Density approximations for multivariate affine jump-diffusion processes
Introduction
Most observed phenomena in financial markets are inherently multivariate: stochastic trends, stochastic volatility, and the leverage effect in equity markets are well-known examples. The theory of affine processes provides multivariate stochastic models with a well established theoretical basis and sufficient degree of tractability to model such empirical attributes. They enjoy much attention and are widely used in practice and academia. Among their best-known proponents are Vasicek’s interest rate model (Vasicek, 1977), the square-root model (Cox et al., 1985), Heston’s model (cf. Heston, 1993), and affine term structure models (Duffie and Kan, 1996, Dai and Singleton, 2000, Collin-Dufresne et al., 2008). Affine models owe their popularity and their name to their key defining property: their characteristic function is of exponential affine form and can be computed by solving a system of generalized Riccati differential equations (cf. Duffie et al. (2003)). This allows for computing transition densities and transition probabilities by means of Fourier inversion (Duffie et al., 2000). Transition densities constitute the likelihood which is an ingredient for both frequentist and Bayesian econometric methodologies.1 Also, they appear in the pricing of financial derivatives. However, Fourier inversion is a very delicate task. Complexity and numerical difficulties increase with the dimensionality of the process. Efficient density approximations avoiding the need for Fourier inversion are therefore desirable.
This paper is concerned with directly approximating the transition density without resorting to Fourier inversion techniques. We pursue a polynomial expansion approach, an idea that has been proposed by Wong and Thomas (1962), Wong (1964), Schoutens (2000), Schaumburg (2001), Aït-Sahalia (2002), and Hurn et al. (2008) among others for univariate diffusion processes. Extensions for multivariate (jump-)diffusions do exist in Aït-Sahalia (2008) (with applications in Aït-Sahalia and Kimmel, 2007, Aït-Sahalia and Kimmel, 2010), and Yu (2007), but they follow a different route by approximating the Kolmogorov forward-, and backward partial differential equations. Our approach exploits a crucial property of affine processes. Under some technical conditions, conditional moments of all orders exist and can be explicitly computed in closed form as the solution of a matrix exponential. This ensures that the coefficients of the polynomial expansions can be computed without an approximation error.
We present a general theory of density approximations with several traits of the affine model class in mind. The assumptions made for the general theory are then justified by proving existence and differentiability of the true, unknown transition densities of affine models. These theoretical results, contrary to the density approximations themselves, do rely on Fourier theory. Specifically, we investigate the asymptotic behavior of the characteristic function with novel ODE techniques.
Specializing to affine models we improve earlier work along several lines. Our method (i) is applicable to multivariate models; (ii) works equally well for reducible and irreducible processes in the sense of Aït-Sahalia (2008),2 in particular stochastic volatility models; (iii) produces density approximations the quality of which is independent of the time interval between observations; (iv) allows for expansions on the “correct” state space. That is, the support of the density approximation agrees with the support of the true, unknown transition density as in Hurn et al. (2008) and Schoutens (2000); (v) produces density approximations that integrate to unity by construction, hence are much more amenable to applications that demand the constant of proportionality than the purely polynomial expansions from Aït-Sahalia (2008).3 This includes Wishart processes (Bru, 1991) and even general affine matrix-valued processes (Cuchiero et al., 2010). This paper therefore provides a unified framework for econometric inference for financial models, because in applications one typically needs to evaluate, both, the transition densities themselves, as well as integrals of payoff functions against the transition densities for model-based asset pricing. This complements the methods recently developed in Chen and Joslin (2011) and Kristensen and Mele (2011), which are aimed at asset pricing.4
The paper proceeds as follows: Section 2 develops a general theory of orthonormal polynomial density approximations in certain weighted spaces under suitable integrability and regularity assumptions. These may be validated by the sufficient criteria presented subsequently in Section 3. The density approximations are then specialized within the context of affine processes: Section 4 reviews the affine transform formula and the polynomial moment formula for affine processes, which in turn allows the aforementioned polynomial approximations. The main theoretical contribution–general results on existence and differentiability of transition densities of affine processes–is elaborated in Section 4.3. In Section 5 we introduce candidate weight functions and the Gram–Schmidt algorithm to compute orthonormal polynomial bases corresponding to these weights, along with important examples. Section 6 details computation of the ingredients to the density expansions using an explicit example. Section 7 relates existing techniques for density approximations to ours. An empirical study is presented in Section 8: Applications in stochastic volatility (Section 8.1), option pricing (Section 8.2), credit risk (Section 8.3), and likelihood inference (Section 8.4), support the tractability and usefulness of the likelihood expansions. Section 9 concludes. The proofs of our main results are given in an Appendix.
In the paper we will use the following notational conventions. The nonnegative integers are denoted by . The length of a multi-index is defined by , and we write for any . The degree of a polynomial in is defined as . For the likelihood ratio functions below we define . The class of -times continuously differentiable (or continuous, if ) functions on is denoted by .
Section snippets
Density approximations
Let denote a probability density on whose polynomial moments of every order exist and are known in closed form. For example, may denote the pricing density in a financial market model. Typically, is not known in explicit form, and needs to be approximated. Let be an auxiliary probability density function on . The aim is to expand the likelihood ratio in terms of orthonormal polynomials of in order to get an explicit approximation of the unknown density
Sufficient conditions for Assumptions 1 and 2
In this section we provide sufficient conditions for Assumption 1, Assumption 2 to hold. The proofs of the following lemmas are deferred to Appendix B. We first provide sufficient conditions on that guarantee that Assumption 1 is satisfied. Lemma 3.1 Suppose that the density functionhas a finite exponential momentfor some. Then the set of polynomials is dense in. Moreover, Assumption 1 is satisfied.
In applications, the auxiliary density function on will often be given
Affine models
The main application of the polynomial density approximation are affine factor models. In this section, we follow the setup of Duffie et al. (2003), which we now briefly recap. Let . We define the index set , and write and , for any vector and matrix . We consider an affine process on the canonical state space with generator
Examples of auxiliary density functions
For applications of the polynomial density approximations to affine models we are free to choose any auxiliary density function as weight, limited only by the requirements of Assumption 1, Assumption 2. For Lévy-driven stochastic differential equations, Schaumburg (2001) uses the densities of the driving Lévy processes. For affine processes there is no such obvious candidate in general. We therefore pursue an approach with auxiliary densities with exponential tails, which are reasonably
Explicit example
In the present section we illustrate the necessary steps to engineer a density expansion to order . We adopt notation used conventionally in finance and econometrics. In particular we deviate from Duffie et al. (2003) notation. From here onward the time interval between observations is denoted by .
Consider as an example the Basic Affine Jump-Diffusion process (BAJD) with domain , solving the SDE Here, is a compound Poisson process with jump intensity ,
Relation to existing approximations
In this section we recall facts about closed-form density approximations from previous literature and relate them to the density expansions of the present paper. A short summary of the capabilities and limitations of the different methods is reported in Table 1. The closest methodology to the one introduced in Section 2 is Aït-Sahalia (2002) (AS02) and Chapter 1 of Schaumburg (2001). To illustrate the differences between AS02 and our approach we devote the next section to a direct comparison
Applications
In the following we present applications which highlight the usefulness of the transition density approximations developed in this paper. For the empirical investigations considered below we find that there is a trade-off between numerical accuracy and the order of the expansion. Higher-order expansions may perform worse than low-order expansions due to numerical errors that are induced by the limited numeric precision of the computer environment in representing very large or very small
Discussion
This paper develops a general framework for density approximations for affine processes using orthonormal polynomial expansions in well-chosen weighted spaces. We also provide novel existence and smoothness results for their true, unknown transition densities.
The approximations are designed to exploit the explicit polynomial moments of affine processes to compute the coefficients of the expansion without approximation error and in closed form; the computational burden is concentrated only in
Acknowledgments
We are thankful to Yacine Aït-Sahalia, Michael Brandt, Anna Cieslak, Pierre Collin-Dufresne, Valentina Corradi, Ron Gallant, Cheng Hsiao (the editor), Aleksandar Mijatović, Alessandro Palandri, Benedikt Pötscher, and Gareth Roberts for helpful discussions. We are indebted to the substantial comments of two referees. We benefited from suggestions from participants of the Workshop on Financial Econometrics at the Fields institute, Toronto, the internal workshop at Warwick Business School,
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