Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 38 / 2 / c / Solution

Let ${\pi }_t$ be the predictable stock holding during $(t-1,t]$. Cash has constant price, so self-financing givesPut $\mathcal{G}={\mathcal{F}}_{t-1}$. The tower property of conditional expectation and ${\mathcal{F}}_t$-measurability of $S_t$ giveSince $X_{t-1}$, $S_{t-1}$ and ${\pi }_t$ are $\mathcal{G}$-measurable, substituting the gains identity yieldsThe denominator is positive on every positive-probability parent atom. If it were zero on such an atom, $S_t$ would be constant there, and the martingale property would force that constant to equal $S_{t-1}$, contradicting the nonzero-increment assumption. HenceThis is conditional covariance hedge ratio. It uses attainability; a regression coefficient alone would not replicate a general unattainable payoff. After maturity one may liquidate into cash, so the same formula gives zero for later dates wherever its denominator remains nonzero. On a finite sample space a martingale cannot have nonzero increments forever; the stated nondegeneracy is naturally a finite-maturity assumption.