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Pricing Discretely Monitored Barrier Options by a Markov Chain

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  • Jin-Chuan Duan
  • Evan Dudley
  • Geneviève Gauthier

    ()

  • Jean-Guy Simonato

Abstract

We propose a Markov chain method for pricing discretely monitored barrier options in both the constant and time-varying volatility valuation frameworks. The method uses a time homogeneous Markov Chain to approximate the underlying asset price process. Our approach provides a natural framework for pricing discretely monitored barrier options because the discrete time step of the Markov chain can be easily matched with the monitoring frequency of the barrier. Furthermore the underlying asset price can also be partitioned to have the barrier suitably placed. Our method is fast, flexible and easy to implement as it reduces the pricing of American and European barrier options to simple matrix operations. Our method can efficiently handle the difficult cases where the barrier is close to the initial asset price. We study both knock-in and knock-out barrier options. Different types of barriers such as single, double and moving barriers are also analyzed. Cette étude propose l'utilisation de chaînes de Markov pour l'évaluation de prix d'options à barrière avec vérification à temps discrets dans des contextes de volatilité constante ou variable. La méthode utilise une chaîne de Markov homogène afin d'approcher le processus stochastique postulé pour l'actif sous-jacent. Cette méthode procure un environnement naturel pour évaluer ce type d'option puisque le pas discret de la chaîne de Markov peut être adapté à la longueur de temps entre les vérifications de la barrière. Le prix du sous-jacent peut aussi être discrétisé de façon optimale par rapport à la barrière. La méthode est rapide, flexible et simple à implanter puisque le calcul de prix d'options européennes et américaines est réalisé à l'aide de multiplications matricielles. De plus, la méthode proposée est précise pour les cas difficiles où la barrière est située près de la valeur du sous-jacent. Les options knock-in et knock-out sont examinées. Différents types de barrières telles les barrières doubles ainsi que les barrières mobiles sont aussi examinés.

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Bibliographic Info

Paper provided by CIRANO in its series CIRANO Working Papers with number 99s-15.

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Date of creation: 01 Apr 1999
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Handle: RePEc:cir:cirwor:99s-15

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Keywords: Barrier options; Markov chain; sparse matrix; American options; knock-in options; knock-out options; Options à barrière; chaînes de Markov; matrices creuses; options américaines; options knock-in; options knock-out;

References

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  1. Boyle, Phelim & Broadie, Mark & Glasserman, Paul, 1997. "Monte Carlo methods for security pricing," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 21(8-9), pages 1267-1321, June.
  2. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  3. K. Sandmann & Reimer, M., 1995. "A Discrete Time Approach for European and American Barrier Options," Discussion Paper Serie B 272, University of Bonn, Germany.
  4. Mark Broadie & Paul Glasserman & Steven Kou, 1997. "A Continuity Correction for Discrete Barrier Options," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 7(4), pages 325-349.
  5. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, Elsevier, vol. 7(3), pages 229-263, September.
  6. Duan, Jin-Chuan & Simonato, Jean-Guy, 2001. "American option pricing under GARCH by a Markov chain approximation," Journal of Economic Dynamics and Control, Elsevier, Elsevier, vol. 25(11), pages 1689-1718, November.
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Citations

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Cited by:
  1. Christian Skaug & Arvid Naess, 2007. "Fast and accurate pricing of discretely monitored barrier options by numerical path integration," Computational Economics, Society for Computational Economics, Society for Computational Economics, vol. 30(2), pages 143-151, September.
  2. Chun-Chou Wu, 2006. "The GARCH Option Pricing Model: A Modification of Lattice Approach," Review of Quantitative Finance and Accounting, Springer, Springer, vol. 26(1), pages 55-66, February.
  3. D. M. Pooley & P. A. Forsyth & K. R. Vetzal & R. B. Simpson, 2000. "Unstructured meshing for two asset barrier options," Applied Mathematical Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 7(1), pages 33-60.
  4. J. C. Ndogmo & D. B. Ntwiga, 2007. "High-order accurate implicit methods for the pricing of barrier options," Papers 0710.0069, arXiv.org.
  5. Duan, Jin-Chuan & Zhang, Hua, 2001. "Pricing Hang Seng Index options around the Asian financial crisis - A GARCH approach," Journal of Banking & Finance, Elsevier, vol. 25(11), pages 1989-2014, November.

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