A Generalization of the Classical Kelly Betting Formula to the Case of Temporal Correlation

Abstract

For sequential betting games, Kelly’s theory, aimed at maximization of the logarithmic growth of one’s account value, involves optimization of the so-called betting fraction $K$. In this paper, we extend the classical formulation to allow for temporal correlation among bets. For the example of successive coin flips with even-money payoff, used throughout the paper as an example demonstrating the potential for this new research direction, we formulate and solve a problem with memory depth $m$. By this, we mean that the outcomes of coin flips are no longer assumed to be i.i.d. random variables. Instead, the probability of heads on flip $k$ depends on previous flips $k-1,k-2,…,k-m$. For the simplest case of $n$ flips, even-money payoffs and $m = 1$, we obtain a closed form solution for the optimal betting fraction, denoted $K_n$, which generalizes the classical result for the memoryless case. That is, instead of fraction $K = 2p-1$ which pervades the literature for a coin with probability of heads $p \ge 1/2$, our new fraction $K_n$ depends on both $n$ and the parameters associated with the temporal correlation model. Subsequently, we obtain a generalization of these results for cases when $m > 1$ and illustrate the use of the theory in the context of numerical simulations.

Joey O'Brien

Senior Quantitative Analyst

Senior Consultant in quantitative finance and data science for Acadia, working in R and Python.