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An explicit finite difference approach to the pricing of barrier options

Author

Listed:
  • Phelim Boyle
  • Yisong Tian

Abstract

A modified explicit finite difference approach to the pricing of barrier options is developed. To obtain accurate prices, the grid is constructed such that the barrier is located in a suitable position relative to horizontal layers of nodes on the grid. This means that the barrier passes through a horizontal layer of nodes for continuous-time barrier options and is located halfway between two horizontal layers of nodes for discrete-time barrier options. Both single and double barrier cases can be accommodated. The option price at each node on the grid may be obtained by implementing a standard trinomial tree procedure. As the initial asset price will generally not lie exactly on the grid, the current value of the option is obtained using a quadratic interpolation of the option prices at the three adjacent nodes. The approach is shown to be robust and to provide accurate option prices and hedge ratios (such as delta, gamma, and theta) regardless of whether or not the barrier is close to the initial asset price, and it works effectively for both continuous-time and discrete-time barrier options. This device of adjusting the grid so that the barrier and the asset price lie on the grid is well known in the numerical analysis area.

Suggested Citation

  • Phelim Boyle & Yisong Tian, 1998. "An explicit finite difference approach to the pricing of barrier options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 5(1), pages 17-43.
  • Handle: RePEc:taf:apmtfi:v:5:y:1998:i:1:p:17-43
    DOI: 10.1080/135048698334718
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    References listed on IDEAS

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    1. Hull, John & White, Alan, 1990. "Valuing Derivative Securities Using the Explicit Finite Difference Method," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 25(01), pages 87-100, March.
    2. Boyle, Phelim P., 1988. "A Lattice Framework for Option Pricing with Two State Variables," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 23(01), pages 1-12, March.
    3. 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.
    4. Schwartz, Eduardo S., 1977. "The valuation of warrants: Implementing a new approach," Journal of Financial Economics, Elsevier, vol. 4(1), pages 79-93, January.
    5. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
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    Cited by:

    1. Simona Sanfelici, 2004. "Galerkin infinite element approximation for pricing barrier options and options with discontinuous payoff," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 27(2), pages 125-151, December.
    2. Emmanuel Gobet, 2009. "Advanced Monte Carlo methods for barrier and related exotic options," Post-Print hal-00319947, HAL.

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