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

Jakša Cvitanić*, Robert Liptser, Boris Rozovskii

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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≥o is given by dSt =m(θt)St dt + ν(θ t)St dBt, where B = (Bt) t≥0 is a Brownian motion, ν is a positive function and θ = (θt)t≥0 is a cádlág strong Markov process. The random process θ is unobservable. We assume also that the asset price St 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 (τk, log Sτk). 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 θt, based on the observations of (τk, 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.

Original languageEnglish
Pages (from-to)1633-1652
Number of pages20
JournalAnnals of Applied Probability
Issue number3
StatePublished - Aug 2006


  • Discrete observations
  • Nonlinear filtering
  • Volatility estimation


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