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On the Numerical Evaluation of Option Prices in Jump Diffusion Processes

Author

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  • Peter Carr
  • Anita Mayo

Abstract

The fair price of a financial option on an asset that follows a Poisson jump diffusion process satisfies a partial integro-differential equation. When numerical methods are used to solve such equations the integrals are usually evaluated using either quadrature methods or fast Fourier methods. Quadrature methods are expensive since the integrals must be evaluated at every point of the mesh. Though less so, Fourier methods are also computationally intensive since in order to avoid wrap around effects they require enlargement of the computational domain. They are also slow to converge when the parameters of the jump process are not smooth, and for efficiency require uniform meshes. We present a different and more efficient class of methods which are based on the fact that the integrals often satisfy differential equations. Depending on the process the asset follows, the equations are either ordinary differential equations or parabolic partial differential equations. Both types of equations can be accurately solved very rapidly. We discuss the methods and present results of numerical experiments.

Suggested Citation

  • Peter Carr & Anita Mayo, 2007. "On the Numerical Evaluation of Option Prices in Jump Diffusion Processes," The European Journal of Finance, Taylor & Francis Journals, vol. 13(4), pages 353-372.
    • Handle: RePEc:taf:eurjfi:v:13:y:2007:i:4:p:353-372
    DOI: 10.1080/13518470701201512
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    Citations

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    Cited by:

    1. Jingtang Ma & Dongya Deng & Harry Zheng, 2016. "Convergence analysis and optimal strike choice for static hedges of general path-independent pay-offs," Quantitative Finance, Taylor & Francis Journals, vol. 16(4), pages 593-603, April.
    2. Olijslagers, Stan & van der Ploeg, Frederick & van Wijnbergen, Sweder, 2023. "On current and future carbon prices in a risky world," Journal of Economic Dynamics and Control, Elsevier, vol. 146(C).
    3. Abootaleb Shirvani & Frank J. Fabozzi & Stoyan V. Stoyanov, 2020. "Option Pricing in an Investment Risk-Return Setting," Papers 2001.00737, arXiv.org.
    4. Andrey Itkin & Peter Carr, 2012. "Using Pseudo-Parabolic and Fractional Equations for Option Pricing in Jump Diffusion Models," Computational Economics, Springer;Society for Computational Economics, vol. 40(1), pages 63-104, June.
    5. Kyriakos Georgiou & Athanasios N. Yannacopoulos, 2023. "Probability of Default modelling with L\'evy-driven Ornstein-Uhlenbeck processes and applications in credit risk under the IFRS 9," Papers 2309.12384, arXiv.org.
    6. Christara, Christina C. & Leung, Nat Chun-Ho, 2016. "Option pricing in jump diffusion models with quadratic spline collocation," Applied Mathematics and Computation, Elsevier, vol. 279(C), pages 28-42.
    7. Hideharu Funahashi & Masaaki Kijima, 2016. "Analytical pricing of single barrier options under local volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 867-886, June.
    8. Karel in 't Hout & Jari Toivanen, 2015. "Application of Operator Splitting Methods in Finance," Papers 1504.01022, arXiv.org.
    9. Ron Chan & Simon Hubbert, 2014. "Options pricing under the one-dimensional jump-diffusion model using the radial basis function interpolation scheme," Review of Derivatives Research, Springer, vol. 17(2), pages 161-189, July.

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