Estimating jump–diffusions using closed-form likelihood expansions

The indispensable role of likelihood expansions in financial econometrics for continuous-time models has been established since the ground-breaking work of Aït-Sahalia (1999, 2002a, 2008). Jump–diffusions play an important role in modeling a variety of economic and financial variables. As a significant generalization of Li (2013), we propose a new closed-form expansion for transition density of Poisson-driven jump–diffusion models and its application in maximum-likelihood estimation based on discretely sampled data. Technically speaking, our expansion is obtained by perturbing paths of the underlying model; correction terms can be calculated explicitly using any symbolic software. Numerical examples and Monte Carlo evidence for illustrating the performance of density expansion and the resulting approximate MLE are provided in order to demonstrate the practical applicability of the method. Using the theoretical results in Hayashi and Ishikawa (2012), some convergence properties related to the density expansion and the approximate MLE method can be justified under some standard sufficient (but not necessary) conditions.