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Stochastic volatility and option pricing with long-memory in discrete and continuous time

Author

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  • Alexandra Chronopoulou
  • Frederi G. Viens

Abstract

It is commonly accepted that certain financial data exhibit long-range dependence. We consider a continuous-time stochastic volatility model in which the stock price is Geometric Brownian Motion with volatility described by a fractional Ornstein--Uhlenbeck process. We also study two discrete-time models: a discretization of the continuous model via a Euler scheme and a discrete model in which the returns are a zero mean i.i.d. sequence where the volatility is a fractional ARIMA process. We implement a particle filtering algorithm to estimate the empirical distribution of the unobserved volatility, which we then use in the construction of a multinomial recombining tree for option pricing. We also discuss appropriate parameter estimation techniques for each model. For the long-memory parameter we compute an implied value by calibrating the model with real data. We compare the performance of the three models using simulated data and we price options on the S&P 500 index.

Suggested Citation

  • Alexandra Chronopoulou & Frederi G. Viens, 2012. "Stochastic volatility and option pricing with long-memory in discrete and continuous time," Quantitative Finance, Taylor & Francis Journals, vol. 12(4), pages 635-649, December.
    • Handle: RePEc:taf:quantf:v:12:y:2012:i:4:p:635-649
    DOI: 10.1080/14697688.2012.664939
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    Citations

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    Cited by:

    1. Archil Gulisashvili, 2017. "Large deviation principle for Volterra type fractional stochastic volatility models," Papers 1710.10711, arXiv.org, revised Aug 2018.
    2. Turiel, Jeremy D. & Aste, Tomaso, 2022. "Heterogeneous criticality in high frequency finance: a phase transition in flash crashes," LSE Research Online Documents on Economics 113892, London School of Economics and Political Science, LSE Library.
    3. Sung Ik Kim, 2022. "ARMA–GARCH model with fractional generalized hyperbolic innovations," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 8(1), pages 1-25, December.
    4. Nourdin, Ivan & Diu Tran, T.T., 2019. "Statistical inference for Vasicek-type model driven by Hermite processes," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3774-3791.
    5. Archil Gulisashvili & Frederi Viens & Xin Zhang, 2019. "Extreme-strike asymptotics for general Gaussian stochastic volatility models," Annals of Finance, Springer, vol. 15(1), pages 59-101, March.
    6. Antoine Jacquier & Alexandre Pannier, 2020. "Large and moderate deviations for stochastic Volterra systems," Papers 2004.10571, arXiv.org, revised Apr 2022.
    7. Xiao, Weilin & Yu, Jun, 2019. "Asymptotic theory for rough fractional Vasicek models," Economics Letters, Elsevier, vol. 177(C), pages 26-29.
    8. Archil Gulisashvili & Frederi Viens & Xin Zhang, 2015. "Small-time asymptotics for Gaussian self-similar stochastic volatility models," Papers 1505.05256, arXiv.org, revised Mar 2016.
    9. Wang, XiaoTian & Yang, ZiJian & Cao, PiYao & Wang, ShiLin, 2021. "The closed-form option pricing formulas under the sub-fractional Poisson volatility models," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    10. Rachid Belfadli & Khalifa Es-Sebaiy & Fatima-Ezzahra Farah, 2022. "Statistical analysis of the non-ergodic fractional Ornstein–Uhlenbeck process with periodic mean," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(7), pages 885-911, October.
    11. Wang, Jixia & Xiao, Xiaofang & Li, Chao, 2023. "Least squares estimations for approximate fractional Vasicek model driven by a semimartingale," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 207-218.
    12. Chih-Chen Hsu & An-Sing Chen & Shih-Kuei Lin & Ting-Fu Chen, 2017. "The affine styled-facts price dynamics for the natural gas: evidence from daily returns and option prices," Review of Quantitative Finance and Accounting, Springer, vol. 48(3), pages 819-848, April.
    13. Jacquier, Antoine & Pannier, Alexandre, 2022. "Large and moderate deviations for stochastic Volterra systems," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 142-187.
    14. Josselin Garnier & Knut Solna, 2015. "Correction to Black-Scholes formula due to fractional stochastic volatility," Papers 1509.01175, arXiv.org, revised Mar 2017.
    15. Sérgio C. Bezerra & Alberto Ohashi & Francesco Russo & Francys Souza, 2020. "Discrete-type Approximations for Non-Markovian Optimal Stopping Problems: Part II," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1221-1255, September.
    16. Archil Gulisashvili & Frederi Viens & Xin Zhang, 2015. "Extreme-Strike Asymptotics for General Gaussian Stochastic Volatility Models," Papers 1502.05442, arXiv.org, revised Feb 2017.
    17. Qi Zhao & Alexandra Chronopoulou, 2023. "Delta-hedging in fractional volatility models," Annals of Finance, Springer, vol. 19(1), pages 119-140, March.
    18. Yicun Li & Yuanyang Teng, 2022. "Estimation of the Hurst Parameter in Spot Volatility," Mathematics, MDPI, vol. 10(10), pages 1-26, May.

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