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Rate of convergence for parametric estimation in a stochastic volatility model

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  • Hoffmann, Marc

Abstract

We consider the following hidden Markov chain problem: estimate the finite-dimensional parameter [theta] in the equation when we observe discrete data Xi/n at times i=0,...,n from the diffusion . The processes (Wt)t[set membership, variant][0,1] and (Bt)t[set membership, variant][0,1] are two independent Brownian motions; asymptotics are taken as n-->[infinity]. This stochastic volatility model has been paid some attention lately, especially in financial mathematics. We prove in this note that the unusual rate n-1/4 is a lower bound for estimating [theta]. This rate is indeed optimal, since Gloter (CR Acad. Sci. Paris, t330, Série I, pp. 243-248), exhibited n-1/4 consistent estimators. This result shows in particular the significant difference between "high frequency data" and the ergodic framework in stochastic volatility models (compare Genon-Catalot, Jeantheau and Laredo (Bernoulli 4 (1998) 283; Bernoulli 5 (2000) 855; Bernoulli 6 (2000) 1051 and also Sørensen (Prediction-based estimating functions. Technical report, Department of Theoretical Statistics, University of Copenhagen, 1998)).

Suggested Citation

  • Hoffmann, Marc, 2002. "Rate of convergence for parametric estimation in a stochastic volatility model," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 147-170, January.
    • Handle: RePEc:eee:spapps:v:97:y:2002:i:1:p:147-170
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    References listed on IDEAS

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    1. Melino, Angelo & Turnbull, Stuart M., 1990. "Pricing foreign currency options with stochastic volatility," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 239-265.
    2. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    3. Wiggins, James B., 1987. "Option values under stochastic volatility: Theory and empirical estimates," Journal of Financial Economics, Elsevier, vol. 19(2), pages 351-372, December.
    4. Chesney, Marc & Scott, Louis, 1989. "Pricing European Currency Options: A Comparison of the Modified Black-Scholes Model and a Random Variance Model," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 24(3), pages 267-284, September.
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    Cited by:

    1. Neil Shephard, 2005. "Stochastic Volatility," Economics Papers 2005-W17, Economics Group, Nuffield College, University of Oxford.
    2. Gloter, A. & Hoffmann, M., 2004. "Stochastic volatility and fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 113(1), pages 143-172, September.
    3. Cecilia Mancini & Vanessa Mattiussi & Roberto Renò, 2015. "Spot volatility estimation using delta sequences," Finance and Stochastics, Springer, vol. 19(2), pages 261-293, April.
    4. Reiß, Markus & Winkelmann, Lars, 2021. "Inference on the maximal rank of time-varying covariance matrices using high-frequency data," Discussion Papers 2021/14, Free University Berlin, School of Business & Economics.
    5. Robert Azencott & Peng Ren & Ilya Timofeyev, 2020. "Realised volatility and parametric estimation of Heston SDEs," Finance and Stochastics, Springer, vol. 24(3), pages 723-755, July.

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