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Scaling and long-range dependence in option pricing II: Pricing European option with transaction costs under the mixed Brownian–fractional Brownian model

Author

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  • Wang, Xiao-Tian
  • Zhu, En-Hui
  • Tang, Ming-Ming
  • Yan, Hai-Gang

Abstract

This paper deals with the problem of discrete-time option pricing by the mixed Brownian–fractional Brownian model with transaction costs. By a mean-self-financing delta hedging argument in a discrete-time setting, a European call option pricing formula is obtained. In particular, the minimal pricing cmin(t,st) of an option under transaction costs is obtained, which shows that timestep δt and Hurst exponent H play an important role in option pricing with transaction costs. In addition, we also show that there exists fundamental difference between the continuous-time trade and discrete-time trade and that continuous-time trade assumption will result in underestimating the value of a European call option.

Suggested Citation

  • Wang, Xiao-Tian & Zhu, En-Hui & Tang, Ming-Ming & Yan, Hai-Gang, 2010. "Scaling and long-range dependence in option pricing II: Pricing European option with transaction costs under the mixed Brownian–fractional Brownian model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(3), pages 445-451.
    • Handle: RePEc:eee:phsmap:v:389:y:2010:i:3:p:445-451
    DOI: 10.1016/j.physa.2009.09.043
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    References listed on IDEAS

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    Cited by:

    1. Shokrollahi, Foad & Sottinen, Tommi, 2017. "Hedging in fractional Black–Scholes model with transaction costs," Statistics & Probability Letters, Elsevier, vol. 130(C), pages 85-91.
    2. Xiao, Weilin & Zhang, Weiguo & Xu, Weijun & Zhang, Xili, 2012. "The valuation of equity warrants in a fractional Brownian environment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(4), pages 1742-1752.
    3. Kim, Kyong-Hui & Kim, Nam-Ung & Ju, Dong-Chol & Ri, Ju-Hyang, 2020. "Efficient hedging currency options in fractional Brownian motion model with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    4. Ballestra, Luca Vincenzo & Pacelli, Graziella & Radi, Davide, 2016. "A very efficient approach for pricing barrier options on an underlying described by the mixed fractional Brownian motion," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 240-248.
    5. Ying Chang & Yiming Wang & Sumei Zhang, 2021. "Option Pricing under Double Heston Jump-Diffusion Model with Approximative Fractional Stochastic Volatility," Mathematics, MDPI, vol. 9(2), pages 1-10, January.
    6. 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.
    7. 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).
    8. M. Rezaei & A. R. Yazdanian & A. Ashrafi & S. M. Mahmoudi, 2022. "Numerically Pricing Nonlinear Time-Fractional Black–Scholes Equation with Time-Dependent Parameters Under Transaction Costs," Computational Economics, Springer;Society for Computational Economics, vol. 60(1), pages 243-280, June.
    9. Foad Shokrollahi, 2017. "The valuation of European option with transaction costs by mixed fractional Merton model," Papers 1702.00152, arXiv.org.
    10. Foad Shokrollahi, 2017. "Fractional delta hedging strategy for pricing currency options with transaction costs," Papers 1702.00037, arXiv.org.
    11. Wang, Wensheng, 2019. "Asymptotics for discrete time hedging errors under fractional Black–Scholes models," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 160-170.
    12. Foad Shokrollahi & Davood Ahmadian & Luca Vincenzo Ballestra, 2021. "Actuarial strategy for pricing Asian options under a mixed fractional Brownian motion with jumps," Papers 2105.06999, arXiv.org.
    13. Foad Shokrollahi, 2016. "Subdiffusive fractional Brownian motion regime for pricing currency options under transaction costs," Papers 1612.06665, arXiv.org, revised Aug 2017.
    14. Foad Shokrollahi & Tommi Sottinen, 2017. "Hedging in fractional Black-Scholes model with transaction costs," Papers 1706.01534, arXiv.org, revised Jul 2017.
    15. Jean-Philippe Aguilar & Jan Korbel & Yuri Luchko, 2019. "Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations," Mathematics, MDPI, vol. 7(9), pages 1-23, September.
    16. Gu, Hui & Liang, Jin-Rong & Zhang, Yun-Xiu, 2012. "Time-changed geometric fractional Brownian motion and option pricing with transaction costs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(15), pages 3971-3977.
    17. Guo, Zhidong & Yuan, Hongjun, 2014. "Pricing European option under the time-changed mixed Brownian-fractional Brownian model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 406(C), pages 73-79.
    18. Tommi Sottinen & Lauri Viitasaari, 2017. "Conditional-Mean Hedging Under Transaction Costs in Gaussian Models," Papers 1708.03242, arXiv.org.
    19. Zhang, Xili & Xiao, Weilin, 2017. "Arbitrage with fractional Gaussian processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 620-628.
    20. Kyong-Hui Kim & Myong-Guk Sin, 2013. "Efficient hedging in general Black-Scholes model," Papers 1308.6387, arXiv.org, revised Mar 2014.
    21. Sun, Lin, 2013. "Pricing currency options in the mixed fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(16), pages 3441-3458.
    22. Omid Jenabi & Nazar Dahmardeh Ghale No, 2018. "Option Pricing in Stochastic Volatility Models Driven by Fractional Jump-Diffusion Processes," International Journal of Finance, Insurance and Risk Management, International Journal of Finance, Insurance and Risk Management, vol. 8(1), pages 1374-1374.
    23. Foad Shokrollahi, 2017. "Pricing compound and extendible options under mixed fractional Brownian motion with jumps," Papers 1708.04829, arXiv.org.

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