Abstract

We examine the optimal hedging of variance derivatives, focussing principally on variance swaps (but, en route, also considering skewness swaps), when the underlying stock price has discontinuous sample paths i.e. jumps. In general, with jumps in the underlying, the market is incomplete and perfect hedging is not possible. We derive easily implementable formulae which give optimal (or nearly optimal) hedges for variance swaps under very general dynamics for the underlying stock which allow for multiple jump processes and stochastic volatility (or, more generally, (possibly, multiple) stochastic time-changes). We also consider special cases when perfect hedging of variance swaps is possible even when the underlying stock price has jumps. We illustrate how, for parameters which are realistic for equity markets, our methodology gives significantly better hedges than the standard log-contract replication approach which assumes continuous sample paths.