GameTheoretic Optimal Portfolios in Continuous Time
Abstract
We consider a twoperson trading game in continuous time whereby each player chooses a constant rebalancing rule $b$ that he must adhere to over $[0,t]$. If $V_t(b)$ denotes the final wealth of the rebalancing rule $b$, then Player 1 (the `numerator player') picks $b$ so as to maximize $\mathbb{E}[V_t(b)/V_t(c)]$, while Player 2 (the `denominator player') picks $c$ so as to minimize it. In the unique Nash equilibrium, both players use the continuoustime Kelly rule $b^*=c^*=\Sigma^{1}(\mur\textbf{1})$, where $\Sigma$ is the covariance of instantaneous returns per unit time, $\mu$ is the drift vector of the stock market, and $\textbf{1}$ 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 (1988) result in discrete time.
 Publication:

arXiv eprints
 Pub Date:
 June 2019
 arXiv:
 arXiv:1906.02216
 Bibcode:
 2019arXiv190602216G
 Keywords:

 Quantitative Finance  Portfolio Management;
 Economics  General Economics;
 Economics  Theoretical Economics;
 Quantitative Finance  General Finance;
 Quantitative Finance  Mathematical Finance
 EPrint:
 14 pages, 1 figure