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Fractional Brownian markets with time-varying volatility and high-frequency data

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  • Lahiri, Ananya
  • Sen, Rituparna

Abstract

Diffusion processes driven by fractional Brownian motion (fBm) have often been considered in modeling stock price dynamics in order to capture the long range dependence of stock prices observed in real markets. Option prices for such models under constant drift and volatility are available. Option prices are obtained under time varying volatility. The expression of option price depends on the volatility and the Hurst parameter of the model, in a complicated manner. A central limit theorem is derived for the quadratic variation as an estimator for volatility for both the cases, constant as well as time varying volatility. The estimator of volatility is useful for finding estimators of option prices and their asymptotic distributions.

Suggested Citation

  • Lahiri, Ananya & Sen, Rituparna, 2020. "Fractional Brownian markets with time-varying volatility and high-frequency data," Econometrics and Statistics, Elsevier, vol. 16(C), pages 91-107.
    • Handle: RePEc:eee:ecosta:v:16:y:2020:i:c:p:91-107
    DOI: 10.1016/j.ecosta.2018.10.004
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    References listed on IDEAS

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    1. Robert J. Elliott & John Van Der Hoek, 2003. "A General Fractional White Noise Theory And Applications To Finance," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 301-330, April.
    2. Erhan Bayraktar & H. Vincent Poor & K. Ronnie Sircar, 2004. "Estimating The Fractal Dimension Of The S&P 500 Index Using Wavelet Analysis," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 7(05), pages 615-643.
    3. Zhang, Lan & Mykland, Per A. & Ait-Sahalia, Yacine, 2005. "A Tale of Two Time Scales: Determining Integrated Volatility With Noisy High-Frequency Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1394-1411, December.
    4. Paolo Guasoni, 2006. "No Arbitrage Under Transaction Costs, With Fractional Brownian Motion And Beyond," Mathematical Finance, Wiley Blackwell, vol. 16(3), pages 569-582, July.
    5. Tomas Björk & Henrik Hult, 2005. "A note on Wick products and the fractional Black-Scholes model," Finance and Stochastics, Springer, vol. 9(2), pages 197-209, April.
    6. Ole E. Barndorff-Nielsen, 2004. "Power and Bipower Variation with Stochastic Volatility and Jumps," Journal of Financial Econometrics, Oxford University Press, vol. 2(1), pages 1-37.
    7. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
    8. Cipian Necula, 2008. "Option Pricing in a Fractional Brownian Motion Environment," Advances in Economic and Financial Research - DOFIN Working Paper Series 2, Bucharest University of Economics, Center for Advanced Research in Finance and Banking - CARFIB.
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    Cited by:

    1. Rama Cont & Purba Das, 2022. "Rough volatility: fact or artefact?," Papers 2203.13820, arXiv.org, revised Jul 2023.
    2. R. Vilela Mendes, 2022. "The fractional volatility model and rough volatility," Papers 2206.02205, arXiv.org.

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