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Peacocks nearby: Approximating sequences of measures

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  • Gerhold, Stefan
  • Gülüm, I. Cetin

Abstract

A peacock is a family of probability measures with finite mean that increases in convex order. It is a classical result, in the discrete time case due to Strassen, that any peacock is the family of one-dimensional marginals of a martingale. We study the problem whether a given sequence of probability measures can be approximated by a peacock. In our main results, the approximation quality is measured by the infinity Wasserstein distance. Existence of a peacock within a prescribed distance is reduced to a countable collection of rather explicit conditions. This result has a financial application (developed in a separate paper), as it allows to check European call option quotes for consistency. The distance bound on the peacock then takes the role of a bound on the bid–ask spread of the underlying. We also solve the approximation problem for the stop-loss distance, the Lévy distance, and the Prokhorov distance.

Suggested Citation

  • Gerhold, Stefan & Gülüm, I. Cetin, 2019. "Peacocks nearby: Approximating sequences of measures," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2406-2436.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:7:p:2406-2436
    DOI: 10.1016/j.spa.2018.07.007
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    References listed on IDEAS

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    1. Marco Scarsini & Alfred Muller, 2006. "Stochastic order relations and lattices of probability measures," Post-Print hal-00539119, HAL.
    2. Kaas, R. & van Heerwaarden, A. E. & Goovaerts, M. J., 1988. "On Stop-Loss Premiums for the Individual Model," ASTIN Bulletin, Cambridge University Press, vol. 18(1), pages 91-97, April.
    3. Mark H. A. Davis & David G. Hobson, 2007. "The Range Of Traded Option Prices," Mathematical Finance, Wiley Blackwell, vol. 17(1), pages 1-14, January.
    4. Alison L. Gibbs & Francis Edward Su, 2002. "On Choosing and Bounding Probability Metrics," International Statistical Review, International Statistical Institute, vol. 70(3), pages 419-435, December.
    5. De Pril, Nelson & Dhaene, Jan, 1992. "Error Bounds for Compound Poisson Approximations of the Individual Risk Model," ASTIN Bulletin, Cambridge University Press, vol. 22(2), pages 135-148, November.
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    Cited by:

    1. Beatrice Acciaio & Julio Backhoff & Gudmund Pammer, 2022. "Quantitative Fundamental Theorem of Asset Pricing," Papers 2209.15037, arXiv.org, revised Jan 2024.

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