AMONG THE MOST widely studied time-series processes in the empirical finance literature is the family of affine processes. Their popularity is attributable to their accommodation of stochastic volatility, jumps, and correlations among the risk factors driving asset returns, while leading to computationally tractable pricing relations and moment equations that can be used in estimation. In this chapter we overview some of the key properties of affine processes, in both their discrete- and continuous-time formulations.
Intuitively, an affine process Y is one for which the conditional mean and variance are affine functions of Y. However, following Duffie et al. (2003a), it is convenient to characterize affine processes more formally in terms of their exponential-affine Fourier (for continuous-time) and Laplace (for discrete-time) transforms. We begin with the case of continuous time and present the family of affine-jump diffusions in their familiar form as a stochastic differential equation with affine drift and instantaneous conditional variance. This is followed by a discussion of the “admissibility” problem: the need to impose restrictions on the parameters of an affine process to ensure that it is well defined.
We then turn to the case of discrete-time affine models. While the special case of a Gaussian vector autoregression has been widely studied in the asset pricing literature, discrete-time affine models with stochastic volatility have received less attention. Drawing upon the work by Darolles et al. (2001) and Dai et al. (2005), among others, we present the discrete-time counterparts to a large subfamily of affine diffusions, including virtually all of the continuous-time models that have been examined in the empirical literature.
The popularity of affine representations of the state variables in DAPMs is in large part because they lead to tractable pricing relations. This tractability derives from the knowledge of closed-form solutions to several “transforms” of affine processes. Accordingly, we introduce two key transforms for