We examine the optimal hedging of derivatives written on realised variance, focussing principally on variance swaps (VS) (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 VS under very general dynamics for the underlying stock which allow for multiple jump processes and stochastic volatility. We illustrate how, for parameters which are realistic for options on the S&P 500 and Nikkei-225 stock indices, our methodology gives significantly better hedges than the standard log-contract replication approach of Neuberger and Dupire which assumes continuous sample paths. Our analysis seeks to emphasise practical implications for financial institutions trading variance derivatives.
