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An analytical approximation for single barrier options under stochastic volatility models

Author

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  • Hideharu Funahashi

    (Mizuho Securities Co. Ltd.)

  • Tomohide Higuchi

    (Mizuho Securities Co. Ltd.)

Abstract

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.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:annopr:v:266:y:2018:i:1:d:10.1007_s10479-017-2559-3
    DOI: 10.1007/s10479-017-2559-3
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    References listed on IDEAS

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    1. Gobet, Emmanuel, 2000. "Weak approximation of killed diffusion using Euler schemes," Stochastic Processes and their Applications, Elsevier, vol. 87(2), pages 167-197, June.
    2. Hideharu Funahashi & Masaaki Kijima, 2016. "Analytical pricing of single barrier options under local volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 867-886, June.
    3. Hideharu Funahashi, 2014. "A chaos expansion approach under hybrid volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 14(11), pages 1923-1936, November.
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    5. Fang, Fang & Oosterlee, Kees, 2008. "A Novel Pricing Method For European Options Based On Fourier-Cosine Series Expansions," MPRA Paper 9319, University Library of Munich, Germany.
    6. Hideharu Funahashi, 2017. "Pricing derivatives with fractional volatility," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 1-28, March.
    7. Emanuel, David C. & MacBeth, James D., 1982. "Further Results on the Constant Elasticity of Variance Call Option Pricing Model," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 17(4), pages 533-554, November.
    8. Hideharu Funahashi & Masaaki Kijima, 2017. "A unified approach for the pricing of options relating to averages," Review of Derivatives Research, Springer, vol. 20(3), pages 203-229, October.
    9. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    10. Carl Chiarella & Boda Kang & Gunter H. Meyer, 2010. "The Evaluation Of Barrier Option Prices Under Stochastic Volatility," Research Paper Series 266, Quantitative Finance Research Centre, University of Technology, Sydney.
    11. 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.
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    Cited by:

    1. Jiling Cao & Xi Li & Wenjun Zhang, 2023. "Pricing Path-Dependent Options under Stochastic Volatility via Mellin Transform," JRFM, MDPI, vol. 16(10), pages 1-17, October.
    2. Frido Rolloos & Kenichiro Shiraya, 2024. "A model‐free approximation for barrier options in a general stochastic volatility framework," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 44(6), pages 923-935, June.
    3. Tristan Guillaume, 2022. "Closed form valuation of barrier options with stochastic barriers," Annals of Operations Research, Springer, vol. 313(2), pages 1021-1050, June.
    4. Weilong Fu & Ali Hirsa, 2022. "Solving barrier options under stochastic volatility using deep learning," Papers 2207.00524, arXiv.org.
    5. Jiling Cao & Jeong-Hoon Kim & Xi Li & Wenjun Zhang, 2022. "Pricing Path-dependent Options under Stochastic Volatility via Mellin Transform," Papers 2205.00573, arXiv.org.
    6. Gurjeet Dhesi & Bilal Shakeel & Marcel Ausloos, 2021. "Modelling and forecasting the kurtosis and returns distributions of financial markets: irrational fractional Brownian motion model approach," Annals of Operations Research, Springer, vol. 299(1), pages 1397-1410, April.
    7. Xin-Jiang He & Sha Lin, 2022. "An Analytical Approximation Formula for Barrier Option Prices Under the Heston Model," Computational Economics, Springer;Society for Computational Economics, vol. 60(4), pages 1413-1425, December.

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