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High-Order Splitting Methods for Forward PDEs and PIDEs

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  • Andrey Itkin

Abstract

This paper is dedicated to the construction of high-order (in both space and time) finite-difference schemes for both forward and backward PDEs and PIDEs, such that option prices obtained by solving both the forward and backward equations are consistent. This approach is partly inspired by Andreasen & Huge, 2011 who reported a pair of consistent finite-difference schemes of first-order approximation in time for an uncorrelated local stochastic volatility model. We extend their approach by constructing schemes that are second-order in both space and time and that apply to models with jumps and discrete dividends. Taking correlation into account in our approach is also not an issue.

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  • Andrey Itkin, 2014. "High-Order Splitting Methods for Forward PDEs and PIDEs," Papers 1403.1804, arXiv.org.
    • Handle: RePEc:arx:papers:1403.1804
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    References listed on IDEAS

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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.
    2. 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.
    3. Samuli Ikonen & Jari Toivanen, 2007. "Componentwise Splitting Methods For Pricing American Options Under Stochastic Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 10(02), pages 331-361.
    4. Carl Chiarella & Boda Kang & Gunter H. Meyer & Andrew Ziogas, 2009. "The Evaluation Of American Option Prices Under Stochastic Volatility And Jump-Diffusion Dynamics Using The Method Of Lines," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(03), pages 393-425.
    5. Alexander Lipton, 2001. "Mathematical Methods for Foreign Exchange:A Financial Engineer's Approach," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4694, February.
    6. Andrey Itkin, 2013. "Efficient Solution of Backward Jump-Diffusion PIDEs with Splitting and Matrix Exponentials," Papers 1304.3159, arXiv.org, revised Apr 2014.
    7. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
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    Cited by:

    1. P. Carr & A. Itkin & D. Muravey, 2022. "Semi-analytical pricing of barrier options in the time-dependent Heston model," Papers 2202.06177, arXiv.org.
    2. Andrey Itkin & Fazlollah Soleymani, 2019. "Four-factor model of Quanto CDS with jumps-at-default and stochastic recovery," Papers 1912.08713, arXiv.org.
    3. 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.
    4. Jan Posp'iv{s}il & Vladim'ir v{S}v'igler, 2019. "Isogeometric analysis in option pricing," Papers 1910.00258, arXiv.org.
    5. Cornelis S. L. de Graaf & Drona Kandhai & Christoph Reisinger, 2016. "Efficient exposure computation by risk factor decomposition," Papers 1608.01197, arXiv.org, revised Feb 2018.
    6. Andrey Itkin & Dmitry Muravey, 2021. "Semi-analytical pricing of barrier options in the time-dependent $\lambda$-SABR model," Papers 2109.02134, arXiv.org.
    7. Maarten Wyns & Karel in 't Hout, 2016. "An adjoint method for the exact calibration of Stochastic Local Volatility models," Papers 1609.00232, arXiv.org.
    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. Andrey Itkin, 2015. "LSV models with stochastic interest rates and correlated jumps," Papers 1511.01460, arXiv.org, revised Nov 2016.

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