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Price modelling under generalized fractional Brownian motion

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  • Axel A. Araneda

Abstract

The Generalized fractional Brownian motion (gfBm) is a stochastic process that acts as a generalization for both fractional, sub-fractional, and standard Brownian motion. Here we study its use as the main driver for price fluctuations, replacing the standard Brownian Brownian motion in the well-known Black-Scholes model. By the derivation of the generalized fractional Ito's lemma and the related effective Fokker-Planck equation, we discuss its application to both the option pricing problem valuing European options, and the computation of Value-at-Risk and Expected Shortfall. Moreover, the option prices are computed for a CEV-type model driven by gfBm.

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  • Axel A. Araneda, 2021. "Price modelling under generalized fractional Brownian motion," Papers 2108.12042, arXiv.org, revised Nov 2023.
    • Handle: RePEc:arx:papers:2108.12042
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    References listed on IDEAS

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    1. Kim, Kyong-Hui & Yun, Sim & Kim, Nam-Ung & Ri, Ju-Hyuang, 2019. "Pricing formula for European currency option and exchange option in a generalized jump mixed fractional Brownian motion with time-varying coefficients," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 522(C), pages 215-231.
    2. 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.
    3. repec:bla:jfinan:v:44:y:1989:i:1:p:211-19 is not listed on IDEAS
    4. Emanuel, David C. & MacBeth, James D., 1982. "Further Results on the Constant Elasticity of Variance Call Option Pricing Model," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 17(4), pages 533-554, November.
    5. Ahmadian, D. & Ballestra, L.V., 2020. "Pricing geometric Asian rainbow options under the mixed fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 555(C).
    6. 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.
    7. Tomasz Bojdecki & Luis G. Gorostiza & Anna Talarczyk, 2004. "Sub-fractional Brownian motion and its relation to occupation times," RePAd Working Paper Series lrsp-TRS376, Département des sciences administratives, UQO.
    8. 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).
    9. 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.
    10. Wang, Wei & Cai, Guanghui & Tao, Xiangxing, 2021. "Pricing geometric asian power options in the sub-fractional brownian motion environment," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    11. Bojdecki, Tomasz & Gorostiza, Luis G. & Talarczyk, Anna, 2004. "Sub-fractional Brownian motion and its relation to occupation times," Statistics & Probability Letters, Elsevier, vol. 69(4), pages 405-419, October.
    12. Axel A. Araneda & Nils Bertschinger, 2020. "The sub-fractional CEV model," Papers 2001.06412, arXiv.org, revised Mar 2021.
    13. Araneda, Axel A. & Bertschinger, Nils, 2021. "The sub-fractional CEV model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 573(C).
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