Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 1 / 28J / Solution

The reservation bid price for a claim $Y$ is the fixed cash payment $p_i(Y)$ which leaves the agent indifferent: $E[U_i(X_i+Y-p_i(Y))]=E[U_i(X_i)]$. For Gaussian wealth $W$, exponential utility is $-\exp [-{\gamma }_{i}E(W)+{\gamma }_i^2\mathrm{Var}(W)/2]$, so its certainty equivalent is $E(W)-{\gamma }_i\mathrm{Var}(W)/2$. For $Y=\lambda X_0$, subtracting the original certainty equivalent givesThe marginal price is positive when the claim hedges the agent's endowment, negative when it adds positively correlated risk, and zero for an uncorrelated zero-mean infinitesimal position.

Assume $v_{00}>0$, so the claim is nontrivial. Buying $\theta$ units at per-unit price $p$ changes the certainty equivalent by $-(p+{\gamma }_i{v}_{0i})\theta -{\gamma }_i{v}_{00}{\theta }^2/2$. This strictly concave quadratic is maximized at ${\theta }_i=-(p+{\gamma }_i{v}_{0i})/({\gamma }_i{v}_{00})$. Market clearing ${\sum }_i{\theta }_i=0$ givesAt the optimum the certainty-equivalent gain isThe maximal expected utility equals the original negative utility multiplied by $e^{-{\gamma }_i{\Delta }_i}$, hence is improved. Paying a fixed amount ${\Delta }_i$ removes exactly that gain, proving it is the fixed-payment value of market access. If $v_{00}=0$, the Gaussian claim vanishes almost surely, all its covariances vanish, and the appropriate price and access value are zero; the printed formulas with a variance denominator require the nontrivial-claim assumption.