Jump-Diffusion Processes and Affine Term Structure Models: Additional Closed-Form Approximate Solutions, Distributional Assumptions for Jumps, and Parameter Estimates

Affine term structure models in which the short rate follows a jump-diffusion process are difficult to solve, and the parameters of such models are hard to estimate. Without analytical answers to the partial difference differential equation (PDDE) for bond prices implied by jumpdiffusion processes, one must find a numerical solution to the PDDE or exactly solve an approximate PDDE. Although the literature focuses on a single linearization technique to estimate the PDDE, this paper outlines alternative methods that seem to improve accuracy. Also, closed-form solutions, numerical estimates, and closed-form approximations of the PDDE each ultimately depend on the presumed distribution of jump sizes, and this paper explores a broader set of possible densities that may be more consistent with intuition, including a bi-modal Gaussian mixture. GMM and MLE of oneand two-factor jump-diffusion models produce some evidence for jumps, but sensitivity analyses suggest sizeable confidence intervals around the parameters. * With no implication whatsoever, the author thanks Sanjiv Das, Henrik Rasmussen, and seminar participants at the Federal Reserve Board for helpful comments and suggestions on this project. The views presented are solely those of the author and do not necessarily represent those of the Federal Reserve Board or its staff.