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Probabilistic aspects of finance


  • Hans Follmer
  • Alexander Schied


In the past decades, advanced probabilistic methods have had significant impact on the field of finance, both in academia and in the financial industry. Conversely, financial questions have stimulated new research directions in probability. In this survey paper, we review some of these developments and point to some areas that might deserve further investigation. We start by reviewing the basics of arbitrage pricing theory, with special emphasis on incomplete markets and on the different roles played by the "real-world" probability measure and its equivalent martingale measures. We then focus on the issue of model ambiguity, also called Knightian uncertainty. We present two case studies in which it is possible to deal with Knightian uncertainty in mathematical terms. The first case study concerns the hedging of derivatives, such as variance swaps, in a strictly pathwise sense. The second one deals with capital requirements and preferences specified by convex and coherent risk measures. In the final two sections we discuss mathematical issues arising from the dramatic increase of algorithmic trading in modern financial markets.

Suggested Citation

  • Hans Follmer & Alexander Schied, 2013. "Probabilistic aspects of finance," Papers 1309.7759,
  • Handle: RePEc:arx:papers:1309.7759

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    References listed on IDEAS

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    Cited by:

    1. Stoyan V. Stoyanov & Svetlozar T. Rachev & Stefan Mittnik & Frank J. Fabozzi, 2017. "Pricing derivatives in Hermite markets," Papers 1709.09068,
    2. Alexander Schied, 2015. "On a class of generalized Takagi functions with linear pathwise quadratic variation," Papers 1501.00837,, revised Aug 2015.
    3. Sebastian Herrmann & Johannes Muhle-Karbe & Frank Thomas Seifried, 2016. "Hedging with Small Uncertainty Aversion," Papers 1605.06429,
    4. Nicolas Perkowski & David J. Promel, 2014. "Local times for typical price paths and pathwise Tanaka formulas," Papers 1405.4421,, revised Apr 2015.
    5. Sebastian Herrmann & Johannes Muhle-Karbe & Frank Thomas Seifried, 2017. "Hedging with small uncertainty aversion," Finance and Stochastics, Springer, vol. 21(1), pages 1-64, January.
    6. Knispel, Thomas & Laeven, Roger J.A. & Svindland, Gregor, 2016. "Robust optimal risk sharing and risk premia in expanding pools," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 182-195.
    7. repec:eee:insuma:v:75:y:2017:i:c:p:180-188 is not listed on IDEAS

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