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On The Relationship Between The Call Price Surface And The Implied Volatility Surface Close To Expiry

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  • MICHAEL ROPER

    ()
    (School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia)

  • MAREK RUTKOWSKI

    ()
    (School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia)

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    Abstract

    We examine the asymptotic behaviour of the call price surface and the associated Black-Scholes implied volatility surface in the small time to expiry limit under the condition of no arbitrage. In the final section, we examine a related question of existence of a market model with non-convergent implied volatility. We show that there exist arbitrage free markets in which implied volatility may fail to converge to any value, finite or infinite.

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    Bibliographic Info

    Article provided by World Scientific Publishing Co. Pte. Ltd. in its journal International Journal of Theoretical and Applied Finance.

    Volume (Year): 12 (2009)
    Issue (Month): 04 ()
    Pages: 427-441

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    Handle: RePEc:wsi:ijtafx:v:12:y:2009:i:04:p:427-441

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    Related research

    Keywords: Option pricing; implied volatility; logarithmic limit; Black-Scholes formula; asymptotic and approximate formulae;

    References

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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 637-54, May-June.
    2. Rama Cont & Jose da Fonseca, 2002. "Dynamics of implied volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 45-60.
    3. 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.
    4. A. Brace & G. Fabbri & B. Goldys, 2007. "An Hilbert space approach for a class of arbitrage free implied volatilities models," Science & Finance (CFM) working paper archive 0712.1343, Science & Finance, Capital Fund Management, revised Dec 2007.
    5. Chambers, Donald R & Nawalkha, Sanjay K, 2001. "An Improved Approach to Computing Implied Volatility," The Financial Review, Eastern Finance Association, vol. 36(3), pages 89-99, August.
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    Cited by:
    1. Stefan Gerhold & Ismail Cetin G\"ul\"um, 2013. "The Small Maturity Implied Volatility Slope for L\'evy Models," Science & Finance (CFM) working paper archive 1310.3061, Science & Finance, Capital Fund Management.
    2. Aleksandar Mijatovi\'c & Peter Tankov, 2012. "A new look at short-term implied volatility in asset price models with jumps," Science & Finance (CFM) working paper archive 1207.0843, Science & Finance, Capital Fund Management, revised Jul 2012.
    3. Kun Gao & Roger Lee, 2014. "Asymptotics of implied volatility to arbitrary order," Finance and Stochastics, Springer, vol. 18(2), pages 349-392, April.
    4. Stefan Gerhold & Max Kleinert & Piet Porkert & Mykhaylo Shkolnikov, 2012. "Small time central limit theorems for semimartingales with applications," Science & Finance (CFM) working paper archive 1208.4282, Science & Finance, Capital Fund Management.
    5. Zhi Guo & Eckhard Platen, 2011. "The Small and Large Time Implied Volatilities in the Minimal Market Model," Science & Finance (CFM) working paper archive 1109.6154, Science & Finance, Capital Fund Management, revised Oct 2011.
    6. Stefan Gerhold, 2012. "Can there be an explicit formula for implied volatility?," Science & Finance (CFM) working paper archive 1211.4978, Science & Finance, Capital Fund Management.

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