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

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  • Peter Buchen
  • Otto Konstandatos

Abstract

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

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    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
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    4. Hans-Peter Bermin & Peter Buchen & Otto Konstandatos, 2008. "Two Exotic Lookback Options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(4), pages 387-402.
    5. 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.
    6. Peter Buchen & Otto Konstandatos, 2005. "A New Method Of Pricing Lookback Options," Mathematical Finance, Wiley Blackwell, vol. 15(2), pages 245-259, April.
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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, arXiv.org, revised Mar 2018.
    3. Konstandatos, Otto, 2020. "Fair-value analytical valuation of reset executive stock options consistent with IFRS9 requirements," Annals of Actuarial Science, Cambridge University Press, vol. 14(1), pages 188-218, March.
    4. Igor V. Kravchenko & Vladislav V. Kravchenko & Sergii M. Torba & José Carlos Dias, 2019. "Pricing Double Barrier Options On Homogeneous Diffusions: A Neumann Series Of Bessel Functions Representation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(06), pages 1-24, September.
    5. Otto Konstandatos & Timothy J Kyng, 2012. "Real Options Analysis for Commodity Based Mining Enterprises with Compound and Barrier Features," Published Paper Series 2012-3, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    6. 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, arXiv.org.
    7. Youngchul Han & Geonwoo Kim, 2016. "Efficient Lattice Method for Valuing of Options with Barrier in a Regime Switching Model," Discrete Dynamics in Nature and Society, Hindawi, vol. 2016, pages 1-14, October.
    8. Jos� 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.
    9. Marianito R. Rodrigo, 2020. "Pricing of Barrier Options on Underlying Assets with Jump-Diffusion Dynamics: A Mellin Transform Approach," Mathematics, MDPI, vol. 8(8), pages 1-20, August.
    10. Shiyu Song & Yongjin Wang, 2017. "Pricing double barrier options under a volatility regime-switching model with psychological barriers," Review of Derivatives Research, Springer, vol. 20(3), pages 255-280, October.
    11. Huang, Min & Luo, Guo, 2022. "A simple and efficient numerical method for pricing discretely monitored early-exercise options," Applied Mathematics and Computation, Elsevier, vol. 422(C).
    12. Ha, Mijin & Kim, Donghyun & Yoon, Ji-Hun, 2024. "Valuing of timer path-dependent options," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 215(C), pages 208-227.
    13. Donghyun Kim & Ji-Hun Yoon, 2023. "Analytic Method for Pricing Vulnerable External Barrier Options," Computational Economics, Springer;Society for Computational Economics, vol. 61(4), pages 1561-1591, April.
    14. Hangsuck Lee & Hongjun Ha & Minha Lee, 2022. "Piecewise linear double barrier options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(1), pages 125-151, January.
    15. Kontosakos, Vasileios E. & Mendonca, Keegan & Pantelous, Athanasios A. & Zuev, Konstantin M., 2021. "Pricing discretely-monitored double barrier options with small probabilities of execution," European Journal of Operational Research, Elsevier, vol. 290(1), pages 313-330.
    16. 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.
    17. 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.
    18. Wang, Heqian & Zhang, Jiayi & Zhou, Ke, 2022. "On pricing of vulnerable barrier options and vulnerable double barrier options," Finance Research Letters, Elsevier, vol. 44(C).
    19. Min Huang & Guo Luo, 2019. "A simple and efficient numerical method for pricing discretely monitored early-exercise options," Papers 1905.13407, arXiv.org, revised Jun 2019.
    20. Lee, Hangsuck & Ha, Hongjun & Kong, Byungdoo & Lee, Minha, 2023. "Pricing multi-step double barrier options by the efficient non-crossing probability," Finance Research Letters, Elsevier, vol. 54(C).

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