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Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing

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  • Leif Andersen
  • Jesper Andreasen
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    Abstract

    This paper discusses extensions of the implied diffusion approach of Dupire (1994) to asset processes with Poisson jumps. We show that this extension yields important model improvements, particularly in the dynamics of the implied volatility surface. The paper derives a forward PIDE (PartialIntegro-Differential Equation) and demonstrates how this equationcan be used to fit the model to European option prices. For numerical pricing of general contingent claims, we develop an ADI finite difference method that is shown to be unconditionally stable and, if combined with Fast Fourier Transform methods, computationally efficient. The paper contains several detailed examples fromthe S&P500 market. Copyright Kluwer Academic Publishers 2000

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    File URL: http://hdl.handle.net/10.1023/A:1011354913068
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    Bibliographic Info

    Article provided by Springer in its journal Review of Derivatives Research.

    Volume (Year): 4 (2000)
    Issue (Month): 3 (October)
    Pages: 231-262

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    Handle: RePEc:kap:revdev:v:4:y:2000:i:3:p:231-262

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    Web page: http://www.springerlink.com/link.asp?id=102989

    Related research

    Keywords: jump-diffusion process; local time; forward equation; volatility smile; ADI finite difference method; fast Fourier transform;

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    Cited by:
    1. Stefano Galluccio & Yann Le Cam, 2005. "Implied Calibration of Stochastic Volatility Jump Diffusion Models," Finance 0510028, EconWPA.
    2. Farshid Jamshidian, 2004. "Numeraire-invariant option pricing and american, bermudan, trigger stream rollover (v1.6)," Finance 0407015, EconWPA.
    3. H. A. Windcliff & P. A. Forsyth & K. R. Vetzal, 2006. "Numerical Methods and Volatility Models for Valuing Cliquet Options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 13(4), pages 353-386.
    4. Carl Chiarella & Andrew Ziogas, 2006. "American Call Options on Jump-Diffusion Processes: A Fourier Transform Approach," Research Paper Series 174, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. Andrey Itkin, 2014. "High-Order Splitting Methods for Forward PDEs and PIDEs," Papers 1403.1804, arXiv.org.

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