Tracking volatility (stock markets)

Abstract

Concerns nonlinear filtering of the volatility coefficient in a Black-Scholes type model that allows stochastic volatility. The asset price process S=(S/sub t/)/sub t/spl ges/0/ is given by dS/sub t/=rS/sub t/dt+/spl radic/v/sub t/S/sub t/dB/sub t/, where B=(B/sub t/)/sub t/spl ges/0/ is a Brownian motion and v/sub t/ is the (stochastic) volatility process. Moreover, assumed that v/sub t/=v(/spl theta//sub t/) where v is a nonnegative function and /spl theta/=(/spl theta//sub t/)/sub t/spl ges/0/ is a homogeneous Markov jump process, taking values in the finite alphabet (a/sub 1/,...,a/sub M/), with the intensity matrix /spl Lambda/=/spl par//spl lambda//sub ij//spl par/ and the initial distribution p/sub q/=P_(/spl theta//sub 0/=a/sub q/), q=1,...,M. The random process /spl theta/ is unobservable. Following Frey and Runggaldier (1999), we assume also that S/sub t/ is measured only at random times 0</spl tau//sub 1/< /spl tau//sub 2/<...,. This assumption reflects the discrete nature of high frequency financial data. The random time moments /spl tau//sub k/ represent instances at which a large trade occurs or at which a market maker updates his quotes in reaction to new definition. In the above setting the problem of volatility estimation is reduced naturally to a special nonlinear filtering problem. We remark that while quite natural, the latter problem does not fit into the standard framework and requires new technical tools. In this paper, we derive a mean-square optimal recursive Bayesian filter for /spl theta//sub t/ based on the observations of S/sub /spl tau/1/,S/sub /spl tau/2/,... for all /spl tau//sub k//spl les/t. In addition we derive Duncan-Mortensen-Zakai and Wonham-Kushner type equations for posterior distributions of /spl theta//sub t/ and prove uniqueness of their solutions.