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The British Knock-Out Put Option

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  • LULUWAH AL-FAGIH

    (School of Mathematics, Kingston University London, KT1 2EE, United Kingdom;
    School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom)

Abstract

Following the economic rationale introduced by Peskir & Samee (2011, 2013) we present a new class of barrier options within the British payoff mechanism where the holder enjoys the early exercise feature of American type options whereupon his payoff (deliverable immediately) is the best prediction of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Should the option holder believe the true drift of the stock price to be unfavorable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimize his losses. In this paper, we focus on the knock-out put option with an up barrier. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterized as the unique solution to a nonlinear integral equation. Using these results, we perform a financial analysis of the British knock-out put option. We spot some of the trends previously seen in Peskir & Samee (2011) but observe some behavior unique to the knock-out case. Finally, we derive the British put-call and up-down symmetry relations which express the arbitrage-free price and the rational exercise boundary of the British down-and-out call option in terms of the arbitrage-free price and the rational exercise boundary of the British up-and-out put option.

Suggested Citation

  • Luluwah Al-Fagih, 2015. "The British Knock-Out Put Option," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(02), pages 1-32.
    • Handle: RePEc:wsi:ijtafx:v:18:y:2015:i:02:n:s0219024915500089
    DOI: 10.1142/S0219024915500089
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    References listed on IDEAS

    as
    1. Kristoffer Glover & Goran Peskir & Farman Samee, 2010. "The British Russian Option," Research Paper Series 269, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. Goran Peskir & Farman Samee, 2013. "The British call option," Quantitative Finance, Taylor & Francis Journals, vol. 13(1), pages 95-109, January.
    3. 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..
    4. Goran Peskir & Farman Samee, 2011. "The British Put Option," Applied Mathematical Finance, Taylor & Francis Journals, vol. 18(6), pages 537-563, April.
    5. Jacques du Toit & Goran Peskir, 2009. "Selling a stock at the ultimate maximum," Papers 0908.1014, arXiv.org.
    6. Pavel V. Gapeev, 2006. "Discounted Optimal Stopping for Maxima of some Jump-Diffusion Processes," SFB 649 Discussion Papers SFB649DP2006-059, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    7. Kristoffer Glover & Goran Peskir & Farman Samee, 2009. "The British Asian Option," Research Paper Series 249, Quantitative Finance Research Centre, University of Technology, Sydney.
    8. Min Dai & Yue Kuen Kwok, 2004. "Knock‐in American options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 24(2), pages 179-192, February.
    9. Peter Carr & Roger Lee, 2009. "Volatility Derivatives," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 319-339, November.
    10. Pavel V. Gapeev, 2006. "Discounted Optimal Stopping for Maxima in Diffusion Models with Finite Horizon," SFB 649 Discussion Papers SFB649DP2006-057, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
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    Cited by:

    1. Min Gao, 2017. "The British Asset-Or-Nothing Put Option," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(04), pages 1-19, June.

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