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The Forward Pde For European Options On Stocks With Fixed Fractional Jumps

Author

Listed:
  • PETER CARR

    (Courant Institute, NYU, 251 Mercer Street, New York, NY 10012, USA;
    Head of Quantitative Financial Research, Bloomberg LP, 731 Lexington Avenue, New York, NY 10021, USA)

  • ALIREZA JAVAHERI

    (Citigroup, 388 Greewich Street, New York, NY 10013, USA;
    Ecole des Mines de Paris, 60 Bd. Saint-Michel, Paris, France)

Abstract

We derive a partial integro differential equation (PIDE) which relates the price of a calendar spread to the prices of butterfly spreads and the functions describing the evolution of the process. These evolution functions are the forward local variance rate and a new concept called the forward local default arrival rate. We then specialize to the case where the only jump which can occur reduces the underlying stock price by a fixed fraction of its pre-jump value. This is a standard assumption when valuing an option written on a stock which can default. We discuss novel strategies for calibrating to a term and strike structure of European options prices. In particular using a few calendar dates, we derive closed form expressions for both the local variance and the local default arrival rate.

Suggested Citation

  • Peter Carr & Alireza Javaheri, 2005. "The Forward Pde For European Options On Stocks With Fixed Fractional Jumps," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(02), pages 239-253.
    • Handle: RePEc:wsi:ijtafx:v:08:y:2005:i:02:n:s0219024905002974
    DOI: 10.1142/S0219024905002974
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    Cited by:

    1. S. Kindermann & P. Mayer, 2011. "On the calibration of local jump-diffusion asset price models," Finance and Stochastics, Springer, vol. 15(4), pages 685-724, December.
    2. Peter Carr & Vadim Linetsky, 2006. "A jump to default extended CEV model: an application of Bessel processes," Finance and Stochastics, Springer, vol. 10(3), pages 303-330, September.

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