Model Misspecification and the Hedging of Exotic Options

Asset pricing models are well established and have been used extensively by practitioners both for pricing options as well as for hedging them. Though Black-Scholes is the original and most commonly communicated asset pricing model, alternative asset pricing models which incorporate additional features have since been developed. We present three asset pricing models here - the Black-Scholes model, the Heston model and the Merton (1976) model. For each asset pricing model we test the hedge effectiveness of delta hedging, minimum variance hedging and static hedging, where appropriate. The options hedged under the aforementioned techniques and asset pricing models are down-and-out call options, lookback options and cliquet options. The hedges are performed over three strikes, which represent At-the-money, Out-the-money and In-the-money options. Stock prices are simulated under the stochastic-volatility double jump diffusion (SVJJ) model, which incorporates stochastic volatility as well as jumps in the stock and volatility process. Simulation is performed under two ’Worlds’. World 1 is set under normal market conditions, whereas World 2 represents stressed market conditions. Calibrating each asset pricing model to observed option prices is performed via the use of a least squares optimisation routine. We find that there is not an asset pricing model which consistently provides a better hedge in World 1. In World 2, however, the Heston model marginally outperforms the Black-Scholes model overall. This can be explained through the higher volatility under World 2, which the Heston model can more accurately describe given the stochastic volatility component. Calibration difficulties are experienced with the Merton model. These difficulties lead to larger errors when minimum variance hedging and alternative calibration techniques should be considered for future users of the optimiser