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

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  • 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
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    References listed on IDEAS

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    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.
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    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).

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