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A Note on Universal Bilinear Portfolios

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  • Alex Garivaltis

Abstract

This note provides a neat and enjoyable expansion and application of the magnificent Ordentlich-Cover theory of "universal portfolios." I generalize Cover's benchmark of the best constant-rebalanced portfolio (or 1-linear trading strategy) in hindsight by considering the best bilinear trading strategy determined in hindsight for the realized sequence of asset prices. A bilinear trading strategy is a mini two-period active strategy whose final capital growth factor is linear separately in each period's gross return vector for the asset market. I apply Cover's ingenious (1991) performance-weighted averaging technique to construct a universal bilinear portfolio that is guaranteed (uniformly for all possible market behavior) to compound its money at the same asymptotic rate as the best bilinear trading strategy in hindsight. Thus, the universal bilinear portfolio asymptotically dominates the original (1-linear) universal portfolio in the same technical sense that Cover's universal portfolios asymptotically dominate all constant-rebalanced portfolios and all buy-and-hold strategies. In fact, like so many Russian dolls, one can get carried away and use these ideas to construct an endless hierarchy of ever more dominant $H$-linear universal portfolios.

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  • Alex Garivaltis, 2019. "A Note on Universal Bilinear Portfolios," Papers 1907.09704, arXiv.org, revised Oct 2022.
    • Handle: RePEc:arx:papers:1907.09704
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    References listed on IDEAS

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    1. Alex Garivaltis, 2018. "Exact Replication of the Best Rebalancing Rule in Hindsight," Papers 1810.02485, arXiv.org, revised Mar 2019.
    2. Henry Allen Latane, 1959. "Criteria for Choice Among Risky Ventures," Journal of Political Economy, University of Chicago Press, vol. 67(2), pages 144-144.
    3. Paul A. Samuelson, 2011. "Why We Should Not Make Mean Log of Wealth Big Though Years to Act Are Long," World Scientific Book Chapters, in: Leonard C MacLean & Edward O Thorp & William T Ziemba (ed.), THE KELLY CAPITAL GROWTH INVESTMENT CRITERION THEORY and PRACTICE, chapter 34, pages 491-493, World Scientific Publishing Co. Pte. Ltd..
    4. Paul A. Samuelson, 2011. "Lifetime Portfolio Selection by Dynamic Stochastic Programming," World Scientific Book Chapters, in: Leonard C MacLean & Edward O Thorp & William T Ziemba (ed.), THE KELLY CAPITAL GROWTH INVESTMENT CRITERION THEORY and PRACTICE, chapter 31, pages 465-472, World Scientific Publishing Co. Pte. Ltd..
    5. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    6. Alex Garivaltis, 2019. "Game-Theoretic Optimal Portfolios for Jump Diffusions," Games, MDPI, vol. 10(1), pages 1-9, February.
    7. Alex Garivaltis, 2019. "Game-Theoretic Optimal Portfolios in Continuous Time," Papers 1906.02216, arXiv.org, revised Oct 2022.
    8. Erik Ordentlich & Thomas M. Cover, 1998. "The Cost of Achieving the Best Portfolio in Hindsight," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 960-982, November.
    9. Alex Garivaltis, 2018. "Multilinear Superhedging of Lookback Options," Papers 1810.02447, arXiv.org, revised Oct 2022.
    10. Thomas M. Cover, 1991. "Universal Portfolios," Mathematical Finance, Wiley Blackwell, vol. 1(1), pages 1-29, January.
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