Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 2 / 28J (Stochastic Financial Models)

An arbitrage is a portfolio with nonpositive initial cost, nonnegative terminal payoff almost surely, and a strict benefit either from a negative initial cost or a positive payoff with positive probability. An equivalent martingale measure is a probability law equivalent to the physical law under which discounted traded prices are martingales. The bond has constant price one, so no discount factor is needed here.

Order the four states as $UU,UD,DU,DD$. Equivalence requires positive probability in every state, since the physical product law has full support. Each stock has risk-neutral up probability $1/2$, obtained by solving $1=(1+ϵ)p+(1-ϵ)(1-p)$. Thus all equivalent martingale laws areThe two risky returns need not be independent under a pricing measure.

A replicating portfolio pays $b+\theta S_2+\phi S_3$, so its four payoffs satisfyConversely this condition suffices: take $\theta =(Y_{UU}-Y_{DU})/(2ϵ)$, $\phi =(Y_{UU}-Y_{UD})/(2ϵ)$, and choose the bond holding to match $Y_{DD}$. The relation then matches the fourth state as well. Its unique price is $(Y_{UU}+Y_{DD})/2=(Y_{UD}+Y_{DU})/2$.

For the same-direction claim, the payoff vector is $(1,0,0,1)$ and its price under $q$ is $2a$. Thus the possible strictly arbitrage-free prices fill $(0,1)$ and the lower arbitrage bound is the infimum zero. It is not achieved by an equivalent measure, and pricing the nonnegative nonzero claim at zero would itself permit arbitrage by buying it. Its payoff is not replicable, since the displayed payoff relation fails. A buyer may pay a positive amount for its insurance against joint downward moves, for a view about dependence, or according to risk preferences. Under the stated physical independent law its expected payoff is $13/25$, illustrating that a zero lower bound does not make the payoff economically worthless.