Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 30K (Stochastic Financial Models)

PutFor a risky holding vector $\theta$ and the remaining wealth in the bank,soWhen $V$ is nonsingular it is positive definite. Cauchy--Schwarz in the
$V$ inner product giveswith equality exactly when $\theta =\lambda V^{-1}q$. Under
${\theta }^{T}q=h$,

For the constraint $\mathbb{E}{X}_1\ge m$, if $h\le 0$ the zero risky portfolio is feasible and the minimum variance is zero. If $h>0$, the inequality binds and the preceding optimizer and minimum apply.

For every nonzero-risk portfolio,Equality holds precisely forThe maximum is therefore $\sqrt{q^T{V}^{-1}q}$. For each resulting mean, part (a) shows that this portfolio has the least possible variance, so every maximizer lies on the mean-variance efficient ray.

For a symmetric matrix,
$\mathrm{range}V=(\ker V{)}^{\perp }$. Since $q\notin \mathrm{range}V$, there is
$\eta \in \ker V$ with ${\eta }^{T}q\ne 0$. For any target excess mean $h$, setThen ${\theta }^{T}q=h$ andThus the required minimum is zero for every $m,x$.

Choose the sign of $\eta$ so that ${\eta }^{T}q>0$, buy the risky portfolio $\eta$, and finance it by borrowing its time-zero cost in the bank. Its initial wealth is zero, while its terminal excess payoff has variance zero and positive mean ${\eta }^{T}q$, so it is a strictly positive constant almost surely. This is an arbitrage.