Option Valuation with Jumps in Returns and Volatility

We price options when there are jumps in the pricing kernel and correlated jumps in returns and volatilities. A limiting case of our GARCH process consists of a model where both asset returns and local volatility follow jump diffusion processes with correlated jump sizes. When the jump processes are shut down our model reduces to Duan’s (1995) GARCH option model; when the stochastic volatility process is shut down, our model reduces to a form that nests Merton’s (1976) jump diffusion model. Our general model permits conditional return distributions that are skewed and have fat tails. Empirical analysis on the S&P 500 index reveals that the incorporation of jumps in returns and volatilities adds significantly to the description of the time series process and improves the precision of option prices. We conduct hedging tests and provide evidence that hedges can be maintained very well over time. In this paper we investigate a family of GARCH models that have limiting processes where returns follow a jump-diffusion process and volatility is stochastic and has jumps as well. The GARCH processes are of interest in their own right, having conditional returns that are able to reflect the desired levels of kurtosis and skewness in the data. We specify a pricing kernel that allows option pricing to proceed under an identifiable risk neutral measure. When the jumps are switched off, our models collapses to the GARCH models of Duan (1995); when the GARCH process is curtailed, the dynamics reduces to jump diffusion models like Merton (1976), or the more general model of Naik and Lee (1990). More generally, however, jumps are allowed in volatility, and volatility could have diffusive elements as well. Just as GARCH processes offer a discrete time filter for stochastic volatility models, our models can be viewed as filters for continuous time stochastic volatility models with jumps in returns and volatility. Why is it important to incorporate jumps in volatility? Empirical research has shown that models which describe returns by a jump-diffusion process with volatility being characterized by a correlated diffusive stochastic process are incapable of capturing empirical features of equity index returns or option prices. For example, both Bates (2000) and Pan (2002) examine such models, and are unable to remove systematic option pricing biases that remain.1 While jumps in the return process can explain large daily shocks, these return shocks are transient and have little effect on future returns. At the same time, with volatility being diffusive, changes occur gradually and with high persistence. These models are unlikely to generate clustering of large returns associated with temporarily high levels of volatility, a feature that is displayed by the data. Both of the above authors recommend considering models with jumps in volatility. In response to these findings, researchers have begun to investigate models that incorporate jumps in both returns and volatility. In general, estimating the parameters of continuous time processes when the volatility is not only non-observable, but the return and volatility processes both contain diffusive and jump elements, is difficult. While in the last decade, significant advances in econometric methodology have been made, these estimation problems are still fairly delicate.2 In a very convincing study, Eraker, Johannes and Polson (2002), for example, examine the jump in volatility models proposed by Duffie, Singleton and Pan (2000), and show that the Stochastic volatility option models have been considered by Hull and White (1987), Heston (1993), Nandi (1998), Scott (1987), among others. Bakshi, Cao and Chen (1997) provide empirical tests of alternative option models. None of these models have jumps in both returns and volatility. Naik (1993) considers a regime switching model where volatility can jump. For additional regime switching models, see Duan, Popova and Ritchken (2002). Eraker, Johannes and Polson (2002) provide an excellent review of the difficulties in adopting standard MLE or GMM approaches. Singleton (2001) discusses an approach using characteristic functions. An alternative approach based on simulation methods using Efficient Method of Moments, and Monte Carlo Markov Chains does resolve some of these issues. For an overview on econometric techniques to estimate continuous time models see Renault (1997), Jacquier, Polson, and Rossi (1994), Eraker, Johannes and Polson (2002), and the references therein.