The Annals of Applied Probability

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

Jakša Cvitanić, Robert Liptser, and Boris Rozovskii

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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


where B=(Bt)t≥0 is a Brownian motion, v 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 Sτk)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.

Article information

Ann. Appl. Probab., Volume 16, Number 3 (2006), 1633-1652.

First available in Project Euclid: 2 October 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 91B28
Secondary: 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 93E11: Filtering [See also 60G35]

Nonlinear filtering discrete observations volatility estimation


Cvitanić, Jakša; Liptser, Robert; Rozovskii, Boris. A filtering approach to tracking volatility from prices observed at random times. Ann. Appl. Probab. 16 (2006), no. 3, 1633--1652. doi:10.1214/105051606000000222.

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