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Barrier Option Under Lévy Model : A PIDE and Mellin Transform Approach

Author

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  • Sudip Ratan Chandra

    (Department of Mathematics, Jadavpur University, West Bengal 700032, India)

  • Diganta Mukherjee

    (Indian Statistical Institute, Kolkata, West Bengal 700108, India)

Abstract

We propose a stochastic model to develop a partial integro-differential equation (PIDE) for pricing and pricing expression for fixed type single Barrier options based on the Itô-Lévy calculus with the help of Mellin transform. The stock price is driven by a class of infinite activity Lévy processes leading to the market inherently incomplete, and dynamic hedging is no longer risk free. We first develop a PIDE for fixed type Barrier options, and apply the Mellin transform to derive a pricing expression. Our main contribution is to develop a PIDE with its closed form pricing expression for the contract. The procedure is easy to implement for all class of Lévy processes numerically. Finally, the algorithm for computing numerically is presented with results for a set of Lévy processes.

Suggested Citation

  • Sudip Ratan Chandra & Diganta Mukherjee, 2016. "Barrier Option Under Lévy Model : A PIDE and Mellin Transform Approach," Mathematics, MDPI, vol. 4(1), pages 1-18, January.
    • Handle: RePEc:gam:jmathe:v:4:y:2016:i:1:p:2-:d:61616
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    References listed on IDEAS

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    1. Marc Jeannin & Martijn Pistorius, 2010. "A transform approach to compute prices and Greeks of barrier options driven by a class of Levy processes," Quantitative Finance, Taylor & Francis Journals, vol. 10(6), pages 629-644.
    2. Luís A. N. Amaral & Vasiliki Plerou & Parameswaran Gopikrishnan & Martin Meyer & H. Eugene Stanley, 2000. "The Distribution Of Returns Of Stock Prices," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 3(03), pages 365-369.
    3. 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.
    4. Rama Cont & Ekaterina Voltchkova, 2005. "A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models," Post-Print halshs-00445645, HAL.
    5. Hélyette Geman & Marc Yor, 1996. "Pricing And Hedging Double‐Barrier Options: A Probabilistic Approach," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 365-378, October.
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