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Polynomial processes in stochastic portfolio theory

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  • Cuchiero, Christa

Abstract

We introduce polynomial processes in the context of stochastic portfolio theory to model simultaneously companies’ market capitalizations and the corresponding market weights. These models substantially extend volatility stabilized market models considered in Fernholz and Karatzas (2005), in particular they allow for correlation between the individual stocks. At the same time they remain remarkably tractable which makes them applicable in practice, especially for estimation and calibration to high dimensional equity index data. In the diffusion case we characterize the polynomial property of the market capitalizations and their weights, exploiting the fact that the transformation between absolute and relative quantities perfectly fits the structural properties of polynomial processes. Explicit parameter conditions assuring the existence of a local martingale deflator and relative arbitrages with respect to the market portfolio are given and the connection to non-attainment of the boundary of the unit simplex is discussed. We also consider extensions to models with jumps and the computation of optimal relative arbitrage strategies.

Suggested Citation

  • Cuchiero, Christa, 2019. "Polynomial processes in stochastic portfolio theory," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1829-1872.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:5:p:1829-1872
    DOI: 10.1016/j.spa.2018.06.007
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    References listed on IDEAS

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    Cited by:

    1. Oleg S. Sukharev, 2020. "Portfolio Theory in Solving the Problem Structural Choice," JRFM, MDPI, vol. 13(9), pages 1-21, September.
    2. David Itkin & Benedikt Koch & Martin Larsson & Josef Teichmann, 2022. "Ergodic robust maximization of asymptotic growth under stochastic volatility," Papers 2211.15628, arXiv.org.
    3. Christa Cuchiero & Sara Svaluto-Ferro, 2021. "Infinite-dimensional polynomial processes," Finance and Stochastics, Springer, vol. 25(2), pages 383-426, April.
    4. Andrew L. Allan & Christa Cuchiero & Chong Liu & David J. Promel, 2021. "Model-free Portfolio Theory: A Rough Path Approach," Papers 2109.01843, arXiv.org, revised Oct 2022.
    5. Christa Cuchiero & Walter Schachermayer & Ting‐Kam Leonard Wong, 2019. "Cover's universal portfolio, stochastic portfolio theory, and the numéraire portfolio," Mathematical Finance, Wiley Blackwell, vol. 29(3), pages 773-803, July.
    6. M.E. Mancino & S. Scotti & G. Toscano, 2020. "Is the Variance Swap Rate Affine in the Spot Variance? Evidence from S&P500 Data," Applied Mathematical Finance, Taylor & Francis Journals, vol. 27(4), pages 288-316, July.
    7. David Itkin & Martin Larsson, 2021. "On A Class Of Rank-Based Continuous Semimartingales," Papers 2104.04396, arXiv.org.
    8. Filipović, Damir & Larsson, Martin & Pulido, Sergio, 2020. "Markov cubature rules for polynomial processes," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 1947-1971.
    9. Christa Cuchiero & Janka Moller, 2023. "Signature Methods in Stochastic Portfolio Theory," Papers 2310.02322, arXiv.org, revised Mar 2024.
    10. Christa Cuchiero & Sara Svaluto-Ferro, 2019. "Infinite dimensional polynomial processes," Papers 1911.02614, arXiv.org.
    11. David Itkin & Martin Larsson, 2020. "Robust Asymptotic Growth in Stochastic Portfolio Theory under Long-Only Constraints," Papers 2009.08533, arXiv.org, revised Aug 2021.
    12. Andrew L. Allan & Christa Cuchiero & Chong Liu & David J. Prömel, 2023. "Model‐free portfolio theory: A rough path approach," Mathematical Finance, Wiley Blackwell, vol. 33(3), pages 709-765, July.
    13. Schmidt, Thorsten & Tappe, Stefan & Yu, Weijun, 2020. "Infinite dimensional affine processes," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7131-7169.
    14. David Itkin & Martin Larsson, 2021. "Open Markets and Hybrid Jacobi Processes," Papers 2110.14046, arXiv.org, revised Mar 2024.
    15. Fred Espen Benth & Silvia Lavagnini, 2019. "Correlators of Polynomial Processes," Papers 1906.11320, arXiv.org, revised Apr 2021.
    16. Paolo Guasoni & Kwok Chuen Wong, 2020. "Asset prices in segmented and integrated markets," Finance and Stochastics, Springer, vol. 24(4), pages 939-980, October.
    17. David Itkin & Martin Larsson, 2022. "Robust asymptotic growth in stochastic portfolio theory under long‐only constraints," Mathematical Finance, Wiley Blackwell, vol. 32(1), pages 114-171, January.

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