Density approximations for multivariate affine jump-diffusion processes

Abstract

We introduce closed-form transition density expansions for multivariate affine jump-diffusion processes. The expansions rely on a general approximation theory which we develop in weighted Hilbert spaces for random variables which possess all polynomial moments. We establish parametric conditions which guarantee existence and differentiability of transition densities of affine models and show how they naturally fit into the approximation framework. Empirical applications in option pricing, credit risk, and likelihood inference highlight the usefulness of our expansions. The approximations are extremely fast to evaluate, and they perform very accurately and numerically stable.

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 ${\mathcal{L}}^2$ 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 ${\mathbb{N}}_0$. The length of a multi-index $\alpha =({\alpha }_1,\dots ,{\alpha }_d)\in {\mathbb{N}}_0^d$ is defined by $\left|\alpha \right|={\alpha }_1+\cdots +{\alpha }_d$, and we write ${\xi }^{\alpha }={\xi }_1^{{\alpha }_1}\cdots {\xi }_d^{{\alpha }_d}$ for any $\xi \in {\mathbb{R}}^d$. The degree of a polynomial $p\left(x\right)={\sum }_{\left|\alpha \right|\ge 0}{p}_{\alpha }{x}^{\alpha }$ in $x\in {\mathbb{R}}^d$ is defined as $degp\left(x\right)=max\{\left|\alpha \right|\mid p_{\alpha }\ne 0\}$. For the likelihood ratio functions below we define $0/0=0$. The class of $p$-times continuously differentiable (or continuous, if $p=0$) functions on ${\mathbb{R}}^d$ is denoted by $C^p$.