Abstract: We consider the hedging of options when the underlying assetprice is exposed to the possibility ofjumps of random size. Working in a single factor Markovian setting, we derive a new, static spanningrelation 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 Carlosimulation to determine the hedging error. We compare this hedging error to that of a delta hedgingstrategy based on daily rebalancing in the underlying futures. The simulation results show that thetwo types of hedging strategies generate comparable performance under purely continuous assetprice dynamics, but that our static hedge strongly outperforms delta hedging when the underlyingasset price process contains random jumps. When we compare the hedging effectiveness of the twotypes of strategies using over six years of data on S&P 500 index options, we find that a static hedgeusing just five call options outperforms daily delta hedgingwith the underlying futures. This resultlends empirical support for the existence of random jumps inthe movement of the S&P 500 index.