Static Hedging of Standard Options

We consider the hedging of derivative securities when the price movement of the underlying asset can exhibit random jumps. Under a one factor Markovian setting, we derive a spanning relation between a long term option and a continuum of short term options. We then apply this spanning relation to the static hedging of long term options with a finite choice of short term, more liquid options based on a quadrature rule. We use Monte Carlo simulation to determine the hedging error introduced by the quadrature approximation and compare this hedging error to the hedging error from a delta hedging strategy based on daily rebalancing in the underlying futures. The simulation results indicate that the two types of strategies have comparable hedging effectiveness in the classic Black-Scholes environment, but that our static hedging strategy strongly outperforms the dynamic delta-hedging strategy when the underlying asset price movement is governed by Merton (1976)’s jump diffusion model. Further simulation exercises indicate that these results are robust to model misspecification, so long as one performs ad hoc adjustments based on the observed implied volatility. We also compare the hedging effectiveness of the two types of strategies using more than six years of data on S&P 500 index options. We find that a static hedge using just five call options outperforms daily rebalancing on the delta hedging with the underlying futures. The consistency of this result with our jump model simulations lends empirical support for the existence of jumps of random size in the movement of the S&P 500 index. We also find that our static strategy performs best when the maturity of the options in the hedging portfolio is close to the maturity of the target option being hedged. As the maturity gap increases, the hedging performance deteriorates moderately, indicating the likely existence of additional random factors such as stochastic volatility.