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

Cvitanić, Jakša and Lipster, Robert and Rozovskii, Boris (2006) A filtering approach to tracking volatility from prices observed at random times. Annals of Applied Probability, 16 (3). pp. 1633-1652. ISSN 1050-5164. https://resolver.caltech.edu/CaltechAUTHORS:CVIaap06

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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)St dt+v(θt)St dBt, 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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1214/105051606000000222DOIArticle
https://arxiv.org/abs/math/0612212arXivDiscussion Paper
ORCID:
AuthorORCID
Cvitanić, Jakša0000-0001-6651-3552
Additional Information:© 2006 The Institute of Mathematical Statistics. Received December 2003; revised February 2006. We are grateful to the anonymous Associate Editor and the referee for their constructive suggestions, especially regarding a simplified presentation of the results. We are very much indebted to Remigijus Mikulevicius for many important suggestions, and to Ilya Zaliapin, whose numerical experiments helped to discover an error in a preprint version of the paper. [J.C. was] [s]upported in part by the NSF Grants DMS-00-99549 and DMS-04-03575. [B.R. was] [s]upported in part by the Army Research Office and the Office of Naval Research under Grants DAAD19-02-1-0374 and N0014-03-0027.
Funders:
Funding AgencyGrant Number
NSFDMS-00-99549
NSFDMS-04-03575
Army Research Office (ARO)DAAD19-02-1-0374
Office of Naval Research (ONR)N00014-03-0027
Subject Keywords:Nonlinear filtering, discrete observations, volatility estimation
Issue or Number:3
Record Number:CaltechAUTHORS:CVIaap06
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:CVIaap06
Official Citation: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. https://projecteuclid.org/euclid.aoap/1159804994
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6977
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:04 Jan 2007
Last Modified:29 Jan 2020 17:14

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