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Stochastic Local Volatility

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Author Info

  • 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.

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Bibliographic Info

Paper provided by Henley Business School, Reading University in its series ICMA Centre Discussion Papers in Finance with number icma-dp2008-02.

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Length: 23 pages
Date of creation: Sep 2004
Date of revision: Mar 2008
Handle: RePEc:rdg:icmadp:icma-dp2008-02

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Related research

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

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References

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  1. 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.
  2. Ledoit, Olivier & Santa-Clara, Pedro & Yan, Shu, 2002. "Relative Pricing of Options with Stochastic Volatility," University of California at Los Angeles, Anderson Graduate School of Management qt7jp8f42t, Anderson Graduate School of Management, UCLA.
  3. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
  4. 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.
  5. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
  6. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
  7. Heath, David & Jarrow, Robert & Morton, Andrew, 1992. "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation," Econometrica, Econometric Society, vol. 60(1), pages 77-105, January.
  8. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-52.
  9. Carr, Peter & Madan, Dilip B., 2005. "A note on sufficient conditions for no arbitrage," Finance Research Letters, Elsevier, vol. 2(3), pages 125-130, September.
  10. Marco Avellaneda & Craig Friedman & Richard Holmes & Dominick Samperi, 1997. "Calibrating volatility surfaces via relative-entropy minimization," Applied Mathematical Finance, Taylor & Francis Journals, vol. 4(1), pages 37-64.
  11. 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 621-51, October.
  12. Jackwerth, Jens Carsten, 1999. "Option Implied Risk-Neutral Distributions and Implied Binomial Trees: A Literature Review," MPRA Paper 11634, University Library of Munich, Germany.
  13. 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.
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Cited by:
  1. Aurélien Alfonsi & Céline Labart & Jérôme Lelong, 2013. "Stochastic Local Intensity Loss Models with Interacting Particle Systems," Working Papers hal-00786239, HAL.
  2. 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.

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