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

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.