IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v130y2020i2p824-844.html
   My bibliography  Save this article

Dyson type formula for pure jump Lévy processes with some applications to finance

Author

Listed:
  • Jin, Sixian
  • Schellhorn, Henry
  • Vives, Josep

Abstract

In this paper we obtain a Dyson type formula for integrable functionals of a pure jump Lévy process. We represent the conditional expectation of a FT-measurable random variable F at a time t≤T as an exponential formula involving Malliavin derivatives evaluated along a frozen path. The series representation of this exponential formula turns out to be useful for different applications, and in particular in quantitative finance, as we show in the following examples: the first one is the pricing of options in the Poisson–Black–Scholes model; the second one is the pricing of discount bonds in the Lévy quadratic model. We also obtain, for the conditional expectation of a function of a finite number of the process values, a backward Taylor expansion, that turns out to be useful for numerical calculations.

Suggested Citation

  • Jin, Sixian & Schellhorn, Henry & Vives, Josep, 2020. "Dyson type formula for pure jump Lévy processes with some applications to finance," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 824-844.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:2:p:824-844
    DOI: 10.1016/j.spa.2019.03.019
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414919301929
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2019.03.019?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Solé, Josep Lluís & Utzet, Frederic & Vives, Josep, 2007. "Canonical Lévy process and Malliavin calculus," Stochastic Processes and their Applications, Elsevier, vol. 117(2), pages 165-187, February.
    2. Li Chen & Damir Filipović & H. Vincent Poor, 2004. "Quadratic Term Structure Models For Risk‐Free And Defaultable Rates," Mathematical Finance, Wiley Blackwell, vol. 14(4), pages 515-536, October.
    3. Josep Vives & Jorge A. León & Frederic Utzet & Josep L. Solé, 2002. "On Lévy processes, Malliavin calculus and market models with jumps," Finance and Stochastics, Springer, vol. 6(2), pages 197-225.
    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. Moualkia, Seyfeddine & Liu, Yang & Qiu, Jianlong & Lu, Jianquan, 2024. "An averaging result for fractional variable-order neutral differential equations with variable delays driven by Markovian switching and Lévy noise," Chaos, Solitons & Fractals, Elsevier, vol. 182(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. El-Khatib, Youssef & Goutte, Stephane & Makumbe, Zororo S. & Vives, Josep, 2023. "A hybrid stochastic volatility model in a Lévy market," International Review of Economics & Finance, Elsevier, vol. 85(C), pages 220-235.
    2. Choe, Hi Jun & Lee, Ji Min & Lee, Jung-Kyung, 2018. "Malliavin calculus for subordinated Lévy process," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 392-401.
    3. Horst Osswald, 2009. "A Smooth Approach to Malliavin Calculus for Lévy Processes," Journal of Theoretical Probability, Springer, vol. 22(2), pages 441-473, June.
    4. Bernardo D'Auria & Jos'e A. Salmer'on, 2021. "Anticipative information in a Brownian-Poissonmarket: the binary information," Papers 2111.01529, arXiv.org.
    5. Ankirchner, Stefan, 2008. "On filtration enlargements and purely discontinuous martingales," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1662-1678, September.
    6. Leonidas S. Rompolis & Elias Tzavalis, 2017. "Pricing and hedging contingent claims using variance and higher order moment swaps," Quantitative Finance, Taylor & Francis Journals, vol. 17(4), pages 531-550, April.
    7. Shaw, Charles, 2020. "Regimes, Non-Linearities, and Price Discontinuities in Indian Energy Stocks," MPRA Paper 104798, University Library of Munich, Germany.
    8. Eden, Richard & Víquez, Juan, 2015. "Nourdin–Peccati analysis on Wiener and Wiener–Poisson space for general distributions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 182-216.
    9. K. Giesecke & H. Kakavand & M. Mousavi, 2011. "Exact Simulation of Point Processes with Stochastic Intensities," Operations Research, INFORMS, vol. 59(5), pages 1233-1245, October.
    10. Jirô Akahori & Takahiro Tsuchiya, 2006. "What is the Natural Scale for a Lévy Process in Modelling Term Structure of Interest Rates?," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 13(4), pages 299-313, December.
    11. Gourieroux, Christian & Sufana, Razvan, 2011. "Discrete time Wishart term structure models," Journal of Economic Dynamics and Control, Elsevier, vol. 35(6), pages 815-824, June.
    12. Eduardo Abi Jaber, 2022. "The Laplace transform of the integrated Volterra Wishart process," Mathematical Finance, Wiley Blackwell, vol. 32(1), pages 309-348, January.
    13. Samson Assefa, 2007. "Pricing of Defaultable Securities in a Multi-Factor Quadratic Gaussian Model," Research Paper Series 202, Quantitative Finance Research Centre, University of Technology, Sydney.
    14. Delong, Lukasz & Imkeller, Peter, 2010. "On Malliavin's differentiability of BSDEs with time delayed generators driven by Brownian motions and Poisson random measures," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1748-1775, August.
    15. Stéphane Goutte, 2013. "Markov switching quadratic term structure models," Working Papers hal-00821745, HAL.
    16. Raphaël Homayoun Boroumand & St�phane Goutte & Thomas Porcher, 2014. "A regime-switching model to evaluate bonds in a quadratic term structure of interest rates," Applied Financial Economics, Taylor & Francis Journals, vol. 24(21), pages 1361-1366, November.
    17. D'Auria, Bernardo & Salmerón Garrido, José Antonio, 2021. "Anticipative information in a Brownian-Poisson market: the binary information," DES - Working Papers. Statistics and Econometrics. WS 33624, Universidad Carlos III de Madrid. Departamento de Estadística.
    18. Jirô Akahori & Hiroki Aoki & Yoshihiko Nagata, 2006. "Generalizations of Ho–Lee’s binomial interest rate model I: from one- to multi-factor," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 13(2), pages 151-179, June.
    19. Likuan Qin & Vadim Linetsky, 2016. "Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery, and Long-Term Pricing," Operations Research, INFORMS, vol. 64(1), pages 99-117, February.
    20. Fujii, Masaaki & Takahashi, Akihiko, 2018. "Quadratic–exponential growth BSDEs with jumps and their Malliavin’s differentiability," Stochastic Processes and their Applications, Elsevier, vol. 128(6), pages 2083-2130.

    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:eee:spapps:v:130:y:2020:i:2:p:824-844. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.