Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 30L (Stochastic Financial Models)

Use the optimizer for $Z_0$ also with $Z_1$. Strict increase of $U$ and a strict inequality on a positive-probability event give $u(Z_1)>u(Z_0)$.

If $X_i$ optimizes $Z_i$, then $p{X}_1+(1-p)X_0$ is attainable because the wealth set is affine. Concavity gives

The time-zero indifference price is the unique $\pi (Y)$ satisfyingThe left side is continuous and strictly decreasing in the cash amount by part (a), and its limiting values bracket $u(0)$; existence follows by the intermediate-value theorem and uniqueness by strict monotonicity.

Differentiate expected utility at the optimum $X_0$ in each portfolio direction:Rearranging gives the claimed vector identity.

The marginal utility price isThe value function is concave; its tangent at zero lies above its graph. Evaluating that supporting inequality at the utility-indifference displacement $Y-(1+r)\pi (Y)$ gives
$E[U^{\prime }(X_0)(Y-(1+r)\pi (Y))]\ge 0$, hence ${\pi }_0(Y)\ge \pi (Y)$.

By Gaussian conditional expectation,Multiply by $U^{\prime }(X_0)$, take expectations, and use part (c):Substitution into part (d) gives