Parameter estimation for discretely observed stochastic volatility models

This paper deals with parameter estimation for stochastic volatility models. We consider a two-dimensional diffusion process (Y-t, V-t). Only (Y-t) is observed at n discrete times with a regular sampling interval. The unobserved coordinate (V-t) rules the diffusion coefficient (volatility) of (Y-t) and is an ergodic diffusion depending on unknown parameters. We build estimators of the parameters present in the stationary distribution of (V-t), based on appropriate functions of the observations. Consistency is proved under the asymptotic framework that the sampling interval tends to 0, while the number of observations and the length of the observation time tend to infinity. Asymptotic normality is obtained under an additional condition on the rate of convergence of the sampling interval. Examples of models from finance are treated, and numerical simulation results are given.

[1]  Jean Jacod,et al.  On the estimation of the diffusion coefficient for multi-dimensional diffusion processes , 1993 .

[2]  Robert C. Blattberg,et al.  A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices: Reply , 1974 .

[3]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .

[4]  Mathieu Kessler.  Estimation of an Ergodic Diffusion from Discrete Observations , 1997 .

[5]  D. Florens-zmirou,et al.  Estimation of the coefficients of a diffusion from discrete observations , 1986 .

[6]  M. Sørensen,et al.  Martingale estimation functions for discretely observed diffusion processes , 1995 .

[7]  E. Seneta,et al.  The Variance Gamma (V.G.) Model for Share Market Returns , 1990 .

[8]  Michael Sørensen,et al.  A Review of Some Aspects of Asymptotic Likelihood Theory for Stochastic Processes , 1994 .

[9]  Mathieu Kessler,et al.  Simple and Explicit Estimating Functions for a Discretely Observed Diffusion Process , 2000 .

[10]  M. Chesney,et al.  Pricing European Currency Options: A Comparison of the Modified Black-Scholes Model and a Random Variance Model , 1989, Journal of Financial and Quantitative Analysis.

[11]  L. Rogers,et al.  Diffusions, Markov processes, and martingales , 1979 .

[12]  C. Laredo,et al.  A Sufficient Condition for Asymptotic Sufficiency of Incomplete Observations of a Diffusion Process , 1990 .

[13]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[14]  C. W. Clenshaw,et al.  The special functions and their approximations , 1972 .

[15]  A. Harvey,et al.  5 Stochastic volatility , 1996 .

[16]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[17]  T. Pitcher,et al.  Parameter estimation for stochastic processes , 1964 .

[18]  On estimating the diffusion coefficient , 1987 .

[19]  Daniel B. Nelson.  ARCH models as diffusion approximations , 1990 .

[20]  Thierry Jeantheau,et al.  Limit theorems for discretely observed stochastic volatility models , 1998 .