Open Access
August 2006 A filtering approach to tracking volatility from prices observed at random times
Jakša Cvitanić, Robert Liptser, Boris Rozovskii
Ann. Appl. Probab. 16(3): 1633-1652 (August 2006). DOI: 10.1214/105051606000000222

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=(St)t≥0 is given by

dSt=m(θt)Stdt+v(θt)StdBt,

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.

Citation

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Jakša Cvitanić. Robert Liptser. Boris Rozovskii. "A filtering approach to tracking volatility from prices observed at random times." Ann. Appl. Probab. 16 (3) 1633 - 1652, August 2006. https://doi.org/10.1214/105051606000000222

Information

Published: August 2006
First available in Project Euclid: 2 October 2006

zbMATH: 1108.62108
MathSciNet: MR2260076
Digital Object Identifier: 10.1214/105051606000000222

Subjects:
Primary: 60G35 , 91B28
Secondary: 62M20 , 93E11

Keywords: Discrete observations , Nonlinear filtering , Volatility estimation

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.16 • No. 3 • August 2006
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