ almost surely. Therefore $H$ is not attainable and the market is not complete.

The same argument applies if $H$ is viewed as a time-$T$ claim: once $H$ is known at time $1$, its required discounted value at time $1$ is fixed, and the first-period portfolio value would still have to reproduce a nonaffine function of $S_1$.