Jump Starting GARCH: Pricing Options with Jumps in Returns and Volatilities

This paper considers the pricing of options when there are jumps in the pricing kernel and correlated jumps in asset returns and volatilities. Our model nests Duan’s GARCH option models where conditional returns are constrained to being normal, as well as extends Merton’s jump-diffusion model by allowing return volatility to exhibit GARCH-like behavior. Empirical analysis on the S&P 500 index returns reveals that the incorporation of jumps in returns and volatilities improves significantly the performance of the GARCH model in capturing the observed time series of the S&P 500 index returns. Moreover, the corresponding GARCH option pricing model with the jump component delivers a better performance on pricing the S&P 500 index options. (GARCH, options, stochastic volatility, jumps) In this paper we introduce a new family of GARCH models driven by compound Poisson innovations and derive the corresponding option pricing theory. Because the compound Poisson innovations are analogous to the increments in a continuous-time compound Poisson process, we refer to this new class as the GARCH-Jump model. These discrete-time processes are of interest since the conditional returns of the underlying asset allow levels of skewness and kurtosis to be matched to the data and option prices can readily be priced in a way to reflect changing volatility and jumps in both returns and volatilities. This GARCH-Jump option pricing model is a natural generalization of the typical GARCH option pricing models with normal innovations, a pricing approach originated in Duan (1995). We empirically test the model, and show that it fits the return data better than the traditional GARCH model with normal innovations and outperforms the inverse Gaussian GARCH model recently proposed by Christoffersen, Heston and Jacobs (2006) (CHJ). Moreover, our model is better in removing more of the biases in option prices. Just as the binomial model serves as a discrete-time approximation for many underlying diffusion processes, the class of GARCH-Jump models can serve as discrete-time approximations for an array of continuous-time jump diffusion models. As shown in Duan, Ritchken and Sun (2006), a variety of continuous-time limiting models can in fact be derived using our GARCH-Jump processes; for example, (1) when the GARCH feature is disabled but jumps allowed, the limiting model nests the jump-diffusion model of Merton (1976), (2) when jumps are suppressed, the limiting model can be made to converge to continuous-time stochastic volatility models, including Heston (1993), Hull and White (1987) and Scott (1987), among others, and (3) when jumps are permitted, the limiting models contain jumps and diffusive elements in both returns and volatilities, along the lines of Eraker, Johannes and Polson (2003) and Duffie, Singleton and Pan (1999). Furthermore, just as the appropriately defined binomial model provides a useful mechanism for pricing American style options under the geometric Brownian motion assumption, our appropriately defined risk neutralized discrete-time GARCH models provide a mechanism for pricing options when returns and/or volatilities experience random jumps. The option theoretical results developed in this paper has in fact been utilized by Duan, Ritchken and Sun (2006) to derive the limiting option pricing models. This paper contributes to the literature in three aspects. First, we propose a new class of GARCH models based on compound Poisson innovations and establish the discrete-time option pricing theory which allows us to price derivatives when the underlying asset’s innovations may be far from normal and when volatility is stochastic. This is important because our approach offers a unique GARCH option model with non-normal innovations that can be naturally linked to stochastic volatility model with jumps.1 Second, we conduct an empirical analysis to demonWe know of three alternative ways of introducing non-normal innovations into the GARCH option pricing