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

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.