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

A trading strategy chooses a vector $H_t$ of holdings for the period $(t-1,t]$, with $H_t$ measurable with respect to ${\mathcal{F}}_{t-1}$. Its end-of-period wealth is $X_t=H_t\cdot P_t$. Rebalancing at date $t$ is self-financing whennew holdings cost exactly the value released by the old holdings. With fixed initial capital $x$, the equivalent gains identity isA European contingent claim is a maturity-$T$ payoff $\xi$ measurable with respect to ${\mathcal{F}}_T$. It is attainable if there exists a predictable self-financing strategy and an initial capital $x$ for which $X_T=\xi$ almost surely. Such a strategy is a replicating strategy, and $x$ is its initial replication cost. Holdings are understood only up to the maturity being replicated.

On a finite sample space, all real-valued holdings over the finite interval $0,\dots ,T$ are bounded after null states are discarded. The gains identity is therefore a bounded predictable martingale transform of the vector martingale $P$, summed over its coordinates. It follows that $X$ is a martingale, soHere the replication cost is a prescribed deterministic initial capital, as in the definition of attainability. The stronger intermediate identity is $X_t=\mathbb{E}[\xi \mid {\mathcal{F}}_t]$. If initial capital is instead allowed to be ${\mathcal{F}}_0$-measurable and random, the corresponding statement is $X_0=\mathbb{E}[\xi \mid {\mathcal{F}}_0]$; its unconditional expectation still equals $\mathbb{E}\xi$.

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.

Work conditionally on a positive-probability atom of ${\mathcal{F}}_{T-1}$. Let $Z$ have the conditional distribution of $S_T$ there, and let $Z^{\prime }$ be an independent copy of that conditional law. The finite sample space makes all expectations finite. ThenStrict increase of $g$ makes the integrand nonnegative, and strictly positive whenever $Z\ne Z^{\prime }$. The conditional variance is positive by part (c), so the conditional distribution is nondegenerate and $\mathbb{P}(Z\ne Z^{\prime })>0$. Thus the numerator of the hedge ratio is strictly positive on every such atom. The denominator is also positive, givingThis is strict positive covariance with an increasing payoff. The independent copy is taken from the conditional distribution, not from an unrelated unconditional distribution.