IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1810.04623.html
   My bibliography  Save this paper

Challenges in approximating the Black and Scholes call formula with hyperbolic tangents

Author

Listed:
  • Michele Mininni
  • Giuseppe Orlando
  • Giovanni Taglialatela

Abstract

In this paper we introduce the concept of standardized call function and we obtain a new approximating formula for the Black and Scholes call function through the hyperbolic tangent. This formula is useful for pricing and risk management as well as for extracting the implied volatility from quoted options. The latter is of particular importance since it indicates the risk of the underlying and it is the main component of the option's price. Further we estimate numerically the approximating error of the suggested solution and, by comparing our results in computing the implied volatility with the most common methods available in literature we discuss the challenges of this approach.

Suggested Citation

  • Michele Mininni & Giuseppe Orlando & Giovanni Taglialatela, 2018. "Challenges in approximating the Black and Scholes call formula with hyperbolic tangents," Papers 1810.04623, arXiv.org.
    • Handle: RePEc:arx:papers:1810.04623
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1810.04623
    File Function: Latest version
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. Peter Christoffersen & Kris Jacobs, 2004. "Which GARCH Model for Option Valuation?," Management Science, INFORMS, vol. 50(9), pages 1204-1221, September.
    2. Mo, Henry & Wu, Liuren, 2007. "International capital asset pricing: Evidence from options," Journal of Empirical Finance, Elsevier, vol. 14(4), pages 465-498, September.
    3. Fan, Jianqing & Mancini, Loriano, 2009. "Option Pricing With Model-Guided Nonparametric Methods," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1351-1372.
    4. Arturo Estrella, 1995. "Taylor, Black and Scholes: series approximations and risk management pitfalls," Research Paper 9501, Federal Reserve Bank of New York.
    5. Engle, Robert F. & Mustafa, Chowdhury, 1992. "Implied ARCH models from options prices," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 289-311.
    6. 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..
    7. Paul Glasserman & Qi Wu, 2011. "Forward And Future Implied Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(03), pages 407-432.
    8. 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.
    9. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    10. Jin‐Chuan Duan, 1995. "The Garch Option Pricing Model," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 13-32, January.
    11. Omar M. Eidous & Rima Abu-Shareefa, 2020. "New approximations for standard normal distribution function," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 49(6), pages 1357-1374, March.
    12. Ivan Matić & Radoš Radoičić & Dan Stefanica, 2017. "Pólya-based approximation for the ATM-forward implied volatility," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(02n03), pages 1-15, June.
    13. Chance, Don M, 1996. "A Generalized Simple Formula to Compute the Implied Volatility," The Financial Review, Eastern Finance Association, vol. 31(4), pages 859-867, November.
    14. Fuchang Gao & Lixing Han, 2012. "Implementing the Nelder-Mead simplex algorithm with adaptive parameters," Computational Optimization and Applications, Springer, vol. 51(1), pages 259-277, January.
    15. Shuaiqiang Liu & Cornelis W. Oosterlee & Sander M. Bohte, 2019. "Pricing Options and Computing Implied Volatilities using Neural Networks," Risks, MDPI, vol. 7(1), pages 1-22, February.
    16. Li, Minqiang, 2008. "Approximate inversion of the Black-Scholes formula using rational functions," European Journal of Operational Research, Elsevier, vol. 185(2), pages 743-759, March.
    17. Corrado, Charles J. & Miller, Thomas Jr., 1996. "A note on a simple, accurate formula to compute implied standard deviations," Journal of Banking & Finance, Elsevier, vol. 20(3), pages 595-603, April.
    18. E. Page, 1977. "Approximations to the Cumulative Normal Function and its Inverse for Use on a Pocket Calculator," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 26(1), pages 75-76, March.
    19. paolo pianca, 2005. "Simple Formulas to Option Pricing and Hedging in the Black- Scholes Model," Finance 0511005, University Library of Munich, Germany.
    20. Millo, Yuval & MacKenzie, Donald, 2009. "The usefulness of inaccurate models: Towards an understanding of the emergence of financial risk management," Accounting, Organizations and Society, Elsevier, vol. 34(5), pages 638-653, July.
    21. L. Ingber, 1982. "Statistical mechanics of neocortical interactions. I. Basic formulation," Lester Ingber Papers 82ni, Lester Ingber.
    22. Peter Ritchken & Rob Trevor, 1999. "Pricing Options under Generalized GARCH and Stochastic Volatility Processes," Journal of Finance, American Finance Association, vol. 54(1), pages 377-402, February.
    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. Giuseppe Orlando & Michele Bufalo, 2021. "Interest rates forecasting: Between Hull and White and the CIR#—How to make a single‐factor model work," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 40(8), pages 1566-1580, December.
    2. Giuseppe Orlando & Michele Bufalo, 2021. "Empirical Evidences on the Interconnectedness between Sampling and Asset Returns’ Distributions," Risks, MDPI, vol. 9(5), pages 1-35, May.
    3. Geon Lee & Tae-Kyoung Kim & Hyun-Gyoon Kim & Jeonggyu Huh, 2022. "Newton Raphson Emulation Network for Highly Efficient Computation of Numerous Implied Volatilities," Papers 2210.15969, arXiv.org.
    4. Daniel Wei-Chung Miao & Xenos Chang-Shuo Lin & Chang-Yao Lin, 2021. "Using Householder’s method to improve the accuracy of the closed-form formulas for implied volatility," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(3), pages 493-528, December.

    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. Zhu, Ke & Ling, Shiqing, 2015. "Model-based pricing for financial derivatives," Journal of Econometrics, Elsevier, vol. 187(2), pages 447-457.
    2. Dan Stefanica & Radoš Radoičić, 2017. "An Explicit Implied Volatility Formula," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(07), pages 1-32, November.
    3. Lars Stentoft, 2008. "American Option Pricing Using GARCH Models and the Normal Inverse Gaussian Distribution," Journal of Financial Econometrics, Oxford University Press, vol. 6(4), pages 540-582, Fall.
    4. Christoffersen, Peter & Jacobs, Kris & Ornthanalai, Chayawat & Wang, Yintian, 2008. "Option valuation with long-run and short-run volatility components," Journal of Financial Economics, Elsevier, vol. 90(3), pages 272-297, December.
    5. Stentoft, Lars, 2011. "American option pricing with discrete and continuous time models: An empirical comparison," Journal of Empirical Finance, Elsevier, vol. 18(5), pages 880-902.
    6. Minqiang Li & Kyuseok Lee, 2011. "An adaptive successive over-relaxation method for computing the Black-Scholes implied volatility," Quantitative Finance, Taylor & Francis Journals, vol. 11(8), pages 1245-1269.
    7. Henri Bertholon & Alain Monfort & Fulvio Pegoraro, 2006. "Pricing and Inference with Mixtures of Conditionally Normal Processes," Working Papers 2006-28, Center for Research in Economics and Statistics.
    8. Ghysels, E. & Harvey, A. & Renault, E., 1995. "Stochastic Volatility," Papers 95.400, Toulouse - GREMAQ.
    9. F. Fornari & A. Mele, 1998. "ARCH Models and Option Pricing : The Continuous Time Connection," THEMA Working Papers 98-30, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    10. Matthias R. Fengler & Helmut Herwartz & Christian Werner, 2012. "A Dynamic Copula Approach to Recovering the Index Implied Volatility Skew," Journal of Financial Econometrics, Oxford University Press, vol. 10(3), pages 457-493, June.
    11. Christoffersen, Peter & Heston, Steve & Jacobs, Kris, 2006. "Option valuation with conditional skewness," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 253-284.
    12. Dan Stefanica & Radoš Radoičić, 2016. "A sharp approximation for ATM-forward option prices and implied volatilites," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(01), pages 1-24, March.
    13. Rombouts, Jeroen V.K. & Stentoft, Lars, 2015. "Option pricing with asymmetric heteroskedastic normal mixture models," International Journal of Forecasting, Elsevier, vol. 31(3), pages 635-650.
    14. Katarzyna Toporek, 2012. "Simple is better. Empirical comparison of American option valuation methods," Ekonomia journal, Faculty of Economic Sciences, University of Warsaw, vol. 29.
    15. Badescu, Alexandru M. & Kulperger, Reg J., 2008. "GARCH option pricing: A semiparametric approach," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 69-84, August.
    16. Peter Christoffersen & Kris Jacobs, 2002. "Which Volatility Model for Option Valuation?," CIRANO Working Papers 2002s-33, CIRANO.
    17. Javier Frutos & Víctor Gatón, 2017. "Chebyshev reduced basis function applied to option valuation," Computational Management Science, Springer, vol. 14(4), pages 465-491, October.
    18. Li, Minqiang, 2008. "Approximate inversion of the Black-Scholes formula using rational functions," European Journal of Operational Research, Elsevier, vol. 185(2), pages 743-759, March.
    19. Choi, Youngsoo, 2005. "An analytical approximation to the option formula for the GARCH model," International Review of Financial Analysis, Elsevier, vol. 14(2), pages 149-164.
    20. Badescu Alex & Kulperger Reg & Lazar Emese, 2008. "Option Valuation with Normal Mixture GARCH Models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 12(2), pages 1-42, May.

    More about this item

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C88 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Other Computer Software

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    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:arx:papers:1810.04623. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.