Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 44 / 3

Let ${\beta }_t$ be the cash position and ${\pi }_t$ the risky shares chosen using information at $t-1$. At the start of the period the portfolio costs ${\beta }_t+{\pi }_t{S}_{t-1}=X_{t-1}$. With cash price fixed at one, its ending value is ${\beta }_t+{\pi }_t{S}_t$. Rebalancing cannot add or remove wealth under a self-financing strategy, soThe holdings are predictable: they are chosen before the current price increment is observed. This is the discounted wealth equation in discrete time with cash as numéraire.

Take the nondegenerate case $\sigma >0$ and an integer trading horizon $T$. For exponential utility, maximize the negative of the exponential loss, equivalently minimize its expectation. Conditional on past information, a predictable choice $\pi$ hasComplete the square:Thus the minimum conditional multiplier is $e^{-\kappa }$ with $\kappa ={\mu }^2/(2{\sigma }^2)$. Backward induction starts with terminal loss $e^{-\gamma x}$ and gives the minimal continuation loss $e^{-\gamma x-\kappa (T-t)}$. Independence of the next increment justifies the same conditional minimization for every history, not just deterministic trading plans. Hence exponential-utility trading with Gaussian increments givesThese are constant numbers of shares, not constant wealth fractions. Strategies with infinite exponential loss have expected utility $-\infty$ and cannot improve this finite value. If $\sigma =0$, the usual formula is inapplicable: with $\mu =0$ trading has no effect, while a nonzero deterministic increment admits unbounded riskless gains and no finite maximizing position.

For a predictable position $\pi$, the current net wealth increment $\pi \Delta S_t+Y_t-y$ is conditionally Gaussian with mean $\pi \mu +a-y$ and variance ${\pi }^2{\sigma }^2+2\pi \rho b\sigma +b^2$. Its exponential-loss multiplier isDifferentiate its quadratic exponent. The unique minimizing position isThe first term is the speculative demand; the second offsets the part of the income correlated with the tradable price increment. Independent period vectors and backward induction justify using this same position at every date. Completing the square gives minimal exponent $-\kappa -\gamma d(y)$, whereTherefore hedging a Gaussian income stream with exponential utility has valueThe residual variance is $b^2(1-{\rho }^2)$; it disappears for perfectly correlated income.

LetPart (c) has $d(y)=\overline{y}-y$. Comparing the optimized values with and without the contract, $V_0^{\mathrm{swap}}=-e^{-\gamma x-T\kappa -\gamma T(\overline{y}-y)}$ and $V_0=-e^{-\gamma x-T\kappa }$, shows that $y>\overline{y}$ makes the swap value strictly more negative. Thus The investor prefers not to enter when $y>\overline{y}$. He is indifferent at equality and prefers entry for a smaller payment.

The periodic utility indifference payment for Gaussian income equals the mean income less the cost of hedging its correlated component and the exponential-utility penalty for the residual unhedgeable variance. Comparing unoptimized strategies would not justify this price threshold.