Model Misspecification and the Hedging of Exotic Options

Abstract

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 inWorld 1. InWorld 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.