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Closed form valuation of barrier options with stochastic barriers

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  • Tristan Guillaume

    (Université de Cergy-Pontoise)

Abstract

This article deals with the computation of the probability, for a GBM (geometric Brownian motion) process, to hit sequences of one-sided stochastic boundaries defined as GBM processes, over a closed time interval. Explicit formulae are obtained, allowing the analytical valuation of all the main kinds of barrier options in a much more general setting than the usual one assuming constant or time-dependent, deterministic barriers. The numerical implementation of all stated formulae is shown to be easy, fast and accurate. The practical applications are potentially substantial, since barrier options play a major role in quantitative finance, not only as intensively traded contracts on their own, but also as the building blocks of a large variety of structured products. Barrier options are also an important tool in financial modelling, used to measure default risk in the so-called “structural” models.

Suggested Citation

  • Tristan Guillaume, 2022. "Closed form valuation of barrier options with stochastic barriers," Annals of Operations Research, Springer, vol. 313(2), pages 1021-1050, June.
    • Handle: RePEc:spr:annopr:v:313:y:2022:i:2:d:10.1007_s10479-020-03860-w
    DOI: 10.1007/s10479-020-03860-w
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    References listed on IDEAS

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    1. Szu-Lang Liao & Hsing-Hua Huang, 2005. "Pricing Black-Scholes options with correlated interest rate risk and credit risk: an extension," Quantitative Finance, Taylor & Francis Journals, vol. 5(5), pages 443-457.
    2. Hideharu Funahashi & Tomohide Higuchi, 2018. "An analytical approximation for single barrier options under stochastic volatility models," Annals of Operations Research, Springer, vol. 266(1), pages 129-157, July.
    3. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    4. Klein, Peter & Inglis, Michael, 2001. "Pricing vulnerable European options when the option's payoff can increase the risk of financial distress," Journal of Banking & Finance, Elsevier, vol. 25(5), pages 993-1012, May.
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