Option Hedging in the Presence of Jump Risk

We examine Schweizer's (1991) locally risk-minimizing (LRM) hedge approach for hedging a European call in the case when the stock price follows a Poisson jump di usion process with lognormally distributed jump sizes. In contrast to Merton's (1976) hedging strategy where di usion risk is perfectly hedged while jump risk remains un-hedged, the locally risk-minimizing strategy hedges both di usion risk as well as jump risk partly. The hedge ratio consists of a di usion component and a jump component. The value of the LRM hedge portfolio is equal to the (discounted) expected terminal payo of the option weighted with a so-called minimal martingale density. It is a weighted sum of Black/Scholes values where some weights may be negative. The latter property is due to our result that the minimal martingale density is negative with positive probability if, e.g., the market price of risk is positive. However, introducing a single call does not admit arbitrage opportunities if its value is smaller than the underlying stock price and larger than the Black/Scholes value based on the di usion volatility. We relate the LRM approach to the so-called locally variance-minimizing (LVM) hedging strategy in Bates' (1991) systematic jump risk model. By numerical analysis we nd that the LRM and LVM hedge ratios are less sensitive to changes in the stock price than delta hedging strategies in the models of Merton, Black/Scholes, and Bates. If the expected jump size is signi cantly di erent from zero and positive (negative), then the LRM and LVM hedge ratios are substantially larger (smaller) than e.g. Merton's for out-of-the-money (in-the-money) calls. Moreover, the worst case behaviour of the LRM strategy and LVM strategy are substantially better: The 99%-, 95%-, as well as the 90%-quantile of the total hedging costs are signi cantly lower than for the alternative strategies.