A filtering approach to tracking volatility from prices observed at random times

Abstract

This paper is concerned with nonlinear filtering of the coefficients in asset price models with stochastic volatility. More specifically, we assume that the asset price process S=(S_t)_t≥0 is given by

dS_t=m(θ_t)S_t dt+v(θ_t)S_t dB_t,

where B=(B_t)_t≥0 is a Brownian motion, v is a positive function and θ=(θ_t)_t≥0 is a cádlág strong Markov process. The random process θ is unobservable. We assume also that the asset price S_t is observed only at random times 0<τ₁<τ₂<⋯. This is an appropriate assumption when modeling high frequency financial data (e.g., tick-by-tick stock prices).

In the above setting the problem of estimation of θ can be approached as a special nonlinear filtering problem with measurements generated by a multivariate point process (τ_k, log S_τk). While quite natural, this problem does not fit into the “standard” diffusion or simple point process filtering frameworks and requires more technical tools. We derive a closed form optimal recursive Bayesian filter for θ_t, based on the observations of (τ_k, log S_τk)_k≥1. It turns out that the filter is given by a recursive system that involves only deterministic Kolmogorov-type equations, which should make the numerical implementation relatively easy.