Static Hedging of Standard Options

We consider the hedging of options when the price of the underlying asset is always exposed to the possibility of jumps of random size. Working in a single factor Markovian setting, we derive a new spanning relation between a given option and a continuum of shorter-term options written on the same asset. In this portfolio of shorter-term options, the portfolio weights do not vary with the underlying asset price or calendar time. We then implement this static relation using a finite set of shorter-term options and use Monte Carlo simulation to determine the hedging error thereby introduced. We compare this hedging error to that of a delta hedging strategy based on daily rebalancing in the underlying futures. The simulation results indicate that the two types of hedging strategies exhibit comparable performance in the classic Black-Scholes environment, but that our static hedge strongly outperforms delta hedging when the underlying asset price is governed by Merton (1976)'s jump-diffusion model. The conclusions are unchanged when we switch to ad hoc static and dynamic hedging practices necessitated by a lack of knowledge of the driving process. Further simulations indicate that the inferior performance of the delta hedge in the presence of jumps cannot be improved upon by increasing the rebalancing frequency. In contrast, the superior performance of the static hedging strategy can be further enhanced by using more strikes or by optimizing on the common maturity in the hedge portfolio. 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 in all cases considered, a static hedge using just five call options outperforms daily 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 the performance of our static hedge deteriorates moderately as we increase the gap between the maturity of the target call option and the common maturity of the call options in the hedge portfolio. We interpret this result as evidence of additional random factors such as stochastic volatility.

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