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Game-Theoretic Optimal Portfolios for Jump Diffusions

Author

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  • Alex Garivaltis

    (Department of Economics, Northern Illinois University, 514 Zulauf Hall, DeKalb, IL 60115, USA)

Abstract

This paper studies a two-person trading game in continuous time that generalizes Garivaltis (2018) to allow for stock prices that both jump and diffuse. Analogous to Bell and Cover (1988) in discrete time, the players start by choosing fair randomizations of the initial dollar, by exchanging it for a random wealth whose mean is at most 1. Each player then deposits the resulting capital into some continuously rebalanced portfolio that must be adhered to over [ 0 , t ] . We solve the corresponding “investment ? -game”, namely the zero-sum game with payoff kernel E [ ? { W 1 V t ( b ) / ( W 2 V t ( c ) ) } ] , where W i is player i ’s fair randomization, V t ( b ) is the final wealth that accrues to a one dollar deposit into the rebalancing rule b , and ? ( • ) is any increasing function meant to measure relative performance. We show that the unique saddle point is for both players to use the (leveraged) Kelly rule for jump diffusions, which is ordinarily defined by maximizing the asymptotic almost-sure continuously compounded capital growth rate. Thus, the Kelly rule for jump diffusions is the correct behavior for practically anybody who wants to outperform other traders (on any time frame) with respect to practically any measure of relative performance.

Suggested Citation

  • Alex Garivaltis, 2019. "Game-Theoretic Optimal Portfolios for Jump Diffusions," Games, MDPI, vol. 10(1), pages 1-9, February.
    • Handle: RePEc:gam:jgames:v:10:y:2019:i:1:p:8-:d:205587
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    References listed on IDEAS

    as
    1. 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..
    2. Alex Garivaltis, 2019. "Game-Theoretic Optimal Portfolios in Continuous Time," Papers 1906.02216, arXiv.org, revised Oct 2022.
    3. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
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    Cited by:

    1. Alex Garivaltis, 2019. "Nash Bargaining Over Margin Loans to Kelly Gamblers," Risks, MDPI, vol. 7(3), pages 1-14, August.
    2. Alex Garivaltis, 2019. "A Note on Universal Bilinear Portfolios," Papers 1907.09704, arXiv.org, revised Oct 2022.
    3. Alex Garivaltis, 2021. "A Note on Universal Bilinear Portfolios," IJFS, MDPI, vol. 9(1), pages 1-17, February.

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    More about this item

    Keywords

    portfolio choice; continuously rebalanced portfolios; Kelly criterion; log-optimal investment; minimax; jump processes;
    All these keywords.

    JEL classification:

    • C - Mathematical and Quantitative Methods
    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

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