We consider the hedging of options when the underlying asset price is exposed to the possibility of jumps of random size. Working in a single factor Markovian setting, we derive a new, static spanning relation between a given option and a continuum of shorter-term options written on the same asset. We implement this static relation using a finite set of shorter-term options and use Monte Carlo simulation to determine the hedging error. We compare this hedging error to that of a delta hedging strategy based on daily rebalancing in the underlying futures.