IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v389y2010i3p445-451.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437109008206
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000

    File URL: https://libkey.io/10.1016/j.physa.2009.09.043?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Darrell Duffie & Chi-Fu Huang, 2005. "Implementing Arrow-Debreu Equilibria By Continuous Trading Of Few Long-Lived Securities," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 4, pages 97-127, World Scientific Publishing Co. Pte. Ltd..
    2. Leland, Hayne E, 1985. "Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December.
    3. Ozdemir, Zeynel Abidin, 2009. "Linkages between international stock markets: A multivariate long-memory approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(12), pages 2461-2468.
    4. Alvarez-Ramirez, Jose & Alvarez, Jesus & Rodriguez, Eduardo & Fernandez-Anaya, Guillermo, 2008. "Time-varying Hurst exponent for US stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(24), pages 6159-6169.
    5. Cajueiro, Daniel O. & Tabak, Benjamin M., 2007. "Long-range dependence and multifractality in the term structure of LIBOR interest rates," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 373(C), pages 603-614.
    6. Mariani, M.C. & Florescu, I. & Beccar Varela, M.P. & Ncheuguim, E., 2009. "Long correlations and Levy models applied to the study of memory effects in high frequency (tick) data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(8), pages 1659-1664.
    7. David M. Kreps, 1982. "Multiperiod Securities and the Efficient Allocation of Risk: A Comment on the Black-Scholes Option Pricing Model," NBER Chapters, in: The Economics of Information and Uncertainty, pages 203-232, National Bureau of Economic Research, Inc.
    8. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    9. Walter Willinger & Murad S. Taqqu, 1991. "Toward A Convergence Theory For Continuous Stochastic Securities Market Models1," Mathematical Finance, Wiley Blackwell, vol. 1(1), pages 55-59, January.
    10. 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.
    11. Cajueiro, Daniel O. & Tabak, Benjamin M., 2007. "Long-range dependence and market structure," Chaos, Solitons & Fractals, Elsevier, vol. 31(4), pages 995-1000.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. 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.
    4. 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.
    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. 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.
    7. Foad Shokrollahi, 2017. "Fractional delta hedging strategy for pricing currency options with transaction costs," Papers 1702.00037, arXiv.org.
    8. 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.
    9. Foad Shokrollahi & Tommi Sottinen, 2017. "Hedging in fractional Black-Scholes model with transaction costs," Papers 1706.01534, arXiv.org, revised Jul 2017.
    10. 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.
    11. 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.
    12. Tommi Sottinen & Lauri Viitasaari, 2017. "Conditional-Mean Hedging Under Transaction Costs in Gaussian Models," Papers 1708.03242, arXiv.org.
    13. Kyong-Hui Kim & Myong-Guk Sin, 2013. "Efficient hedging in general Black-Scholes model," Papers 1308.6387, arXiv.org, revised Mar 2014.
    14. 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).
    15. 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.
    16. Foad Shokrollahi, 2017. "The valuation of European option with transaction costs by mixed fractional Merton model," Papers 1702.00152, arXiv.org.
    17. Wang, Wensheng, 2019. "Asymptotics for discrete time hedging errors under fractional Black–Scholes models," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 160-170.
    18. Foad Shokrollahi, 2016. "Subdiffusive fractional Brownian motion regime for pricing currency options under transaction costs," Papers 1612.06665, arXiv.org, revised Aug 2017.
    19. 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.
    20. Zhang, Xili & Xiao, Weilin, 2017. "Arbitrage with fractional Gaussian processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 620-628.
    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Wang, Xiao-Tian & Wu, Min & Zhou, Ze-Min & Jing, Wei-Shu, 2012. "Pricing European option with transaction costs under the fractional long memory stochastic volatility model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(4), pages 1469-1480.
    2. Wang, Xiao-Tian, 2011. "Scaling and long-range dependence in option pricing V: Multiscaling hedging and implied volatility smiles under the fractional Black–Scholes model with transaction costs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(9), pages 1623-1634.
    3. Wang, Xiao-Tian, 2010. "Scaling and long-range dependence in option pricing I: Pricing European option with transaction costs under the fractional Black–Scholes model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(3), pages 438-444.
    4. Wang, Xiao-Tian, 2010. "Scaling and long range dependence in option pricing, IV: Pricing European options with transaction costs under the multifractional Black–Scholes model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(4), pages 789-796.
    5. Foad Shokrollahi, 2017. "The valuation of European option with transaction costs by mixed fractional Merton model," Papers 1702.00152, arXiv.org.
    6. Suresh M. Sundaresan, 2000. "Continuous‐Time Methods in Finance: A Review and an Assessment," Journal of Finance, American Finance Association, vol. 55(4), pages 1569-1622, August.
    7. Duffie, Darrell, 2003. "Intertemporal asset pricing theory," Handbook of the Economics of Finance, in: G.M. Constantinides & M. Harris & R. M. Stulz (ed.), Handbook of the Economics of Finance, edition 1, volume 1, chapter 11, pages 639-742, Elsevier.
    8. repec:dau:papers:123456789/5374 is not listed on IDEAS
    9. Foad Shokrollahi, 2017. "Fractional delta hedging strategy for pricing currency options with transaction costs," Papers 1702.00037, arXiv.org.
    10. Madan, Dilip B & Milne, Frank & Shefrin, Hersh, 1989. "The Multinomial Option Pricing Model and Its Brownian and Poisson Limits," Review of Financial Studies, Society for Financial Studies, vol. 2(2), pages 251-265.
    11. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742, Decembrie.
    12. Lim, Terence & Lo, Andrew W. & Merton, Robert C. & Scholes, Myron S., 2006. "The Derivatives Sourcebook," Foundations and Trends(R) in Finance, now publishers, vol. 1(5–6), pages 365-572, April.
    13. Gerlich, Nikolas & Rostek, Stefan, 2015. "Estimating serial correlation and self-similarity in financial time series—A diversification approach with applications to high frequency data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 434(C), pages 84-98.
    14. Elyès Jouini & Clotilde Napp, 2002. "Arbitrage Pricing And Equilibrium Pricing: Compatibility Conditions," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar(Volume III), chapter 6, pages 131-158, World Scientific Publishing Co. Pte. Ltd..
    15. Jos'e Manuel Corcuera, 2021. "The Golden Age of the Mathematical Finance," Papers 2102.06693, arXiv.org, revised Mar 2021.
    16. Wang, Xiao-Tian & Yan, Hai-Gang & Tang, Ming-Ming & Zhu, En-Hui, 2010. "Scaling and long-range dependence in option pricing III: A fractional version of the Merton model with transaction costs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(3), pages 452-458.
    17. Olivier Guéant, 2016. "The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making," Post-Print hal-01393136, HAL.
    18. Carolyn W. Chang, 1995. "A No-Arbitrage Martingale Analysis For Jump-Diffusion Valuation," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 18(3), pages 351-381, September.
    19. Boyle, Phelim & Tian, Weidong, 2008. "The design of equity-indexed annuities," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 303-315, December.
    20. 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.
    21. Jouini, Elyes, 2001. "Arbitrage and control problems in finance: A presentation," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 167-183, April.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:389:y:2010:i:3:p:445-451. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.