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Using pseudo-parabolic and fractional equations for option pricing in jump diffusion models

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  • Andrey Itkin
  • Peter Carr

Abstract

In mathematical finance a popular approach for pricing options under some Levy model is to consider underlying that follows a Poisson jump diffusion process. As it is well known this results in a partial integro-differential equation (PIDE) that usually does not allow an analytical solution while numerical solution brings some problems. In this paper we elaborate a new approach on how to transform the PIDE to some class of so-called pseudo-parabolic equations which are known in mathematics but are relatively new for mathematical finance. As an example we discuss several jump-diffusion models which Levy measure allows such a transformation.

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File URL: http://arxiv.org/pdf/1002.1995
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Paper provided by arXiv.org in its series Papers with number 1002.1995.

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Date of creation: Feb 2010
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Handle: RePEc:arx:papers:1002.1995

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  1. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, University of Chicago Press, vol. 75(2), pages 305-332, April.
  2. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, Elsevier, vol. 71(1), pages 113-141, January.
  3. Peter Carr & Hélyette Geman & Dilip Madan & Marc Yor, 2005. "Pricing options on realized variance," Finance and Stochastics, Springer, Springer, vol. 9(4), pages 453-475, October.
  4. N. Hilber & N. Reich & C. Schwab & C. Winter, 2009. "Numerical methods for Lévy processes," Finance and Stochastics, Springer, Springer, vol. 13(4), pages 471-500, September.
  5. Cartea, Álvaro & del-Castillo-Negrete, Diego, 2007. "Fractional diffusion models of option prices in markets with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, Elsevier, vol. 374(2), pages 749-763.
  6. Amin, Kaushik I, 1993. " Jump Diffusion Option Valuation in Discrete Time," Journal of Finance, American Finance Association, American Finance Association, vol. 48(5), pages 1833-63, December.
  7. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, University of Chicago Press, vol. 63(4), pages 511-24, October.
  8. Peter Carr & Anita Mayo, 2007. "On the Numerical Evaluation of Option Prices in Jump Diffusion Processes," The European Journal of Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 13(4), pages 353-372.
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Cited by:
  1. Andrey Itkin & Alexander Lipton, 2014. "Efficient solution of structural default models with correlated jumps. A fractional PDE approach," Papers 1408.6513, arXiv.org.
  2. Andrey Itkin, 2014. "High-Order Splitting Methods for Forward PDEs and PIDEs," Papers 1403.1804, arXiv.org.
  3. Andrey Itkin, 2014. "Splitting and Matrix Exponential approach for jump-diffusion models with Inverse Normal Gaussian, Hyperbolic and Meixner jumps," Papers 1405.6111, arXiv.org, revised May 2014.
  4. I. Halperin & A. Itkin, 2012. "Pricing Illiquid Options with $N+1$ Liquid Proxies Using Mixed Dynamic-Static Hedging," Papers 1209.3503, arXiv.org.
  5. Leif Andersen & Alexander Lipton, 2012. "Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results," Papers 1206.6787, arXiv.org.

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