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Stochastic Local Volatility

Author

Listed:
  • Carol Alexander

    (ICMA Centre, University of Reading)

  • Leonardo Nogueira

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

Abstract

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, University of Reading, revised Mar 2008.
    • Handle: RePEc:rdg:icmadp:icma-dp2008-02
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    File URL: http://www.icmacentre.ac.uk/files/dp2008_02.pdf
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    References listed on IDEAS

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    Cited by:

    1. Aur'elien Alfonsi & C'eline Labart & J'er^ome Lelong, 2013. "Stochastic Local Intensity Loss Models with Interacting Particle Systems," Papers 1302.2009, arXiv.org.
    2. Julio Guerrero & Giuseppe Orlando, 2022. "Stochastic Local Volatility models and the Wei-Norman factorization method," Papers 2201.11241, arXiv.org.
    3. Antonie Kotzé & Rudolf Oosthuizen & Edson Pindza, 2015. "Implied and Local Volatility Surfaces for South African Index and Foreign Exchange Options," JRFM, MDPI, vol. 8(1), pages 1-40, January.
    4. repec:hal:wpaper:hal-00786239 is not listed on IDEAS

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    More about this item

    Keywords

    Local volatility; stochastic volatility; unified theory of volatility; local volatility dynamics;
    All these keywords.

    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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