Stochastic Volatility, Jumps,
and Asset Returns
THIS CHAPTER EXPLORES the shapes of the conditional distributions of asset returns from two complementary perspectives. First, we present various descriptive statistics of the historical data that will be useful in assessing the goodness-of-fit of DAPMs. Second, we explore how alternative choices of probability models for the risk factors affect the model-implied shapes of return distributions.
That the distributions of most returns are “fat tailed” and often “skewed” has been extensively documented in the finance literature. We begin this chapter with some descriptive evidence on the nonzero skewness and excess kurtosis of the unconditional distributions of equity and bond market yields. Subsequently, we examine the conditional third and forth moments of return distributions and illustrate how these moments can be used to discriminate among alternative time-series models for returns.
Potentially important sources of these nonnormal shapes are recurrent periods of volatile and quiet financial markets. Accordingly, a central focus of this chapter is on how alternative parameterizations of time-varying volatility—“stochastic volatility”—and sudden infrequent price moves— “jumps”—affect the shapes of return distributions. Jumplike behavior can be induced by a classical jump process (e.g., a Poisson process), shocks that are drawn from a mixture of distributions, or a “switching-regime” process. Accordingly, we first highlight the conceptual differences among these alternative formulations and then add in stochastic volatility, all in discrete time.
This discussion is followed by a review of continuous-time models with stochastic volatility and jumps.1 We also review some of the key empirical
1 There is also a continuous-time counterpart to switching-regime models. See, e.g., Dai and Singleton (2003b) for a discussion of regime-switching versions of several popular continuous-time models of interest rates. Most of the empirical literature has focused on discrete-time versions of these models and we do as well.