A Differentiable Particle Filter for Jump-Diffusion Stochastic Volatility Models

Abstract

Stochastic volatility with jumps has emerged as a crucial tool for understanding and modelling the stochastic and intermittently discontinuous nature of many processes in finance. Due to the highly nonlinear structure of these models, their likelihood functions are often unavailable in closed-form. A common numerical approach is to reformulate the original model as a state-space model, where under this framework, the marginal likelihood of the parameters can be estimated efficiently by integrating the latent variables via particle filtering. A combination of such particle-estimated likelihood and Markov Chain Monte Carlo can be used to sample from parameter posteriors, but imposes a substantial computational burden in multi-dimensional parameter space. Bayesian normal approximation serves as a more efficient alternative, if the mode and quadrature of the stochastic approximation of the posterior can be obtained via a gradient-based method. This is not immediately possible, however, as the particle-estimated marginal posterior is not differentiable due to (1) the inherent discontinuity of jumps in the model, and (2) the widely used multinomial resampling technique in particle filtering. This thesis presents a novel construction of a particle filter that incorporates a multivariate normal resampler and circumvents the jump-induced discontinuity with a customized proposal density, thereby attaining full differentiability of the marginal posterior estimate. A comprehensive simulation study and application to S&P 500 Index data are provided to investigate the performance of the differentiable particle filter for parameter inference and volatility recovery.