Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 40 / 3 / c

Use the conditional second-moment matrix $V_t=\mathbb{E}[P_t{P}_t^{\top }\mid {\mathcal{F}}_{t-1}]$ from the PDF. By part (b), the filtration is finite-state at every finite time. On a time-$(t-1)$ atom let $p_1,\dots ,p_m$ be the successor price vectors and $q_1,\dots ,q_m>0$ their conditional probabilities. Then $m\le n$ andPositive definiteness gives rank $n$, so $m\ge n$. Hence $m=n$, and the $n$ successor vectors have linear independence.

We use the finite-horizon fundamental theorem of asset pricing in deflator form: an arbitrage-free finite market admits a strictly positive adapted process $D$, with $D_0=1$, such that $DP$ is a martingale. Equivalently, on every step,Fix a finite horizon containing the step in question. On the parent atom, write $d_i=D_t/D_{t-1}>0$ on successor $i$ and $p_0=P_{t-1}$. ThenThe proposed values $z_i=p_i^{\top }{V}^{-1}{p}_0$ satisfy precisely the same equation:Because the $p_i$ have linear independence and the $q_i$ are positive, this linear system has a unique solution. Thus $z_i=d_i>0$, provingThis is the positive regression deflator in a complete finite market. It also proves the suggested conclusion: with $Y_0=1$ and $Y_t={\prod }_{u=1}^t{Z}_u$,Hence $Y$ is a strictly positive martingale deflator. Finite-state structure makes these expectations integrable on each finite horizon. Positive definiteness alone would not ensure positivity of the regression factor; market completeness and the positive deflator are essential.