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Spectral Decomposition of Option Prices in Fast Mean-Reverting Stochastic Volatility Models

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  • Jean-Pierre Fouque
  • Sebastian Jaimungal
  • Matthew Lorig

Abstract

Using spectral decomposition techniques and singular perturbation theory, we develop a systematic method to approximate the prices of a variety of options in a fast mean-reverting stochastic volatility setting. Four examples are provided in order to demonstrate the versatility of our method. These include: European options, up-and-out options, double-barrier knock-out options, and options which pay a rebate upon hitting a boundary. For European options, our method is shown to produce option price approximations which are equivalent to those developed in [5]. [5] Jean-Pierre Fouque, George Papanicolaou, and Sircar Ronnie. Derivatives in Financial Markets with Stochas- tic Volatility. Cambridge University Press, 2000.

Suggested Citation

  • Jean-Pierre Fouque & Sebastian Jaimungal & Matthew Lorig, 2010. "Spectral Decomposition of Option Prices in Fast Mean-Reverting Stochastic Volatility Models," Papers 1007.4361, arXiv.org, revised Apr 2012.
    • Handle: RePEc:arx:papers:1007.4361
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    References listed on IDEAS

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    1. Antoon Pelsser, 2000. "Pricing double barrier options using Laplace transforms," Finance and Stochastics, Springer, vol. 4(1), pages 95-104.
    2. Eric Hillebrand, 2005. "Overlaying Time Scales in Financial Volatility Data," Econometrics 0501015, University Library of Munich, Germany.
    3. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv.
    4. 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.
    5. Dmitry Davydov & Vadim Linetsky, 2003. "Pricing Options on Scalar Diffusions: An Eigenfunction Expansion Approach," Operations Research, INFORMS, vol. 51(2), pages 185-209, April.
    6. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    7. Vadim Linetsky, 2004. "The Spectral Decomposition Of The Option Value," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 7(03), pages 337-384.
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