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Taylor, Black and Scholes: series approximations and risk management pitfalls

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  • Arturo Estrella

Abstract

Risk managers make frequent use of finite Taylor approximations to option pricing formulas, particularly of first and second order (delta and gamma). This paper shows that for a plausible range of parameter values, the Taylor series for the Black-Scholes formula diverges. Using a numerical technique developed in the paper, it is also shown that even when the series converges, finite approximations of very large order are generally necessary to achieve acceptable levels of accuracy. Implications for risk management and stress testing are discussed.

Suggested Citation

  • Arturo Estrella, 1995. "Taylor, Black and Scholes: series approximations and risk management pitfalls," Research Paper 9501, Federal Reserve Bank of New York.
    • Handle: RePEc:fip:fednrp:9501
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Matthew Pritsker, 1996. "Evaluating Value-at-Risk Methodologies: Accuracy versus Computational Time," Center for Financial Institutions Working Papers 96-48, Wharton School Center for Financial Institutions, University of Pennsylvania.
    2. Giuseppe Di Graziano & Lorenzo Torricelli, 2012. "Target Volatility Option Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(01), pages 1-17.
    3. P.J.G. Vlaar, 1996. "Methods to determine capital requirements for options," BNL Quarterly Review, Banca Nazionale del Lavoro, vol. 49(198), pages 351-373.
    4. Michele Mininni & Giuseppe Orlando & Giovanni Taglialatela, 2021. "Challenges in approximating the Black and Scholes call formula with hyperbolic tangents," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 44(1), pages 73-100, June.
    5. Patricia Jackson & David Maude & William Perraudin, 1998. "Bank Capital and Value at Risk," Bank of England working papers 79, Bank of England.
    6. Michel Aglietta, 1996. "Financial Market Failures and Systemic Risk," Working Papers 1996-01, CEPII research center.
    7. Jean-Philippe Aguilar, 2018. "On expansions for the Black-Scholes prices and hedge parameters," Papers 1809.06736, arXiv.org, revised Jun 2019.

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    Keywords

    options; Risk;

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