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Subleading correction to the Asian options volatility in the Black-Scholes model

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  • Dan Pirjol

Abstract

The short maturity limit $T\to 0$ for the implied volatility of an Asian option in the Black-Scholes model is determined by the large deviations property for the time-average of the geometric Brownian motion. In this note we derive the subleading $O(T)$ correction to this implied volatility, using an asymptotic expansion for the Hartman-Watson distribution. The result is used to compute subleading corrections to Asian options prices in a small maturity expansion, sharpening the leading order result obtained using large deviations theory. We demonstrate good numerical agreement with precise benchmarks for Asian options pricing in the Black-Scholes model.

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  • Dan Pirjol, 2024. "Subleading correction to the Asian options volatility in the Black-Scholes model," Papers 2407.05142, arXiv.org, revised Aug 2024.
    • Handle: RePEc:arx:papers:2407.05142
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    References listed on IDEAS

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    1. Yuecai Han & Chunyang Liu, 2018. "Asian Option Pricing under Uncertain Volatility Model," Papers 1808.00656, arXiv.org.
    2. Hélyette Geman & Marc Yor, 1993. "Bessel Processes, Asian Options, And Perpetuities," Mathematical Finance, Wiley Blackwell, vol. 3(4), pages 349-375, October.
    3. Daniel Dufresne, 2000. "Laguerre Series for Asian and Other Options," Mathematical Finance, Wiley Blackwell, vol. 10(4), pages 407-428, October.
    4. Louis-Pierre Arguin & Nien-Lin Liu & Tai-Ho Wang, 2017. "Most-likely-path in Asian option pricing under local volatility models," Papers 1706.02408, arXiv.org, revised Aug 2018.
    5. Dan Pirjol & Jing Wang & Lingjiong Zhu, 2019. "Short Maturity Forward Start Asian Options in Local Volatility Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 26(3), pages 187-221, May.
    6. Dan Pirjol & Lingjiong Zhu, 2017. "Asymptotics for the Discrete-Time Average of the Geometric Brownian Motion and Asian Options," Papers 1706.09659, arXiv.org.
    7. Dan Pirjol & Lingjiong Zhu, 2018. "Sensitivities Of Asian Options In The Black–Scholes Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(01), pages 1-25, February.
    8. Louis-Pierre Arguin & Nien-Lin Liu & Tai-Ho Wang, 2018. "Most-Likely-Path In Asian Option Pricing Under Local Volatility Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(05), pages 1-32, August.
    9. Vadim Linetsky, 2004. "Spectral Expansions for Asian (Average Price) Options," Operations Research, INFORMS, vol. 52(6), pages 856-867, December.
    10. Dan Pirjol & Lingjiong Zhu, 2016. "Short Maturity Asian Options in Local Volatility Models," Papers 1609.07559, arXiv.org.
    11. Kun Gao & Roger Lee, 2014. "Asymptotics of implied volatility to arbitrary order," Finance and Stochastics, Springer, vol. 18(2), pages 349-392, April.
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