TY - JOUR
T1 - A filtering approach to tracking volatility from prices observed at random times
AU - Cvitanić, Jakša
AU - Liptser, Robert
AU - Rozovskii, Boris
PY - 2006/8
Y1 - 2006/8
N2 - 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≥o is given by dSt =m(θt)St dt + ν(θ t)St dBt, where B = (Bt) t≥0 is a Brownian motion, ν 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 S τk)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.
AB - 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≥o is given by dSt =m(θt)St dt + ν(θ t)St dBt, where B = (Bt) t≥0 is a Brownian motion, ν 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 S τk)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.
KW - Discrete observations
KW - Nonlinear filtering
KW - Volatility estimation
UR - http://www.scopus.com/inward/record.url?scp=33750522200&partnerID=8YFLogxK
U2 - 10.1214/105051606000000222
DO - 10.1214/105051606000000222
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AN - SCOPUS:33750522200
SN - 1050-5164
VL - 16
SP - 1633
EP - 1652
JO - Annals of Applied Probability
JF - Annals of Applied Probability
IS - 3
ER -