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Pricing Perpetual American Put Options with Asset-Dependent Discounting

Author

Listed:
  • Jonas Al-Hadad

    (Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, ul. Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
    These authors contributed equally to this work.)

  • Zbigniew Palmowski

    (Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, ul. Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
    These authors contributed equally to this work.)

Abstract

The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as V A Put ω ( s ) = sup τ ∈ T E s [ e − ∫ 0 τ ω ( S w ) d w ( K − S τ ) + ] , where T is a family of stopping times, ω is a discount function and E is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process S t is a geometric Lévy process with negative exponential jumps, i.e., S t = s e ζ t + σ B t − ∑ i = 1 N t Y i . The asset-dependent discounting is reflected in the ω function, so this approach is a generalisation of the classic case when ω is constant. It turns out that under certain conditions on the ω function, the value function V A Put ω ( s ) is convex and can be represented in a closed form. We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of ω such that V A Put ω ( s ) takes a simplified form.

Suggested Citation

  • Jonas Al-Hadad & Zbigniew Palmowski, 2021. "Pricing Perpetual American Put Options with Asset-Dependent Discounting," JRFM, MDPI, vol. 14(3), pages 1-19, March.
    • Handle: RePEc:gam:jjrfmx:v:14:y:2021:i:3:p:130-:d:520879
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    References listed on IDEAS

    as
    1. Marzia De Donno & Zbigniew Palmowski & Joanna Tumilewicz, 2020. "Double continuation regions for American and Swing options with negative discount rate in Lévy models," Mathematical Finance, Wiley Blackwell, vol. 30(1), pages 196-227, January.
    2. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    3. Loeffen, Ronnie L. & Renaud, Jean-François & Zhou, Xiaowen, 2014. "Occupation times of intervals until first passage times for spectrally negative Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 124(3), pages 1408-1435.
    4. Jonas Al-Hadad & Zbigniew Palmowski, 2020. "Perpetual American options with asset-dependent discounting," Papers 2007.09419, arXiv.org, revised Jan 2021.
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    1. Jonas Al-Hadad & Zbigniew Palmowski, 2021. "Pricing Perpetual American put options with asset-dependent discounting," Papers 2103.02948, arXiv.org.

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