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Stochastic Local Volatility


  • Carol Alexander

    () (ICMA Centre, University of Reading)

  • Leonardo Nogueira

    (ICMA Centre, University of Reading and Banco Central do Brasil)


There are two unique volatility surfaces associated with any arbitrage-free set of standard European option prices, the implied volatility surface and the local volatility surface. Several papers have discussed the stochastic differential equations for implied volatilities that are consistent with these option prices but the static and dynamic no-arbitrage conditions are complex, mainly due to the large (or even infinite) dimensions of the state probability space. These no-arbitrage conditions are also instrument-specific and have been specified for some simple classes of options. However, the problem is easier to resolve when we specify stochastic differential equations for local volatilities instead. And the option prices and hedge ratios that are obtained by making local volatility stochastic are identical to those obtained by making instantaneous volatility or implied volatility stochastic. After proving that there is a one-to-one correspondence between the stochastic implied volatility and stochastic local volatility approaches, we derive a simple dynamic no-arbitrage condition for the stochastic local volatility model that is model-specific. The condition is very easy to check in local volatility models having only a few stochastic parameters.

Suggested Citation

  • Carol Alexander & Leonardo Nogueira, 2004. "Stochastic Local Volatility," ICMA Centre Discussion Papers in Finance icma-dp2008-02, Henley Business School, Reading University, revised Mar 2008.
  • Handle: RePEc:rdg:icmadp:icma-dp2008-02

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    References listed on IDEAS

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    5. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters,in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305 World Scientific Publishing Co. Pte. Ltd..
    6. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    7. Jackwerth, Jens Carsten, 1999. "Option Implied Risk-Neutral Distributions and Implied Binomial Trees: A Literature Review," MPRA Paper 11634, University Library of Munich, Germany.
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    9. 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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    11. Panigirtzoglou, Nikolaos & Skiadopoulos, George, 2004. "A new approach to modeling the dynamics of implied distributions: Theory and evidence from the S&P 500 options," Journal of Banking & Finance, Elsevier, vol. 28(7), pages 1499-1520, July.
    12. Martin Schweizer & Johannes Wissel, 2008. "Term Structures Of Implied Volatilities: Absence Of Arbitrage And Existence Results," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 77-114.
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    Cited by:

    1. Antonie Kotzé & Rudolf Oosthuizen & Edson Pindza, 2015. "Implied and Local Volatility Surfaces for South African Index and Foreign Exchange Options," Journal of Risk and Financial Management, MDPI, Open Access Journal, vol. 8(1), pages 1-40, January.
    2. Aur'elien Alfonsi & C'eline Labart & J'er^ome Lelong, 2013. "Stochastic Local Intensity Loss Models with Interacting Particle Systems," Papers 1302.2009,
    3. repec:hal:wpaper:hal-00786239 is not listed on IDEAS

    More about this item


    Local volatility; stochastic volatility; unified theory of volatility; local volatility dynamics;

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions


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