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

Jaksa Cvitanic·Robert Liptser·Boris Rozovskii
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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=(S_{t})_{t\geq0} $ is given by \[ dS_{t}=r(θ_{t})S_{t}dt+v(θ_{t})S_{t}dB_{t}, \] where $B=(B_{t})_{t\geq0}$ is a Brownian motion, $v$ is a positive function, and $θ=(θ_{t})_{t\geq0}$ is a cádlág strong Markov process. The random process $θ$ is unobservable. We assume also that the asset price $S_{t}$ is observed only at random times $0<τ_{1}<τ_{2}<....$ This is an appropriate assumption when modelling 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 $(τ_{k},\log S_{τ_{k}})$. 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 $θ_{t}$, based on the observations of $(τ_{k},\log S_{τ_{k}})_{k\geq1}$. 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.

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BibTeXRIS

Jaksa Cvitanic, Robert Liptser, Boris Rozovskii. 2005-09-22. A filtering approach to tracking volatility from prices observed at random times. https://arxiv.org/abs/math/0509503

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