Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 30K (Stochastic Financial Models)

A risky-asset portfolio $\phi$ financed through the risk-free asset has zero-cost discounted gainIt is an arbitrage when $G_{\phi }\ge 0$ almost surely and $\mathbb{P}(G_{\phi }>0)>0$.

If $G\ge 0$ were an arbitrage gain, then $X_1^{\ast }+tG$ would be attainable from the same initial wealth for every $t>0$. Since $U$ is increasing,almost surely, with strict inequality on a set of positive probability under the usual strictly increasing meaning of utility. This contradicts optimality of $X_1^{\ast }$. Hence existence of an optimizer implies no-arbitrage.

Let $u(x)$ be maximal expected utility without the claim. The utility indifference price $\pi (Y)$ is determined byFor claims $Y_1,Y_2$, mix optimal portfolios for their indifference problems with weights $\lambda$ and $1-\lambda$. Concavity of $U$ shows that buying $\lambda Y_1+(1-\lambda )Y_2$ at pricegives expected utility at least $u(x)$. Since maximal utility decreases with purchase price,Thus $\pi$ is concave.

At the optimal terminal wealth, normalized marginal utility defines an equivalent martingale measure $Q$ and hence a marginal utility priceThis is an arbitrage-free price, so $p_Q(Y)\in I$. Concavity of utility implies that the finite-quantity indifference bid does not exceed its marginal price:Consequently