Pricing and Hedging of Derivatives in Contagious Markets

Abstract

It is well documented that stock markets are contagious. A negative shock to one market increases the probability of adverse shocks to other markets. We model this contagion effect by including mutually exciting jump processes in the dynamics of the index's log-returns, so that a jump in one market increases the intensities of more jumps in the same market and in other markets. Between jumps the intensities revert to their long-run means. On top of this we add a stochastic volatility component to the dynamics. It is important to take the contagion effect into account if derivatives written on a basket of assets are to be priced or hedged. Due to the affine model specification the joint characteristic function of the log-returns is known analytically, and for two specifications we detail how the model can be calibrated efficiently to option prices using Fourier transform methods. In total we calibrate over an extended period of time the specifications to options data on four US stock indexes; the Amex Biotechnology Index, the Morgan Stanley Technology index, the Securities Broker/Dealer index, and the Natural Gas index, and investigate the effect of contagion on multi-asset derivatives prices. Moreover, we compute the model hedge ratios for put and call options and investigate the historical hedging performances of the contagion specifications. Mutually exciting processes have been analyzed for multivariate intensity modeling for the purpose of credit derivatives pricing, but have not been used for pricing/hedging options on equity indexes.