Cvitani., Jak.a et al., A filtering approach to tracking volatility from prices observed at random times, Annals of applied probability , 16(3), 2006, pp. 1633-1652
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=(St)t.0 is given by dSt=m(.t)St.dt+v(.t)St.dBt, where B=(Bt)t.0 is a Brownian motion, v is a positive function and .=(.t)t.0 is a cádlág strong Markov process. The random process . is unobservable. We assume also that the asset price St is observed only at random times 0<.1<.2<.. 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 . can be approached as a special nonlinear filtering problem with measurements generated by a multivariate point process (.k, log.S.k). 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 .t, based on the observations of (.k, log.S.k)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.