Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 4 / 26K (Stochastic Financial Models)

Use the buyer's utility indifference price: $\pi (Y)$ is the cash payment satisfyingAssume the utilities are well-defined, the finite indifference price exists, and the suprema are attained as allowed. Write the common reference value as $M$. For claims $Y_1,Y_2$, choose optimizers $X_1,X_2$ at their indifference prices $p_1,p_2$. Because $V$ is a vector space, $X=\theta X_1+(1-\theta )X_2$ is admissible. The concavity of $U$ givesThe optimized utility decreases with the payment; strict increase of $U$ and attainment make the comparison strict whenever a smaller payment is substituted into an attained optimizer. Hence the payment restoring utility $M$ is at least $\theta p_1+(1-\theta )p_2$. ThusThe sign corresponds to buying the payoff. A seller's price uses a different convention.

Here $\alpha >0$ and ${\gamma }_j>0$. The asset payoff has a centered Laplace distribution. Its moment-generating function is $\mathbb{E}{e}^{tX}={\alpha }^2/({\alpha }^2-t^2)$ for $|t|<\alpha$, and diverges otherwise. Expected constant absolute risk aversion utility for a holding $\theta$ isTo maximize it, minimize the logarithm of the positive factor on $|{\gamma }_j\theta |<\alpha$. Its derivative is ${\gamma }_{j}p+2{\gamma }_j^2\theta /({\alpha }^2-{\gamma }_j^2{\theta }^2)$, and its second derivative is strictly positive. It diverges at the endpoints, so the unique critical point is the global optimizer:At $p=0$ use the continuous limit ${\theta }_j=0$. The other quadratic root is outside the finite-utility domain.

Let ${\Gamma }^{-1}={\sum }_j{\gamma }_j^{-1}$. Aggregate demand is $D(p)=-(\sqrt{1+p^2{\alpha }^2}-1)/(\Gamma p)$. Unit positive supply requires $p<0$ and $\sqrt{1+p^2{\alpha }^2}=1-\Gamma p$. Squaring and excluding $p=0$ givesThis negative price satisfies the unsquared equation and every individual finite-utility constraint. When $\alpha \le \Gamma$, there is no finite clearing price. Indeed $D(p)<\alpha /\Gamma$ for every finite negative $p$, with limit $\alpha /\Gamma$ as $p\to -\infty$. If $\alpha <\Gamma$ this maximum limiting demand is below supply; if equal, demand only approaches one at an infinitely negative price. A formal positive solution obtained by squaring when $\alpha <\Gamma$ violates the original demand equation.