Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 39 / 1

A one-period arbitrage is a portfolio of initial-information-measurable holdings $(h_B,h_S)$ with value $V_t=h_B{B}_t+h_S{S}_t$ such thatThe holdings are fixed during the period, so this is a self-financing strategy. A portfolio costing a negative amount with nonnegative terminal value also gives an arbitrage: invest its initial surplus in the strictly positive bank account to obtain a zero-cost portfolio with a strict gain. The same definition applies when further traded assets are added.

An equivalent martingale measure is a probability measure $Q$ equivalent to the objective measure $P$, meaning the two have exactly the same null events, under which the discounted stock ${\tilde{S}}_t=S_t/B_t$ is a martingale. In this one-period model the condition isThe discounted bank account is identically one. Integrability of the discounted stock is part of the martingale condition. In an augmented market the discounted price of every additional asset must satisfy the same condition.

Suppose $Q$ is an equivalent martingale measure and a zero-cost self-financing strategy has holdings $(h_B,h_S)$. Its discounted terminal value has conditional expectationIf it were an arbitrage, $V_1/B_1$ would be nonnegative because $B_1>0$. Equivalence implies that it is strictly positive on an event of positive $Q$-probability. A nonnegative random variable with zero conditional expectation must be zero almost surely, a contradiction. Thus $\boxed{\text{an equivalent martingale measure excludes arbitrage}}$. The calculation uses the usual integrable one-period strategies, and also works conditionally for finite initial-information-measurable holdings.

The European call option pays one in the state $S_1=5$ and zero in the other two states. Short one call, buy one third of a stock, and borrow one half unit of cash. Writing the option quote as $c=1/2$, the holdings and initial value areThe terminal value is $-1/2+S_1/3-(S_1-4{)}^{+}$. At stock prices $5,3,2$ it is respectivelyIt is strictly positive in every state, so this portfolio is an arbitrage.

Let $c$ be the initial call price. Write the risk-neutral probabilities of stock prices $5,3,2$ as $q_5,q_3,q_2$. The stock's martingale equation and the normalization give $q_2=2q_5$ and $q_3=1-3q_5$. The call pays only in the first state, so its martingale equation is $q_5=c$. ThusAll three objective probabilities are positive. This defines an equivalent martingale measure precisely when $0<c<1/3$, and part (iii) proves that every such price excludes arbitrage.

To prove necessity without appealing to an existence theorem, consider the other prices explicitly. If $c\le 0$, buy one call and hold $-c$ units of cash. Its initial value is zero and terminal value is $(S_1-4{)}^{+}-c$, nonnegative and positive with positive probability. If $c\ge 1/3$, take holdings $(h_B,h_S,h_C)=(c-1,1/3,-1)$. The initial value is $c-1+1-c=0$ and terminal values at $5,3,2$ are $(c-1/3,c,c-1/3)$. They are nonnegative and the middle-state value is strictly positive. This includes the endpoint $c=1/3$. Consequently