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On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

Author

Listed:
  • D. Madan

    (University of Maryland)

  • M. Pistorius

    (Imperial College London)

  • M. Stadje

    (Universität Ulm)

Abstract

In this paper, we propose the notion of continuous-time dynamic spectral risk measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions of iterated spectral risk measures, which are obtained by iterating a given spectral risk measure (such as expected shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk measures driven by lattice random walks, under suitable scaling and vanishing temporal and spatial mesh sizes. To illustrate its use in financial optimisation problems, we analyse a dynamic portfolio optimisation problem under a DSR.

Suggested Citation

  • D. Madan & M. Pistorius & M. Stadje, 2017. "On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation," Finance and Stochastics, Springer, vol. 21(4), pages 1073-1102, October.
    • Handle: RePEc:spr:finsto:v:21:y:2017:i:4:d:10.1007_s00780-017-0339-1
    DOI: 10.1007/s00780-017-0339-1
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    Cited by:

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    3. Dilip B. Madan & Wim Schoutens & King Wang, 2020. "Bilateral multiple gamma returns: Their risks and rewards," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 7(01), pages 1-27, March.
    4. Dilip B. Madan & Wim Schoutens, 2019. "Equilibrium Asset Returns In Financial Markets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(02), pages 1-43, March.
    5. Yoshihiro Shirai, 2022. "Extreme Measures in Continuous Time Conic Finace," Papers 2210.13671, arXiv.org, revised Oct 2023.
    6. Weiping Wu & Yu Lin & Jianjun Gao & Ke Zhou, 2023. "Mean-variance hybrid portfolio optimization with quantile-based risk measure," Papers 2303.15830, arXiv.org, revised Apr 2023.

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    More about this item

    Keywords

    Spectral risk measure; Dynamic risk measure; g $g$ -expectation; Choquet expectation; Distortion; (Strong) Time-consistency; Limit theorem; Dynamic portfolio optimisation;
    All these keywords.

    JEL classification:

    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill

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