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A New Approach to Pricing Double-Barrier Options with Arbitrary Payoffs and Exponential Boundaries


  • Peter Buchen
  • Otto Konstandatos


We consider in this article the arbitrage free pricing of double knock-out barrier options with payoffs that are arbitrary functions of the underlying asset, where we allow exponentially time-varying barrier levels in an otherwise standard Black-Scholes model. Our approach, reminiscent of the method of images of electromagnetics, considerably simplifies the derivation of analytical formulae for this class of exotics by reducing the pricing of any double-barrier problem to that of pricing a related European option. We illustrate the method by reproducing the well-known formulae of Kunitomo and Ikeda (1992) for the standard knock-out double-barrier call and put options. We give an explanation for the rapid rate of convergence of the doubly infinite sums for affine payoffs in the stock price, as encountered in the pricing of double-barrier call and put options first observed by Kunitomo and Ikeda (1992).

Suggested Citation

  • Peter Buchen & Otto Konstandatos, 2009. "A New Approach to Pricing Double-Barrier Options with Arbitrary Payoffs and Exponential Boundaries," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(6), pages 497-515.
  • Handle: RePEc:taf:apmtfi:v:16:y:2009:i:6:p:497-515
    DOI: 10.1080/13504860903075480

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    References listed on IDEAS

    1. Goldman, M Barry & Sosin, Howard B & Gatto, Mary Ann, 1979. "Path Dependent Options: "Buy at the Low, Sell at the High"," Journal of Finance, American Finance Association, vol. 34(5), pages 1111-1127, December.
    2. Peter Buchen & Otto Konstandatos, 2005. "A New Method Of Pricing Lookback Options," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 245-259.
    3. 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..
    4. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    5. Peter Buchen, 2004. "The pricing of dual-expiry exotics," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 101-108.
    6. Hélyette Geman & Marc Yor, 1996. "Pricing And Hedging Double-Barrier Options: A Probabilistic Approach," Mathematical Finance, Wiley Blackwell, vol. 6(4), pages 365-378.
    7. Hans-Peter Bermin & Peter Buchen & Otto Konstandatos, 2008. "Two Exotic Lookback Options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(4), pages 387-402.
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    Cited by:

    1. Kyng, T. & Konstandatos, O. & Bienek, T., 2016. "Valuation of employee stock options using the exercise multiple approach and life tables," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 17-26.
    2. Keegan Mendonca & Vasileios E. Kontosakos & Athanasios A. Pantelous & Konstantin M. Zuev, 2018. "Efficient Pricing of Barrier Options on High Volatility Assets using Subset Simulation," Papers 1803.03364,, revised Mar 2018.
    3. Choe, Geon Ho & Koo, Ki Hwan, 2014. "Probability of multiple crossings and pricing of double barrier options," The North American Journal of Economics and Finance, Elsevier, vol. 29(C), pages 156-184.
    4. Thorsten Upmann, 2013. "Pricing Onion Options: A Probabilistic Approach," International Journal of Financial Research, International Journal of Financial Research, Sciedu Press, vol. 4(4), pages 11-25, October.
    5. Igor V. Kravchenko & Vladislav V. Kravchenko & Sergii M. Torba & Jos'e Carlos Dias, 2017. "Pricing double barrier options on homogeneous diffusions: a Neumann series of Bessel functions representation," Papers 1712.08247,
    6. José Carlos Dias & João Pedro Vidal Nunes & João Pedro Ruas, 2015. "Pricing and static hedging of European-style double barrier options under the jump to default extended CEV model," Quantitative Finance, Taylor & Francis Journals, vol. 15(12), pages 1995-2010, December.


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