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Fractional Diffusion Models of Option Prices in Markets with Jumps

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  • Alvaro Cartea

    (Department of Economics, Mathematics & Statistics, Birkbeck)

  • Diego del-Castillo-Negrete

Abstract

Most of the recent literature dealing with the modeling of financial assets assumes that the underlying dynamics of equity prices follow a jump process or a Levy process. This is done to incorporate rare or extreme events not captured by Gaussian models. Of those financial models proposed, the most interesting include the CGMY, KoBoL and FMLS. All of these capture some of the most important characteristics of the dynamics of stock prices. In this article we show that for these particular Levy processes, the prices of financial derivatives, such as European-style options, satisfy a fractional partial differential equation (FPDE). As an application, we use numerical techniques to price exotic options, in particular barrier options, by solving the corresponding FPDEs derived.

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File URL: http://www.bbk.ac.uk/ems/research/wp/PDF/BWPEF0604.pdf
File Function: First version, 2006
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Bibliographic Info

Paper provided by Birkbeck, Department of Economics, Mathematics & Statistics in its series Birkbeck Working Papers in Economics and Finance with number 0604.

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Date of creation: Apr 2006
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Handle: RePEc:bbk:bbkefp:0604

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Keywords: Fractional-Black-Scholes; Levy-Stable processes; FMLS; KoBoL; CGMY; fractional calculus; Riemann-Liouville fractional derivative; barrier options; down-and-out; up-and-out; double knock-out.;

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  1. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
  2. Enrico Scalas & Rudolf Gorenflo & Francesco Mainardi, 2004. "Fractional calculus and continuous-time finance," Finance 0411007, EconWPA.
  3. Peter Carr & Liuren Wu, 2002. "The Finite Moment Log Stable Process and Option Pricing," Finance 0207012, EconWPA.
  4. 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.
  5. Alvaro Cartea, 2005. "Dynamic Hedging of Financial Instruments When the Underlying Follows a Non-Gaussian Process," Birkbeck Working Papers in Economics and Finance 0508, Birkbeck, Department of Economics, Mathematics & Statistics.
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Cited by:
  1. Andrey Itkin & Peter Carr, 2010. "Using pseudo-parabolic and fractional equations for option pricing in jump diffusion models," Papers 1002.1995, arXiv.org.
  2. Sun, Lin, 2013. "Pricing currency options in the mixed fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(16), pages 3441-3458.

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