Explicit form of approximate transition probability density functions of diffusion processes

A continuous-time diffusion process is very popular in modeling and provides useful tools to analyze particularly, but not restricted to, a variety of economic and financial variables. The transition probability density function (TPDF) of a diffusion process plays an important role in understanding and explaining the dynamics of the process. A new way to find closed-form approximate TPDFs for multivariate diffusions is proposed in this paper. This method can be applied to general multivariate time-inhomogeneous diffusion processes, as long as, roughly speaking, they have smooth drift and volatility functions. A diffusion process is said to be reducible if it can be converted into a unit diffusion process where the volatility is the identity matrix. We have established how to obtain the approximate TPDF of a reducible diffusion explicitly. When a diffusion process is not reducible, an explicit form of approximate TPDF can be obtained by using the results in Aït-Sahalia (2008) and Choi (2013). The TPDF expansion suggested here enables us to obtain a recursive formula for the coefficient of the approximate TPDF for a multivariate jump diffusion. Monte Carlo simulation studies of conducting maximum likelihood estimation (MLE) using our approximations provide convincing evidence that our TPDF expansion can be used for the MLE when the true TPDF is unavailable. We also applied our approximate TPDFs to option pricing. The differences between our option prices and those from the Extended Black-Scholes formula are shown to be quite small. This implies that our methods can be employed to price assets whose underlying state variables follow general diffusion models.