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

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

2006-12-08

arxiv.org/abs/math/0612212v1

Annals of Applied Probability 2006, Vol. 16, No. 3, 1633-1652

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051606000000222 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051606000000222

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 \[ dS_{t}=m(\theta_{t})S_{t} dt+v(\theta_{t})S_{t} dB_{t}, \] where $B=(B_{t})_{t\geq0}$ is a Brownian motion, $v$ is a positive function and $\theta=(\theta_{t})_{t\geq0}$ is a c\'{a}dl\'{a}g strong Markov process. The random process $\theta$ is unobservable. We assume also that the asset price $S_{t}$ is observed only at random times $0<\tau_{1}<\tau_{2}<....$ 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 $\theta$ can be approached as a special nonlinear filtering problem with measurements generated by a multivariate point process $(\tau_{k},\log S_{\tau_{k}})$. 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 $\theta_{t}$, based on the observations of $(\tau_{k},\log S_{\tau_{k}})_{k\geq1}$. 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.

Affiliated with

Also associated with

No associations

Rating

A filtering approach to tracking volatility from prices observed at random times does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A filtering approach to tracking volatility from prices observed at random times, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A filtering approach to tracking volatility from prices observed at random times will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-573400