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Nash Bargaining Over Margin Loans to Kelly Gamblers

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  • Alex Garivaltis

    (Department of Economics, School of Public and Global Affairs, College of Liberal Arts and Sciences, Northern Illinois University, 514 Zulauf Hall, DeKalb, IL 60115, USA)

Abstract

I derive practical formulas for optimal arrangements between sophisticated stock market investors (continuous-time Kelly gamblers or, more generally, CRRA investors) and the brokers who lend them cash for leveraged bets on a high Sharpe asset (i.e., the market portfolio). Rather than, say, the broker posting a monopoly price for margin loans, the gambler agrees to use a greater quantity of margin debt than he otherwise would in exchange for an interest rate that is lower than the broker would otherwise post. The gambler thereby attains a higher asymptotic capital growth rate and the broker enjoys a greater rate of intermediation profit than would be obtained under non-cooperation. If the threat point represents a complete breakdown of negotiations (resulting in zero margin loans), then we get an elegant rule of thumb: r L * = 3 / 4 r + 1 / 4 ν − σ 2 / 2 , where r is the broker’s cost of funds, ν is the compound-annual growth rate of the market index, and σ is the annual volatility. We show that, regardless of the particular threat point, the gambler will negotiate to size his bets as if he himself could borrow at the broker’s call rate.

Suggested Citation

  • Alex Garivaltis, 2019. "Nash Bargaining Over Margin Loans to Kelly Gamblers," Risks, MDPI, vol. 7(3), pages 1-14, August.
  • Handle: RePEc:gam:jrisks:v:7:y:2019:i:3:p:93-:d:261355
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    References listed on IDEAS

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    1. Nash, John, 1950. "The Bargaining Problem," Econometrica, Econometric Society, vol. 18(2), pages 155-162, April.
    2. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    3. Alex Garivaltis, 2019. "Game-Theoretic Optimal Portfolios for Jump Diffusions," Games, MDPI, vol. 10(1), pages 1-9, February.
    4. Alex Garivaltis, 2019. "Game-Theoretic Optimal Portfolios in Continuous Time," Papers 1906.02216, arXiv.org, revised Oct 2022.
    5. 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.
    6. Alex Garivaltis, 2018. "Exact Replication of the Best Rebalancing Rule in Hindsight," Papers 1810.02485, arXiv.org, revised Mar 2019.
    7. 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.
    8. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    9. Alex Garivaltis, 2019. "Cover's Rebalancing Option With Discrete Hindsight Optimization," Papers 1903.00829, arXiv.org, revised Oct 2022.
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    Cited by:

    1. Alex Garivaltis, 2019. "The Laws of Motion of the Broker Call Rate in the United States," IJFS, MDPI, vol. 7(4), pages 1-23, October.

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    More about this item

    Keywords

    Nash bargaining; margin loans; Kelly betting; log-optimal portfolios; continuously-rebalanced portfolios; net interest margin;
    All these keywords.

    JEL classification:

    • C - Mathematical and Quantitative Methods
    • G0 - Financial Economics - - General
    • G1 - Financial Economics - - General Financial Markets
    • G2 - Financial Economics - - Financial Institutions and Services
    • G3 - Financial Economics - - Corporate Finance and Governance
    • M2 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Business Economics
    • M4 - Business Administration and Business Economics; Marketing; Accounting; Personnel Economics - - Accounting
    • K2 - Law and Economics - - Regulation and Business Law

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