An approximation of small-time probability density functions in a general jump diffusion model

We propose a method for approximating probability density functions related to multidimensional jump diffusion processes. For small-time horizons, a closed-form approximation of the characteristic function is derived based on the Itô–Taylor expansion. The probability density function is then approximated numerically by inverting the characteristic function using fast Fourier transform. As application we consider a general stochastic volatility model, which involves time-/state-dependent drift and diffusion functions as well as jump components. We test our approach under the Heston model and the Bates model and show that our method provides accurate approximations.