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 credit risk, likelihood inference, and option pricing highlight the usefulness of our expansions. The approximations are extremely fast to evaluate, and they perform very accurately and numerically stable.

Stationarity and Ergodicity for an Affine Two-Factor Model

“…in Filipović et al [14]): if S : R + × R → R is a continuous function which is locally Lipschitz continuous in its second variable and p, q : R + → R are differentiable functions satisfying…”

“…Finally, we note that for calculating the moments E(Y n ∞ X p ∞ ) < ∞, n, p ∈ Z + , we could have used formula (4.4) in Filipović et al [14] which gives a formal representation of the polynomial moments of (Y t , X t ), t ∈ R + . The idea behind this formal representation is that the infinitesimal generator of the affine process (Y, X) formally maps the finite-dimensional linear space of all polynomials in (y, x) ∈ R + × R of degree less than or equal to k into itself, where k ∈ N. For a more general class of time-homogeneous Markov processes having this property, for the so-called polynomial processes, see Cuchiero et al [10].…”

Realized Laplace Transforms for Estimation of Jump Diffusive Volatility Models

“…Filipovic et al (2010) employ special routines QAWF and QAWO from the GNU scientific library to numerically compute the Fourier integral of a characteristic function. However, those routines assume that the main source of oscillations in the integrand is the sine (or cosine) factor coming from the Fourier transform.…”

Exponential Ergodicity of the Jump-Diffusion CIR Process

“…We remark that the results in [10,11] are very general and hold for a large class of affine process with state space R + , where R + denotes the set of all non-negative real numbers. The existence and some approximations of the transition densities of the BAJD can be found in [6]. A closed formula of the transition densities of the BAJD was recently derived in [9].…”