Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 211 / 1

A portfolio $H\in {\mathbb{R}}^n$ is an arbitrage when $H\cdot P_0\le 0$, $H\cdot P_1\ge 0$ almost surely, and at least one inequality supplies a strict gain: either $H\cdot P_0<0$ or $\mathbb{P}(H\cdot P_1>0)>0$. It is a terminal-consumption arbitrage when $H\cdot P_0=0$, $H\cdot P_1\ge 0$ almost surely, and the terminal inequality is strict with positive probability.

A numéraire portfolio $\eta$ satisfies $\eta \cdot P_0>0$ and $\eta \cdot P_1>0$ almost surely. If an arbitrage $H$ already has zero initial cost, it is a terminal-consumption arbitrage. Otherwise $H\cdot P_0<0$; setThen $\tilde{H}\cdot P_0=0$, while its terminal payoff is the nonnegative payoff of $H$ plus a strictly positive multiple of $\eta \cdot P_1$. Hence it is strictly positive almost surely and is a terminal-consumption arbitrage.

If $A=\mathrm{Im}V={\mathbb{R}}^n$, the symmetric covariance matrix $V$ is positive definite. For every nonzero $H$, the scalar $H\cdot P_1$ is normal with variance $H^{T}VH>0$, so it has positive probability of being negative. It cannot be an arbitrage payoff. The zero portfolio provides no strict gain, proving absence of arbitrage.

For a numéraire portfolio $\eta$, the normal random variable $\eta \cdot P_1$ is strictly positive almost surely. A nondegenerate normal variable has support on all of $\mathbb{R}$, so it must be degenerate: ${\eta }^{T}V\eta =0$, equivalently $V\eta =0$. Thus $\eta \cdot P_1=\eta \cdot \mu >0$ deterministically. The scaled portfoliohas terminal value $B\cdot P_1=1$ almost surely and therefore replicates a risk-free bond.

Let $d=\mu -(1+r)P_0$. Since $V$ is symmetric, $A=\mathrm{Im}V=(\ker V{)}^{\perp }$. If $d\in A$, every $H\in \ker V$ satisfiesAny $H\notin \ker V$ has a nondegenerate normal terminal value and cannot be nonnegative almost surely. Any $H\in \ker V$ has the displayed deterministic relation, which excludes an arbitrage because $1+r=(\eta \cdot \mu )/(\eta \cdot P_0)>0$.

Conversely, if $d\notin A$, choose $h\in \ker V$ with $h\cdot d>0$. The zero-cost portfoliohas deterministic terminal payoffIt is a terminal-consumption arbitrage. Thus no arbitrage is equivalent to $d\in A$, the Arbitrage in a one-period Gaussian market criterion.