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Viscosity Solutions of Integro-Differential Equations and Passport Options in a Jump-Diffusion Model

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  • Yang Wang

    (Shanghai Normal University)

  • Baojun Bian

    (Tongji University)

  • Jizhou Zhang

    (Shanghai Normal University)

Abstract

We study the viscosity solutions of integro-differential Hamilton–Jacobi–Bellman equations of degenerate parabolic type. These equations are from the pricing problem for the European passport options in a jump-diffusion model. The passport option is a call option on a trading account. We discuss the mathematical model for pricing problem. We prove the comparison principle, uniqueness and convexity preserving for the viscosity solutions of related pricing equations.

Suggested Citation

  • Yang Wang & Baojun Bian & Jizhou Zhang, 2014. "Viscosity Solutions of Integro-Differential Equations and Passport Options in a Jump-Diffusion Model," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 122-144, April.
    • Handle: RePEc:spr:joptap:v:161:y:2014:i:1:d:10.1007_s10957-013-0382-9
    DOI: 10.1007/s10957-013-0382-9
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    References listed on IDEAS

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    1. Vicky Henderson & David Hobson & Glenn Kentwell, 2002. "A New Class Of Commodity Hedging Strategies: A Passport Options Approach," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(03), pages 255-278.
    2. Steven E. Shreve & Jan Vecer, 2000. "Options on a traded account: Vacation calls, vacation puts and passport options," Finance and Stochastics, Springer, vol. 4(3), pages 255-274.
    3. H. Ahn & Jeff Dewynne & P. Hua & Antony Penaud & Paul Wilmott, 2002. "The End-Of-The-Year Bonus: How To Optimally Reward A Trader?," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(03), pages 279-306.
    4. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    5. Naik, Vasanttilak & Lee, Moon, 1990. "General Equilibrium Pricing of Options on the Market Portfolio with Discontinuous Returns," The Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 493-521.
    6. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
    7. Vicky Henderson & David Hobson, 2001. "Passport options with stochastic volatility," Applied Mathematical Finance, Taylor & Francis Journals, vol. 8(2), pages 97-118.
    8. Aase, Knut K., 1988. "Contingent claims valuation when the security price is a combination of an Ito process and a random point process," Stochastic Processes and their Applications, Elsevier, vol. 28(2), pages 185-220, June.
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    Cited by:

    1. Zakaria Marah, 2023. "American Passport options in an exponential L\'evy model," Papers 2307.16649, arXiv.org.
    2. Yang Wang & Baojun Bian & Zijiang Yang & Jizhou Zhang, 2019. "The Valuation of American Passport Options: A Viscosity Solution Approach," Journal of Optimization Theory and Applications, Springer, vol. 180(2), pages 608-633, February.

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