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On Lévy processes, Malliavin calculus and market models with jumps

Author

Listed:
  • Josep Vives

    (Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Bellaterra Spain Manuscript)

  • Jorge A. León

    (Departamento de Matemáticas, CINVESTAV-IPN, Apartado Postal 14-740, 07000 México, D.F., México)

  • Frederic Utzet

    (Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Bellaterra Spain Manuscript)

  • Josep L. Solé

    (Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Bellaterra Spain Manuscript)

Abstract

Recent work by Nualart and Schoutens (2000), where a kind of chaotic property for Lévy processes has been proved, has enabled us to develop a Malliavin calculus for Lévy processes. For simple Lévy processes some useful formulas for computing Malliavin derivatives are deduced. Applications for option hedging in a jump-diffusion model are given.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:finsto:v:6:y:2002:i:2:p:197-225
    Note: received: November 2000; final version received: May 2001
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    Citations

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    Cited by:

    1. Geiss, Christel & Labart, Céline, 2016. "Simulation of BSDEs with jumps by Wiener Chaos expansion," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 2123-2162.
    2. Ankirchner, Stefan, 2008. "On filtration enlargements and purely discontinuous martingales," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1662-1678, September.
    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. 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.
    5. 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.
    6. Bernardo D'Auria & Jos'e A. Salmer'on, 2021. "Anticipative information in a Brownian-Poissonmarket: the binary information," Papers 2111.01529, arXiv.org.
    7. El Otmani, Mohamed, 2009. "BSDEs driven by Lévy process with enlarged filtration and applications in finance," Statistics & Probability Letters, Elsevier, vol. 79(1), pages 44-49, January.
    8. Gurjeet Dhesi & Bilal Shakeel & Marcel Ausloos, 2021. "Modelling and forecasting the kurtosis and returns distributions of financial markets: irrational fractional Brownian motion model approach," Annals of Operations Research, Springer, vol. 299(1), pages 1397-1410, April.
    9. 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.
    10. 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.
    11. Davis, Mark H.A. & Johansson, Martin P., 2006. "Malliavin Monte Carlo Greeks for jump diffusions," Stochastic Processes and their Applications, Elsevier, vol. 116(1), pages 101-129, January.
    12. 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.
    13. 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.

    More about this item

    Keywords

    Lévy processes; Malliavin calculus; jump diffusion model; option hedging;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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