Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 2 / 28I (Stochastic Financial Models)

An arbitrage is a portfolio with nonpositive initial cost, nonnegative terminal payoff almost surely and a positive payoff with positive probability, allowing unused initial cash to be invested in the riskless asset. Equivalently it is a zero-cost portfolio with those terminal properties. An equivalent martingale measure $Q$ has the same null sets as the physical measure and satisfies $E_Q[S_1^i]=(1+r)S_0^i$ for every traded asset, with appropriate integrability.

For an agent with differentiable increasing concave utility and an optimal portfolio, let $W^{\ast }$ be its terminal wealth. The first-order conditions for variations in risky holdings are $E[U^{\prime }(W^{\ast })(S_1^i-(1+r)S_0^i)]=0$. If $U^{\prime }(W^{\ast })>0$ and integrable, normalize it to the probability density $Z=U^{\prime }(W^{\ast })/E[U^{\prime }(W^{\ast })]$. The first-order conditions make $Q$ with $dQ=ZdP$ an equivalent martingale measure. A small additional payoff $H$ then has marginal priceThis follows by setting the first variation of expected utility for buying a small claim at price $p$ equal to zero.

A zero-cost stock holding $\theta$ has terminal payoff $\theta (S_1-1)$. If $a\le 1$, shorting the share and holding a bond gives $1-S_1\ge 0$ and a strictly positive payoff almost surely. If $a>1$, both signs of $S_1-1$ occur with positive probability, so no nonzero holding is an arbitrage. Hence arbitrage freedom is equivalent to $a>1$.

The equivalent martingale measures are exactly the densities $Z(s)>0$ almost everywhere on $(0,a)$ satisfyingThey exist for $a>1$: exponential tilts of the uniform law have means ranging continuously from zero to $a$, so one has mean one.

Under the usual utility assumptions $U^{\prime }>0$, $U^{\prime \prime }<0$, and existence of an admissible optimum, $J(\theta )=E[U(w+\theta (S_1-1))]$ is strictly concave and $J^{\prime }(0)=U^{\prime }(w)(a/2-1)$. Its decreasing derivative implies that the optimizer is positive exactly when $a>2$, negative when $a<2$, and zero when $a=2$, provided the feasible interval contains zero in its interior. The hypothesis $C^2$ alone is insufficient: a linear increasing utility can give an unbounded objective rather than a finite optimizer. Concavity and existence are implicit in the standard utility formulation.

For $a=2$, $\theta =0$ makes marginal utility constant, so the pricing measure is the original uniform law andFor any equivalent pricing measure, the call expectation is positive and strictly less than $E_Q{S}_1/2=1/2$, since $(s-1{)}^{+}<s/2$ for $0<s<2$. Both endpoints can be approached. Symmetric densities concentrated in $[1-\delta ,1+\delta ]$ give prices tending to zero; symmetric densities concentrated near zero and two give prices tending to one half. Mix either density with an arbitrarily small amount of the uniform law to keep it everywhere positive; symmetry keeps its mean one. The set of positive mean-one densities is convex and the call price is linear, so every intermediate value is attained. Thus the exact price range is $(0,1/2)$, with neither endpoint attained.