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Game-theoretic optimal portfolios in continuous time

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  • Alex Garivaltis

    (Northern Illinois University)

Abstract

We consider a two-person trading game in continuous time where each player chooses a constant rebalancing rule b that he must adhere to over [0, t]. If $$V_t(b)$$ V t ( b ) denotes the final wealth of the rebalancing rule b, then Player 1 (the “numerator player”) picks b so as to maximize $$E[V_t(b)/V_t(c)]$$ E [ V t ( b ) / V t ( c ) ] , while Player 2 (the “denominator player”) picks c so as to minimize it. In the unique Nash equilibrium, both players use the continuous-time Kelly rule $$b^*=c^*=\varSigma ^{-1}(\mu -r\mathbf 1 )$$ b ∗ = c ∗ = Σ - 1 ( μ - r 1 ) , where $$\varSigma $$ Σ is the covariance of instantaneous returns per unit time, $$\mu $$ μ is the drift vector, and $$\mathbf 1 $$ 1 is a vector of ones. Thus, even over very short intervals of time [0, t], the desire to perform well relative to other traders leads one to adopt the Kelly rule, which is ordinarily derived by maximizing the asymptotic exponential growth rate of wealth. Hence, we find agreement with Bell and Cover’s ( Manag Sci 34(6):724–733, 1988) result in discrete time.

Suggested Citation

  • Alex Garivaltis, 2019. "Game-theoretic optimal portfolios in continuous time," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 7(2), pages 235-243, December.
  • Handle: RePEc:spr:etbull:v:7:y:2019:i:2:d:10.1007_s40505-018-0156-5
    DOI: 10.1007/s40505-018-0156-5
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    References listed on IDEAS

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    1. Robert M. Bell & Thomas M. Cover, 1980. "Competitive Optimality of Logarithmic Investment," Mathematics of Operations Research, INFORMS, vol. 5(2), pages 161-166, May.
    2. Robert Bell & Thomas M. Cover, 1988. "Game-Theoretic Optimal Portfolios," Management Science, INFORMS, vol. 34(6), pages 724-733, June.
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    More about this item

    Keywords

    Portfolio choice; Constant rebalanced portfolios; Continuous-time Kelly rule; Minimax;
    All these keywords.

    JEL classification:

    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory
    • D80 - Microeconomics - - Information, Knowledge, and Uncertainty - - - General
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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