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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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  • 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.
    • Handle: RePEc:kap:compec:v:40:y:2012:i:1:p:63-104
    DOI: 10.1007/s10614-011-9269-8
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    1. Leif Andersen & Jesper Andreasen, 2000. "Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing," Review of Derivatives Research, Springer, vol. 4(3), pages 231-262, October.
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    7. 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, vol. 374(2), pages 749-763.
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    Cited by:

    1. Darae Jeong & Minhyun Yoo & Changwoo Yoo & Junseok Kim, 2019. "A Hybrid Monte Carlo and Finite Difference Method for Option Pricing," Computational Economics, Springer;Society for Computational Economics, vol. 53(1), pages 111-124, January.
    2. Zhang, Meihui & Jia, Jinhong & Zheng, Xiangcheng, 2023. "Numerical approximation and fast implementation to a generalized distributed-order time-fractional option pricing model," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    3. Andrey Itkin, 2023. "Semi-analytic pricing of American options in time-dependent jump-diffusion models with exponential jumps," Papers 2308.08760, arXiv.org, revised Feb 2024.
    4. Yuanda Chen & Zailei Cheng & Haixu Wang, 2023. "Option Pricing for the Variance Gamma Model: A New Perspective," Papers 2306.10659, arXiv.org.
    5. Igor Halperin & Andrey Itkin, 2013. "Pricing Illiquid Options With N + 1 Liquid Proxies Using Mixed Dynamic-Static Hedging," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(07), pages 1-17.
    6. Leif Andersen & Alexander Lipton, 2012. "Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results," Papers 1206.6787, arXiv.org.
    7. Andrey Itkin, 2017. "Modelling stochastic skew of FX options using SLV models with stochastic spot/vol correlation and correlated jumps," Applied Mathematical Finance, Taylor & Francis Journals, vol. 24(6), pages 485-519, November.
    8. Andrey Itkin & Alexander Lipton, 2014. "Efficient solution of structural default models with correlated jumps and mutual obligations," Papers 1408.6513, arXiv.org, revised Nov 2014.
    9. Maximilian Ga{ss} & Kathrin Glau, 2016. "A Flexible Galerkin Scheme for Option Pricing in L\'evy Models," Papers 1603.08216, arXiv.org.
    10. Andrey Itkin, 2015. "HIGH ORDER SPLITTING METHODS FOR FORWARD PDEs AND PIDEs," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(05), pages 1-24.
    11. Itkin, Andrey, 2014. "Splitting and matrix exponential approach for jump-diffusion models with Inverse Normal Gaussian, Hyperbolic and Meixner jumps," Algorithmic Finance, IOS Press, vol. 3(3-4), pages 233-250.
    12. Peter Carr & Roger Lee & Matthew Lorig, 2017. "Pricing Variance Swaps on Time-Changed Markov Processes," Papers 1705.01069, arXiv.org, revised Nov 2019.

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