Stochastic Volatility with Reset at Jumps

This paper presents a model for asset returns incorporating both stochastic volatility and jump e ects. The return process is driven by two types of randomness: small random shocks and large jumps. The stochastic volatility process is a ected by both types of randomness in returns. Speci cally, in the absence of large jumps, volatility is driven by the small random shocks in returns through a GARCH(1,1) model, while the occurrence of a jump event breaks the persistence in the volatility process, and resets it to an unknown deterministic level. Model estimation is performed on daily returns of S&P 500 index using the maximum-likelihood method. The empirical results are discussed. Recently, there has been fair amount of work in the asset pricing literature that studies models with both jump and stochastic volatility dynamics. For Graduate School of Business, Stanford, CA 94305. [email protected]. I am grateful for extensive discussions with Darrell Du e and Kenneth Singleton. I would also like to thank Geert Bekaert for helpful comments. example, in the discrete-time setting, Jorion [1989] employs a jump model with ARCH(1) to test equity returns and exchange rates, and Bekaert and Gray [1996] use a jump with GARCH(1,1) model to study the target zones and exchange rates. In the continuous setting, jump-di usion models with stochastic volatility can be found in Bates [1997] and Bakshi, Cao, and Chen [1997], among others. Although existing empirical work has clearly shown the importance of characterizing the dynamics of both jumps and stochastic volatility in asset returns, it still remains an open question as how these two dynamics interact with each other. To be more speci c, one important question is: how does a jump in return a ect the dynamics of the stochastic volatility process? In Jorion [1989], the volatility dynamics are a ected only through the mixture of random shock and jumps, and as a consequence, this volatility process can not di erentiate large jumps from small random movements. Moreover, because of the ARCH dynamics, the volatility process can \overshoot" to an unreasonably high level due to jumps in returns. Bekaert and Gray [1996] employ a \pressure relief" mechanism to capture the phenomenon that certain types of jumps in exchange rates break up the persistence in the volatility process. However, in their paper, the arrival of such \pressure reliefs" is given exogenously. Bates [1997] and Bakshi, Cao, and Chen [1997] model the stochastic volatility with an autonomous process, which is independent of the return process, hence one can not rely on these models to study the jump e ect on a stochastic volatility process. In this paper, we develop a simple model for asset return that incorporates For details on ARCH and GARCH models, see Engle [1982] and Bollerslev [1986].