IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2102.08338.html
   My bibliography  Save this paper

Multilayer heat equations: application to finance

Author

Listed:
  • A. Itkin
  • A. Lipton
  • D. Muravey

Abstract

In this paper, we develop a Multilayer (ML) method for solving one-factor parabolic equations. Our approach provides a powerful alternative to the well-known finite difference and Monte Carlo methods. We discuss various advantages of this approach, which judiciously combines semi-analytical and numerical techniques and provides a fast and accurate way of finding solutions to the corresponding equations. To introduce the core of the method, we consider multilayer heat equations, known in physics for a relatively long time but never used when solving financial problems. Thus, we expand the analytic machinery of quantitative finance by augmenting it with the ML method. We demonstrate how one can solve various problems of mathematical finance by using our approach. Specifically, we develop efficient algorithms for pricing barrier options for time-dependent one-factor short-rate models, such as Black-Karasinski and Verhulst. Besides, we show how to solve the well-known Dupire equation quickly and accurately. Numerical examples confirm that our approach is considerably more efficient for solving the corresponding partial differential equations than the conventional finite difference method by being much faster and more accurate than the known alternatives.

Suggested Citation

  • A. Itkin & A. Lipton & D. Muravey, 2021. "Multilayer heat equations: application to finance," Papers 2102.08338, arXiv.org.
    • Handle: RePEc:arx:papers:2102.08338
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2102.08338
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Peter Carr & Sergey Nadtochiy, 2017. "Local Variance Gamma And Explicit Calibration To Option Prices," Mathematical Finance, Wiley Blackwell, vol. 27(1), pages 151-193, January.
    2. P. Carr & A. Itkin, 2021. "An Expanded Local Variance Gamma Model," Computational Economics, Springer;Society for Computational Economics, vol. 57(4), pages 949-987, April.
    3. Andrey Itkin & Dmitry Muravey, 2020. "Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit," Papers 2009.09342, arXiv.org, revised Oct 2020.
    4. Peter Carr & Andrey Itkin & Dmitry Muravey, 2020. "Semi-closed form prices of barrier options in the time-dependent CEV and CIR models," Papers 2005.05459, arXiv.org.
    5. Andrey Itkin, 2020. "Fitting Local Volatility:Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 11623, January.
    6. Alexander Lipton & Vadim Kaushansky, 2020. "On the first hitting time density for a reducible diffusion process," Quantitative Finance, Taylor & Francis Journals, vol. 20(5), pages 723-743, May.
    7. Beáta Stehlíková & Luca Capriotti, 2014. "An Effective Approximation For Zero-Coupon Bonds And Arrow–Debreu Prices In The Black–Karasinski Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(06), pages 1-16.
    8. Andrey Itkin & Dmitry Muravey, 2020. "Semi-closed form prices of barrier options in the Hull-White model," Papers 2004.09591, arXiv.org, revised Sep 2020.
    9. Alexander Lipton & Marcos Lopez de Prado, 2020. "A closed-form solution for optimal mean-reverting trading strategies," Papers 2003.10502, arXiv.org.
    10. Carr, Elliot J. & March, Nathan G., 2018. "Semi-analytical solution of multilayer diffusion problems with time-varying boundary conditions and general interface conditions," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 286-303.
    11. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Jan Barlak & Matus Bakon & Martin Rovnak & Martina Mokrisova, 2022. "Heat Equation as a Tool for Outliers Mitigation in Run-Off Triangles for Valuing the Technical Provisions in Non-Life Insurance Business," Risks, MDPI, vol. 10(9), pages 1-17, August.
    2. Itkin, Andrey & Lipton, Alexander & Muravey, Dmitry, 2022. "Multilayer heat equations and their solutions via oscillating integral transforms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 601(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. A. Itkin & A. Lipton & D. Muravey, 2020. "From the Black-Karasinski to the Verhulst model to accommodate the unconventional Fed's policy," Papers 2006.11976, arXiv.org, revised Jan 2021.
    2. Alexander Lipton, 2020. "Old Problems, Classical Methods, New Solutions," Papers 2003.06903, arXiv.org.
    3. Andrey Itkin & Dmitry Muravey, 2023. "American options in time-dependent one-factor models: Semi-analytic pricing, numerical methods and ML support," Papers 2307.13870, arXiv.org.
    4. P. Carr & A. Itkin & D. Muravey, 2022. "Semi-analytical pricing of barrier options in the time-dependent Heston model," Papers 2202.06177, arXiv.org.
    5. Alexander Lipton & Artur Sepp, 2022. "Toward an efficient hybrid method for pricing barrier options on assets with stochastic volatility," Papers 2202.07849, arXiv.org.
    6. Andrey Itkin, 2020. "Geometric Local Variance Gamma Model," World Scientific Book Chapters, in: Fitting Local Volatility Analytic and Numerical Approaches in Black-Scholes and Local Variance Gamma Models, chapter 6, pages 137-173, World Scientific Publishing Co. Pte. Ltd..
    7. Andrey Itkin & Dmitry Muravey, 2020. "Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit," Papers 2009.09342, arXiv.org, revised Oct 2020.
    8. Fabien Le Floc'h, 2020. "An arbitrage-free interpolation of class $C^2$ for option prices," Papers 2004.08650, arXiv.org, revised May 2020.
    9. Andrey Itkin & Dmitry Muravey, 2021. "Semi-analytical pricing of barrier options in the time-dependent $\lambda$-SABR model," Papers 2109.02134, arXiv.org.
    10. Peter Carr & Andrey Itkin & Dmitry Muravey, 2020. "Semi-closed form prices of barrier options in the time-dependent CEV and CIR models," Papers 2005.05459, arXiv.org.
    11. Alexander Lipton & Marcos Lopez de Prado, 2020. "A closed-form solution for optimal mean-reverting trading strategies," Papers 2003.10502, arXiv.org.
    12. Chenxu Li, 2014. "Closed-Form Expansion, Conditional Expectation, and Option Valuation," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 487-516, May.
    13. Alexander Lipton, 2020. "Physics and Derivatives -- Interview Questions and Answers," Papers 2003.11471, arXiv.org.
    14. Fazlollah Soleymani & Andrey Itkin, 2019. "Pricing foreign exchange options under stochastic volatility and interest rates using an RBF--FD method," Papers 1903.00937, arXiv.org.
    15. Ilya Molchanov & Michael Schmutz, 2009. "Exchangeability type properties of asset prices," Papers 0901.4914, arXiv.org, revised Apr 2011.
    16. Moisés A. C. Lemos & Camilla T. Baran & André L. B. Cavalcante & Ennio M. Palmeira, 2023. "A Semi-Analytical Model of Contaminant Transport in Barrier Systems with Arbitrary Numbers of Layers," Sustainability, MDPI, vol. 15(23), pages 1-18, November.
    17. Andrey Itkin, 2023. "The ATM implied skew in the ADO-Heston model," Papers 2309.15044, arXiv.org.
    18. Simonella, Roberta & Vázquez, Carlos, 2023. "XVA in a multi-currency setting with stochastic foreign exchange rates," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 59-79.
    19. Viatcheslav Gorovoi & Vadim Linetsky, 2004. "Black's Model of Interest Rates as Options, Eigenfunction Expansions and Japanese Interest Rates," Mathematical Finance, Wiley Blackwell, vol. 14(1), pages 49-78, January.
    20. Marcos Escobar & Christoph Gschnaidtner, 2018. "A multivariate stochastic volatility model with applications in the foreign exchange market," Review of Derivatives Research, Springer, vol. 21(1), pages 1-43, April.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2102.08338. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.