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Pricing Options With Curved Boundaries

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  • Naoto Kunitomo
  • Masayuki Ikeda
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    Abstract

    This paper provides a general valuation method for the European options whose payoff is restricted by curved boundaries contractually set on the underlying asset price process when it follows the geometric Brownian motion. Our result is based on the generalization of the Levy formula on the Brownian motion by T. W. Anderson in sequential analysis. We give the explicit probability formula that the geometric Brownian motion reaches in an interval at the maturity date without hitting either the lower or the upper curved boundaries. Although the general pricing formulae for options with boundaries are expressed as infinite series in the general case, our numerical study suggests that the convergence of the series is rapid. Our results include the formulae for options with a lower boundary by Merton (1973), for path-dependent options by Goldman, Sossin, and Gatto (1979), and for some corporate securities as special cases. Copyright 1992 Blackwell Publishers.

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    Bibliographic Info

    Article provided by Wiley Blackwell in its journal Mathematical Finance.

    Volume (Year): 2 (1992)
    Issue (Month): 4 ()
    Pages: 275-298

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    Handle: RePEc:bla:mathfi:v:2:y:1992:i:4:p:275-298

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    Cited by:
    1. Yuh-Dauh Lyuu & Huei-Wen Teng, 2011. "Unbiased and efficient Greeks of financial options," Finance and Stochastics, Springer, vol. 15(1), pages 141-181, January.
    2. Jean-Luc PRIGENT & Olivier RENAULT & Olivier SCAILLET, 2002. "Option Pricing with Discrete Rebalancing," FAME Research Paper Series rp55, International Center for Financial Asset Management and Engineering.
    3. Massimo Costabile, 2001. "A discrete-time algorithm for pricing double barrier options," Decisions in Economics and Finance, Springer, vol. 24(1), pages 49-58, 05.
    4. Fusai, Gianluca & Recchioni, Maria Cristina, 2007. "Analysis of quadrature methods for pricing discrete barrier options," Journal of Economic Dynamics and Control, Elsevier, vol. 31(3), pages 826-860, March.
    5. Tristan Guillaume, 2011. "Some sequential boundary crossing results for geometric Brownian motion and their applications in financial engineering," Post-Print hal-00924277, HAL.
    6. Li, Minqiang & Pearson, Neil D. & Poteshman, Allen M., 2004. "Conditional estimation of diffusion processes," Journal of Financial Economics, Elsevier, vol. 74(1), pages 31-66, October.
    7. repec:dgr:uvatin:2097015 is not listed on IDEAS
    8. Hieber, Peter & Scherer, Matthias, 2012. "A note on first-passage times of continuously time-changed Brownian motion," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 165-172.
    9. Antoon Pelsser, 1997. "Pricing Double Barrier Options: An Analytical Approach," Tinbergen Institute Discussion Papers 97-015/2, Tinbergen Institute.
    10. Farid MKAOUAR & Jean-luc PRIGENT, 2014. "Constant Proportion Portfolio Insurance under Tolerance and Transaction Costs," Working Papers 2014-303, Department of Research, Ipag Business School.
    11. Dietmar P.J. Leisen, 1999. "Valuation of Barrier Options in a Black--Scholes Setup with Jump Risk," Discussion Paper Serie B 446, University of Bonn, Germany.
    12. Pavel V. Shevchenko & Pierre Del Moral, 2014. "Valuation of Barrier Options using Sequential Monte Carlo," Papers 1405.5294, arXiv.org.
    13. Norland, Erik & Wilford, D. Sykes, 2002. "Leverage, liquidity, volatility, time horizon, and the risk of ruin: A barrier option approach," Review of Financial Economics, Elsevier, vol. 11(3), pages 225-239.
    14. Antoon Pelsser, 1997. "Pricing Double Barrier Options: An Analytical Approach," Tinbergen Institute Discussion Papers 97-015/2, Tinbergen Institute.
    15. Doobae Jun & Hyejin Ku, 2013. "Valuation of American partial barrier options," Review of Derivatives Research, Springer, vol. 16(2), pages 167-191, July.
    16. M. Krivko & M. V. Tretyakov, 2012. "Application of simplest random walk algorithms for pricing barrier options," Papers 1211.5726, arXiv.org.
    17. Zvan, R. & Vetzal, K. R. & Forsyth, P. A., 2000. "PDE methods for pricing barrier options," Journal of Economic Dynamics and Control, Elsevier, vol. 24(11-12), pages 1563-1590, October.
    18. Sbuelz, A., 2000. "Hedging Double Barriers with Singles," Discussion Paper 2000-112, Tilburg University, Center for Economic Research.

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