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Semi-closed form solutions for barrier and American options written on a time-dependent Ornstein Uhlenbeck process

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  • Peter Carr
  • Andrey Itkin

Abstract

In this paper we develop a semi-closed form solutions for the barrier (perhaps, time-dependent) and American options written on the underlying stock which follows a time-dependent OU process with a log-normal drift. This model is equivalent to the familiar Hull-White model in FI, or a time dependent OU model in FX. Semi-closed form means that given the time-dependent interest rate, continuous dividend and volatility functions, one need to solve numerically a linear (for the barrier option) or nonlinear (for the American option) Fredholm equation of the first kind. After that the option prices in all cases are presented as one-dimensional integrals of combination of the above solutions and Jacobi theta functions. We also demonstrate that computationally our method is more efficient than the backward finite difference method used for solving these problems, and can also be as efficient as the forward finite difference solver while providing better accuracy and stability.

Suggested Citation

  • Peter Carr & Andrey Itkin, 2020. "Semi-closed form solutions for barrier and American options written on a time-dependent Ornstein Uhlenbeck process," Papers 2003.08853, arXiv.org, revised Mar 2020.
    • Handle: RePEc:arx:papers:2003.08853
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    File URL: http://arxiv.org/pdf/2003.08853
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    References listed on IDEAS

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    1. Alexander Lipton & Marcos Lopez de Prado, 2020. "A closed-form solution for optimal mean-reverting trading strategies," Papers 2003.10502, arXiv.org.
    2. Aleksandar Mijatović, 2010. "Local time and the pricing of time-dependent barrier options," Finance and Stochastics, Springer, vol. 14(1), pages 13-48, January.
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    Cited by:

    1. A. Itkin & A. Lipton & D. Muravey, 2020. "From the Black-Karasinski to the Verhulst model to accommodate the unconventional Fed's policy," Papers 2006.11976, arXiv.org, revised Jan 2021.
    2. Andrey Itkin & Dmitry Muravey, 2020. "Semi-analytic pricing of double barrier options with time-dependent barriers and rebates at hit," Papers 2009.09342, arXiv.org, revised Oct 2020.
    3. Peter Carr & Andrey Itkin & Dmitry Muravey, 2020. "Semi-closed form prices of barrier options in the time-dependent CEV and CIR models," Papers 2005.05459, arXiv.org.
    4. Andrey Itkin & Dmitry Muravey, 2020. "Semi-closed form prices of barrier options in the Hull-White model," Papers 2004.09591, arXiv.org, revised Sep 2020.
    5. Lazar, Emese & Qi, Shuyuan, 2022. "Model risk in the over-the-counter market," European Journal of Operational Research, Elsevier, vol. 298(2), pages 769-784.

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