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On peacocks and lyrebirds: Australian options, Brownian bridges, and the average of submartingales

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  • Christian†Oliver Ewald
  • Marc Yor

Abstract

We introduce a class of stochastic processes, which we refer to as lyrebirds. These extend a class of stochastic processes, which have recently been coined peacocks, but are more commonly known as processes that are increasing in the convex order. We show how these processes arise naturally in the context of Asian and Australian options and consider further applications, such as the arithmetic average of a Brownian bridge and the average of submartingales, including the case of Asian and Australian options where the underlying features constant elasticity of variance or is of Merton jump diffusion type.

Suggested Citation

  • Christian†Oliver Ewald & Marc Yor, 2018. "On peacocks and lyrebirds: Australian options, Brownian bridges, and the average of submartingales," Mathematical Finance, Wiley Blackwell, vol. 28(2), pages 536-549, April.
    • Handle: RePEc:bla:mathfi:v:28:y:2018:i:2:p:536-549
    DOI: 10.1111/mafi.12144
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    Cited by:

    1. Erhan Bayraktar & Shuoqing Deng & Dominykas Norgilas, 2023. "Supermartingale Brenier’s Theorem with Full-Marginal Constraint," World Scientific Book Chapters, in: Robert A Jarrow & Dilip B Madan (ed.), Peter Carr Gedenkschrift Research Advances in Mathematical Finance, chapter 17, pages 569-636, World Scientific Publishing Co. Pte. Ltd..
    2. Michael R. Tehranchi, 2020. "A Black–Scholes inequality: applications and generalisations," Finance and Stochastics, Springer, vol. 24(1), pages 1-38, January.
    3. Liu, Yating & Pagès, Gilles, 2022. "Monotone convex order for the McKean–Vlasov processes," Stochastic Processes and their Applications, Elsevier, vol. 152(C), pages 312-338.

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