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An introduction to L\'{e}vy processes with applications in finance


  • Antonis Papapantoleon


These lectures notes aim at introducing L\'{e}vy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of L\'{e}vy processes. We analyze a `toy' example of a L\'{e}vy process, viz. a L\'{e}vy jump-diffusion, which yet offers significant insight into the distributional and path structure of a L\'{e}vy process. Then, we present several important results about L\'{e}vy processes, such as infinite divisibility and the L\'{e}vy-Khintchine formula, the L\'{e}vy-It\^{o} decomposition, the It\^{o} formula for L\'{e}vy processes and Girsanov's transformation. Some (sketches of) proofs are presented, still the majority of proofs is omitted and the reader is referred to textbooks instead. In the second part, we turn our attention to the applications of L\'{e}vy processes in financial modeling and option pricing. We discuss how the price process of an asset can be modeled using L\'{e}vy processes and give a brief account of market incompleteness. Popular models in the literature are presented and revisited from the point of view of L\'{e}vy processes, and we also discuss three methods for pricing financial derivatives. Finally, some indicative evidence from applications to market data is presented.

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  • Antonis Papapantoleon, 2008. "An introduction to L\'{e}vy processes with applications in finance," Papers 0804.0482,, revised Nov 2008.
  • Handle: RePEc:arx:papers:0804.0482

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    References listed on IDEAS

    1. Ernst Eberlein & Jean Jacod, 1997. "On the range of options prices (*)," Finance and Stochastics, Springer, vol. 1(2), pages 131-140.
    2. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters,in: Theory Of Valuation, chapter 8, pages 229-288 World Scientific Publishing Co. Pte. Ltd..
    3. Thomas Goll & Ludger Rüschendorf, 2001. "Minimax and minimal distance martingale measures and their relationship to portfolio optimization," Finance and Stochastics, Springer, vol. 5(4), pages 557-581.
    4. Ole E. Barndorff-Nielsen & Karsten Prause, 2001. "Apparent scaling," Finance and Stochastics, Springer, vol. 5(1), pages 103-113.
    5. Ernst Eberlein & Antonis Papapantoleon & Albert Shiryaev, 2008. "On the duality principle in option pricing: semimartingale setting," Finance and Stochastics, Springer, vol. 12(2), pages 265-292, April.
    6. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    7. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
    8. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    9. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
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    Cited by:

    1. Kakushadze, Zura, 2017. "Volatility smile as relativistic effect," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 475(C), pages 59-76.
    2. D. J. Manuge, 2015. "L\'evy Processes For Finance: An Introduction In R," Papers 1503.03902,
    3. Watson, John G. & Scott, Jason S., 2014. "Ratchet consumption over finite and infinite planning horizons," Journal of Mathematical Economics, Elsevier, vol. 54(C), pages 84-96.

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