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Financial Modeling in a Fast Mean-Reverting Stochastic Volatility Environment

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Author Info

  • Jean-Pierre Fouque

    ()

  • George Papanicolaou

    ()

  • K. Sircar

    ()

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    Abstract

    We present a derivative pricing and estimation methodology for a class of stochastic volatility models that exploits the observed 'bursty' or persistent nature of stock price volatility. Empirical analysis of high-frequency S&P 500 index data confirms that volatility reverts slowly to its mean in comparison to the tick-by- tick fluctuations of the index value, but it is fast mean- reverting when looked at over the time scale of a derivative contract (many months). This motivates an asymptotic analysis of the partial differential equation satisfied by derivative prices, utilizing the distinction between these time scales. The analysis yields pricing and implied volatility formulas, and the latter provides a simple procedure to 'fit the skew' from European index option prices. The theory identifies the important group parameters that are needed for the derivative pricing and hedging problem for European-style securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the log- moneyness-to-maturity-ratio. The results considerably simplify the estimation procedure. The remaining parameters, including the growth rate of the underlying, the correlation between asset price and volatility shocks, the rate of mean-reversion of the volatility and the market price of volatility risk are not needed for the asymptotic pricing formulas for European derivatives, and we derive the formula for a knock-out barrier option as an example. The extension to American and path-dependent contingent claims is the subject of future work. Copyright Kluwer Academic Publishers 1999

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    File URL: http://hdl.handle.net/10.1023/A:1010010626460
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    Bibliographic Info

    Article provided by Springer in its journal Asia-Pacific Financial Markets.

    Volume (Year): 6 (1999)
    Issue (Month): 1 (January)
    Pages: 37-48

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    Handle: RePEc:kap:apfinm:v:6:y:1999:i:1:p:37-48

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    Web page: http://springerlink.metapress.com/link.asp?id=102851

    Related research

    Keywords: incomplete markets; option pricing; stochastic equations; stochastic volatility;

    References

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    1. Ghysels, E. & Harvey, A. & Renault, E., 1996. "Stochastic Volatility," Cahiers de recherche 9613, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
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    Cited by:
    1. Collan, Mikael, 2004. "Giga-Investments: Modelling the Valuation of Very Large Industrial Real Investments," MPRA Paper 4328, University Library of Munich, Germany.
    2. Kyo Yamamoto & Akihiko Takahashi, 2009. "A Remark on a Singular Perturbation Method for Option Pricing Under a Stochastic Volatility Model," Asia-Pacific Financial Markets, Springer, vol. 16(4), pages 333-345, December.

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