IDEAS home Printed from https://ideas.repec.org/a/wsi/ijfexx/v03y2016i01ns242478631650002x.html
   My bibliography  Save this article

A sharp approximation for ATM-forward option prices and implied volatilites

Author

Listed:
  • Dan Stefanica

    (Baruch College, City University of New York, USA)

  • Radoš Radoičić

    (Baruch College, City University of New York, USA)

Abstract

In this paper, we provide an approximation formula for at-the-money forward options based on a Pólya approximation of the cumulative density function of the standard normal distribution, and prove that the relative error of this approximation is uniformly bounded for options with arbitrarily large (or small) maturities and implied volatilities. This approximation is viable in practice: for options with implied volatility less than 95% and maturity less than three years, which includes the large majority of traded options, the values given by the approximation formula fall within the tightest typical implied vol bid–ask spreads. The relative errors of the corresponding approximate option values are also uniformly bounded for all maturities and implied volatilities. The error bounds established here are the first results in the literature holding for all integrated volatilities, and are vastly superior to those of two other approximation formulas analyzed in this paper, including the Brenner–Subrahmanyam formula.

Suggested Citation

  • 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.
    • Handle: RePEc:wsi:ijfexx:v:03:y:2016:i:01:n:s242478631650002x
    DOI: 10.1142/S242478631650002X
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S242478631650002X
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S242478631650002X?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. 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..
    2. Patrick Hagan & Diana Woodward, 1999. "Equivalent Black volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(3), pages 147-157.
    3. 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.
    4. 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.
    5. Latane, Henry A & Rendleman, Richard J, Jr, 1976. "Standard Deviations of Stock Price Ratios Implied in Option Prices," Journal of Finance, American Finance Association, vol. 31(2), pages 369-381, May.
    6. Kun Gao & Roger Lee, 2014. "Asymptotics of implied volatility to arbitrary order," Finance and Stochastics, Springer, vol. 18(2), pages 349-392, April.
    7. James S. Ang & Gwoduan David Jou & Tsong-Yue Lai, 2009. "Alternative Formulas to Compute Implied Standard Deviation," Review of Pacific Basin Financial Markets and Policies (RPBFMP), World Scientific Publishing Co. Pte. Ltd., vol. 12(02), pages 159-176.
    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. paolo pianca, 2005. "Simple Formulas to Option Pricing and Hedging in the Black- Scholes Model," Finance 0511005, University Library of Munich, Germany.
    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. 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.
    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.

    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. 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.
    2. 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.
    3. 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.
    4. Michael R. Tehranchi, 2015. "Uniform bounds for Black--Scholes implied volatility," Papers 1512.06812, arXiv.org, revised Aug 2016.
    5. Jacquier, Antoine & Roome, Patrick, 2016. "Large-maturity regimes of the Heston forward smile," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1087-1123.
    6. Yibing Chen & Cheng-Few Lee & John Lee & Jow-Ran Chang, 2018. "Alternative Methods to Estimate Implied Variance: Review and Comparison," Review of Pacific Basin Financial Markets and Policies (RPBFMP), World Scientific Publishing Co. Pte. Ltd., vol. 21(04), pages 1-28, December.
    7. Steven Li, 2003. "The estimation of implied volatility from the Black-Scholes model: some new formulas and their applications," School of Economics and Finance Discussion Papers and Working Papers Series 141, School of Economics and Finance, Queensland University of Technology.
    8. Don M. Chance & Thomas A. Hanson & Weiping Li & Jayaram Muthuswamy, 2017. "A bias in the volatility smile," Review of Derivatives Research, Springer, vol. 20(1), pages 47-90, April.
    9. Antoine Jacquier & Patrick Roome, 2015. "Black-Scholes in a CEV random environment," Papers 1503.08082, arXiv.org, revised Nov 2017.
    10. Chenxu Li, 2014. "Closed-Form Expansion, Conditional Expectation, and Option Valuation," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 487-516, May.
    11. Richard Jordan & Charles Tier, 2016. "Asymptotic Approximations For Pricing Derivatives Under Mean-Reverting Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(05), pages 1-31, August.
    12. Ghysels, E. & Harvey, A. & Renault, E., 1995. "Stochastic Volatility," Papers 95.400, Toulouse - GREMAQ.
    13. Guan Wang & Pierre Yourougou & Yue Wang, 2012. "Which implied volatility provides the best measure of future volatility?," Journal of Economics and Finance, Springer;Academy of Economics and Finance, vol. 36(1), pages 93-105, January.
    14. Giulia Di Nunno & Kk{e}stutis Kubilius & Yuliya Mishura & Anton Yurchenko-Tytarenko, 2023. "From constant to rough: A survey of continuous volatility modeling," Papers 2309.01033, arXiv.org, revised Sep 2023.
    15. Smith, Paul & Gronewoller, Paul & Rose, Lawrence C., 1998. "Pricing efficiency on the New Zealand Futures and Options Exchange," Journal of Multinational Financial Management, Elsevier, vol. 8(1), pages 49-62, January.
    16. Birkelund, Ole Henrik & Haugom, Erik & Molnár, Peter & Opdal, Martin & Westgaard, Sjur, 2015. "A comparison of implied and realized volatility in the Nordic power forward market," Energy Economics, Elsevier, vol. 48(C), pages 288-294.
    17. Dicle, Mehmet F. & Levendis, John, 2020. "Historic risk and implied volatility," Global Finance Journal, Elsevier, vol. 45(C).
    18. Butler, J. S. & Schachter, Barry, 1996. "The statistical properties of parameters inferred from the black-scholes formula," International Review of Financial Analysis, Elsevier, vol. 5(3), pages 223-235.
    19. William Pedersen, 1998. "Capturing all the information in foreign currency option prices: solving for one versus two implied variables," Applied Economics, Taylor & Francis Journals, vol. 30(12), pages 1679-1683.
    20. Lyons, Richard K., 1988. "Tests of the foreign exchange risk premium using the expected second moments implied by option pricing," Journal of International Money and Finance, Elsevier, vol. 7(1), pages 91-108, March.

    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:wsi:ijfexx:v:03:y:2016:i:01:n:s242478631650002x. 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: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscientific.com/worldscinet/ijfe .

    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.