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Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time Observations

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  • John-Fritz Thony

    (Laboratoire de Mathématiques et Informatique et Applications (LAMIA), Université des Antilles, 97157 Pointe-à-Pitre, France
    Ecole Normale Supérieure, Université d’Etat d’Haiti, Port-au-Prince HT6110, Haiti)

  • Jean Vaillant

    (Laboratoire de Mathématiques et Informatique et Applications (LAMIA), Université des Antilles, 97157 Pointe-à-Pitre, France)

Abstract

We consider a stochastic differential equation (SDE) governed by a fractional Brownian motion ( B t H ) and a Poisson process ( N t ) associated with a stochastic process ( A t ) such that: d X t = μ X t d t + σ X t d B t H + A t X t − d N t , X 0 = x 0 > 0 . The solution of this SDE is analyzed and properties of its trajectories are presented. Estimators of the model parameters are proposed when the observations are carried out in discrete time. Some convergence properties of these estimators are provided according to conditions concerning the value of the Hurst index and the nonequidistance of the observation dates.

Suggested Citation

  • John-Fritz Thony & Jean Vaillant, 2022. "Parameter Estimation for a Fractional Black–Scholes Model with Jumps from Discrete Time Observations," Mathematics, MDPI, vol. 10(22), pages 1-17, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4190-:d:967517
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    References listed on IDEAS

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    1. Hu, Yaozhong & Nualart, David & Song, Xiaoming, 2008. "A singular stochastic differential equation driven by fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2075-2085, October.
    2. Arne Henningsen & Ott Toomet, 2011. "maxLik: A package for maximum likelihood estimation in R," Computational Statistics, Springer, vol. 26(3), pages 443-458, September.
    3. Coeurjolly, Jean-Francois, 2000. "Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 5(i07).
    4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    5. El Mehdi Haress & Yaozhong Hu, 2021. "Estimation of all parameters in the fractional Ornstein–Uhlenbeck model under discrete observations," Statistical Inference for Stochastic Processes, Springer, vol. 24(2), pages 327-351, July.
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