Approximating GARCH-Jump Models, Jump-Diffusion Processes, and Option Pricing

This paper considers the pricing of options when there are jumps in the pricing kernel and correlated jumps in asset prices and volatilities. We extend theory developed by Nelson (1990) and Duan (1997) by considering limiting models for our resulting approximating GARCH-Jump process. Limiting cases of our processes consist of models where both asset price and local volatility follow jump diffusion processes with correlated jump sizes. Convergence of a few GARCH models to their continuous time limits are evaluated and the benefits of the models explored. (GARCH option models, stochastic volatility models with jumps, limiting GARCH with Jump processes) Up until the 1990s the literature on continuous time models, used in theoretical finance and especially in derivative modeling, and discrete time models, often favored in empirical studies, developed along very separate lines. Most of the discrete time models were of the generalized autoregressive conditionally heteroskedastic (GARCH) type, while the continuous time models were based on diffusion models. In the early 1990s researchers began to reconcile the two approaches. In particular, Nelson (1990) showed that as the sampling frequency increased, the volatility process generated within some GARCH models converged in distribution towards well defined solutions of stochastic differential equations. Duan (1997) extended this work and showed that most of the existing bivariate diffusion models that had been used to model asset returns and volatility could be represented as limits of a family of GARCH models. As a result, even if one prefers modeling prices and volatilities by a bivariate process, there may be advantages in considering GARCH techniques. For example, by suitably curtailing the parameters of generalized GARCH processes, we can obtain European and American option prices under the stochastic volatility models of Hull and White (1987), Scott (1987), Wiggins (1987), Stein and Stein (1991), and Heston (1993). Unfortunately, there is now overwhelming empirical evidence that indicates that stock price processes cannot be adequately characterized by bivariate diffusions, and that incorporating jump components in both price and in volatility is necessary. For example, Bates (2000) and Pan (2002) consider models where prices follow a jump-diffusion process with volatility being characterized by a correlated diffusive stochastic process. Both authors show that such models are incapable of capturing empirical features of equity index returns or option prices, and they attribute this to the fact that volatility itself may contain jumps. More recently, Eraker, Johannes and Polson (2003) examine the jump in volatility models proposed by Duffie, Singleton and Pan (1999), and provide a study that shows that the addition of jumps in volatility provide a significant improvement to explaining the returns data on the S&P 500 and Nasdaq 100 index, beyond a stochastic volatility model with just jumps in prices. To date, the GARCH approximating models that have been considered in the literature are set up for stochastic volatility diffusions. In light of the importance of jumps, both in price and volatility, the current GARCH approximating models are inadequate. The primary purpose of this paper is to propose a new set of GARCH models that include, as limiting cases, processes characterized by stochastic volatility with jumps in price and volatility. The secondary purpose of this paper is to explore the potential advantages of these GARCH with jump models over their continuous time counterparts. The discrete time model on which we rely in constructing our limiting models is that of Duan, Ritchken and Sun (2004) (hereafter DRS). The DRS (2004) model has the property that the conditional returns have fat tails and are skewed. As a result, local skewness and kurtosis present in data can easily be matched. Further, much of the volatility smile observed in option