Annals of Operations Research

, Volume 266, Issue 1–2, pp 129–157 | Cite as

An analytical approximation for single barrier options under stochastic volatility models

  • Hideharu FunahashiEmail author
  • Tomohide Higuchi
Analytical Models for Financial Modeling and Risk Management


The aim of this paper is to derive an approximation formula for a single barrier option under local volatility models, stochastic volatility models, and their hybrids, which are widely used in practice. The basic idea of our approximation is to mimic a target underlying asset process by a polynomial of the Wiener process. We then translate the problem of solving first hit probability of the asset process into that of a Wiener process whose distribution of passage time is known. Finally, utilizing the Girsanov’s theorem and the reflection principle, we show that single barrier option prices can be approximated in a closed-form. Furthermore, ample numerical examples will show the accuracy of our approximation is high enough for practical applications.


Single barrier option Analytical approximation Local and stochastic volatility models Wiener–Ito chaos expansion 



The authors is grateful to the anonymous referees for invaluable comments that improved the original manuscript considerably. Funahashi also thanks Tetsuhiro Takeshita, QDS Consulting, for his careful reading of this manuscript. Needless to say, all errors and confusions are ours.


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Mizuho Securities Co. Ltd.TokyoJapan
  2. 2.Mizuho Securities Co. Ltd.TokyoJapan

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