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The application of fractional derivatives in stochastic models driven by fractional Brownian motion

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  • Longjin, Lv
  • Ren, Fu-Yao
  • Qiu, Wei-Yuan

Abstract

In this paper, in order to establish connection between fractional derivative and fractional Brownian motion (FBM), we first prove the validity of the fractional Taylor formula proposed by Guy Jumarie. Then, by using the properties of this Taylor formula, we derive a fractional Itô formula for H∈[1/2,1), which coincides in form with the one proposed by Duncan for some special cases, whose formula is based on the Wick Product. Lastly, we apply this fractional Itô formula to the option pricing problem when the underlying of the option contract is supposed to be driven by a geometric fractional Brownian motion. The case that the drift, volatility and risk-free interest rate are all dependent on t is also discussed.

Suggested Citation

  • Longjin, Lv & Ren, Fu-Yao & Qiu, Wei-Yuan, 2010. "The application of fractional derivatives in stochastic models driven by fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(21), pages 4809-4818.
    • Handle: RePEc:eee:phsmap:v:389:y:2010:i:21:p:4809-4818
    DOI: 10.1016/j.physa.2010.06.016
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    References listed on IDEAS

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    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    2. Bender, Christian, 2003. "An Itô formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter," Stochastic Processes and their Applications, Elsevier, vol. 104(1), pages 81-106, March.
    3. 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.
    4. Jumarie, Guy, 2008. "Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 271-287, February.
    5. Jonathan Lewellen, 2002. "Momentum and Autocorrelation in Stock Returns," The Review of Financial Studies, Society for Financial Studies, vol. 15(2), pages 533-564, March.
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    Cited by:

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    3. Xiao, Wei-Lin & Zhang, Wei-Guo & Zhang, Xili & Zhang, Xiaoli, 2012. "Pricing model for equity warrants in a mixed fractional Brownian environment and its algorithm," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(24), pages 6418-6431.
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    7. Song, Wanqing & Li, Ming & Li, Yuanyuan & Cattani, Carlo & Chi, Chi-Hung, 2019. "Fractional Brownian motion: Difference iterative forecasting models," Chaos, Solitons & Fractals, Elsevier, vol. 123(C), pages 347-355.
    8. Yue Qi & Yue Wang, 2023. "Innovating and Pricing Carbon-Offset Options of Asian Styles on the Basis of Jump Diffusions and Fractal Brownian Motions," Mathematics, MDPI, vol. 11(16), pages 1-22, August.
    9. Liu, He & Song, Wanqing & Li, Ming & Kudreyko, Aleksey & Zio, Enrico, 2020. "Fractional Lévy stable motion: Finite difference iterative forecasting model," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    10. Song, Wanqing & Cattani, Carlo & Chi, Chi-Hung, 2020. "Multifractional Brownian motion and quantum-behaved particle swarm optimization for short term power load forecasting: An integrated approach," Energy, Elsevier, vol. 194(C).
    11. Farshid Mehrdoust & Ali Reza Najafi, 2018. "Pricing European Options under Fractional Black–Scholes Model with a Weak Payoff Function," Computational Economics, Springer;Society for Computational Economics, vol. 52(2), pages 685-706, August.

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