Hermite Expansion for Transition Densities of Irreducible Diffusions with An Application to Option Pricing

Abstract

This paper shows that a small-time Hermite expansion is feasible for multivariate diffusions. By introducing an innovative quasi-Lamperti transform, which unitizes the diffusion matrix at the initial time, we derive explicit recursive formulas for the expansion coefficients of transition densities and European option prices for multivariate diffusions with jumps in return. These immediately available explicit formulas, particularly for the irreducbile, nonaffine, time-inhomogeneous model with different types of jump-size distribution, is new to the literature. The explicit formulas can lead to real-time derivatives pricing and hedging as well as model calibration. Extensive numerical experiments illustrate the accuracy and effectiveness of our approach.