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Barrier options and their static hedges: simple derivations and extensions

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  • Rolf Poulsen

Abstract

We use a reflection result to give simple proofs of (well-known) valuation formulas and static hedge portfolio constructions for zero-rebate single-barrier options in the Black-Scholes model. We then illustrate how to extend the ideas to other model types giving (at least) easy-to-program numerical methods and other option types such as options with rebates, and double-barrier and lookback options.

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  • Rolf Poulsen, 2006. "Barrier options and their static hedges: simple derivations and extensions," Quantitative Finance, Taylor & Francis Journals, vol. 6(4), pages 327-335.
    • Handle: RePEc:taf:quantf:v:6:y:2006:i:4:p:327-335
    DOI: 10.1080/14697680600690331
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    1. Alessandro Sbuelz, 2005. "Hedging Double Barriers With Singles," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(03), pages 393-407.
    2. 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..
    3. G. O. Roberts & C. F. Shortland, 1997. "Pricing Barrier Options with Time–Dependent Coefficients," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 83-93, January.
    4. Peter Carr & Katrina Ellis & Vishal Gupta, 1998. "Static Hedging of Exotic Options," Journal of Finance, American Finance Association, vol. 53(3), pages 1165-1190, June.
    5. Emanuel Derman & Iraj Kani, 1998. "Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 1(01), pages 61-110.
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    Cited by:

    1. Jun, Doobae & Ku, Hyejin, 2015. "Static hedging of chained-type barrier options," The North American Journal of Economics and Finance, Elsevier, vol. 33(C), pages 317-327.
    2. Yuri Imamura & Katsuya Takagi, 2013. "Semi-Static Hedging Based on a Generalized Reflection Principle on a Multi Dimensional Brownian Motion," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 20(1), pages 71-81, March.
    3. 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.
    4. Lee, Hangsuck & Ko, Bangwon & Lee, Minha, 2023. "The pricing and static hedging of multi-step double barrier options," Finance Research Letters, Elsevier, vol. 55(PA).
    5. Lee, Hangsuck & Choi, Yang Ho & Lee, Gaeun, 2022. "Multi-step barrier products and static hedging," The North American Journal of Economics and Finance, Elsevier, vol. 61(C).
    6. Tsai, Wei-Che, 2014. "Improved method for static replication under the CEV model," Finance Research Letters, Elsevier, vol. 11(3), pages 194-202.
    7. Petra Andrlikova, 2014. "Is Barrier version of Merton model more realistic? Evidence from Europe," Proceedings of International Academic Conferences 0801868, International Institute of Social and Economic Sciences.
    8. Ilya Molchanov & Michael Schmutz, 2008. "Geometric extension of put-call symmetry in the multiasset setting," Papers 0806.4506, arXiv.org, revised Mar 2009.
    9. Bernd Engelmann & Matthias Fengler & Morten Nalholm & Peter Schwendner, 2006. "Static versus dynamic hedges: an empirical comparison for barrier options," Review of Derivatives Research, Springer, vol. 9(3), pages 239-264, November.
    10. Johannes Siven & Rolf Poulsen, 2009. "Auto-static for the people: risk-minimizing hedges of barrier options," Review of Derivatives Research, Springer, vol. 12(3), pages 193-211, October.
    11. Gabriel Drimus, 2010. "A forward started jump-diffusion model and pricing of cliquet style exotics," Review of Derivatives Research, Springer, vol. 13(2), pages 125-140, July.
    12. Hansjörg Albrecher & Philipp Mayer, 2010. "Semi-Static Hedging Strategies For Exotic Options," World Scientific Book Chapters, in: Rüdiger Kiesel & Matthias Scherer & Rudi Zagst (ed.), Alternative Investments And Strategies, chapter 14, pages 345-373, World Scientific Publishing Co. Pte. Ltd..

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    Keywords

    Barrier option; Static hedging;

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