At What Frequency Should the Kelly Bettor Bet?

Abstract

We study the problem of optimizing the betting frequency in a dynamic game setting using Kelly's celebrated expected logarithmic growth criterion as the performance metric. The game is defined by a sequence of bets with independent and identically distributed returns X(k). The bettor selects the fraction of wealth K wagered at k = 0 and waits n steps before updating the bet size. Between updates, the proceeds from the previous bets remain at risk in the spirit of "buy and hold." Within this context, the main questions we consider are as follows: How does the optimal performance, we call it g * n , change with n? Does the highfrequency case, n = 1, always lead to the best performance? What are the effects of accrued interest and transaction costs? First, we provide rather complete answers to these questions for the important special case when X(k) ∈ {−1, 1} is a Bernoulli random variable with probability p that X(k) = 1. This serves as an entry point for future research using a binomial lattice model for stock trading. The latter sections focus on more general probability distributions for X(k) and two conjectures. The first conjecture is simple to state: Absent transaction costs, g * n is non-increasing in n. The second conjecture involves the technical condition which we call the sufficient attractiveness inequality. We first prove that satisfaction of this inequality is sufficient to guarantee that the low-frequency bettor using large n can match the performance of the high-frequency bettor using n = 1. Subsequently, we conjecture, and provide supporting evidence that this condition is also necessary.

Rebalancing Frequency Considerations for Kelly-Optimal Stock Portfolios in a Control-Theoretic Framework

“…However, for the zero transaction-cost case, we also showed that it is possible to obtain g * 1 = g * n for all n ≥ 1 although it is still an open question whether g * n > g * 1 is possible. For this case, in the sequel, we generalize these results in [21] to the multiple-risky-asset case, and prove that there are many scenarios where the lowfrequency trader's performance can actually match that of the high-frequency trader -the extreme case with n very large corresponding to buy and hold. This performance matching is proven when at least one of the assets in the portfolio is dominant in the sense that it is relatively more attractive than every other potential asset under consideration.…”

“…Most closely related to this paper is our recent work [21] which considers a repeated betting game and the impact on expected logarithmic growth resulting from making a wager and letting it ride for several steps in lieu of updating. This can be interpreted as weight selection for a two-asset portfolio and "letting it ride" to capture the effect of the frequency of rebalancing.…”

In-game betting and the Kelly criterion

“…The reason is that the mean is much more important for determining the optimal fractions than the variance [14]. This was also indicated by Hsieh et al [12], who considered the optimal frequency for updating the Kelly fractions in the case where the sequence of games corresponds to i.i.d. random variables.…”

The Impact of Execution Delay on Kelly-Based Stock Trading: High-Frequency Versus Buy and Hold

“…Together, these imply that we also require V (k) ≥ 0. Now in the Kelly framework, discussed later in this section, the trader's investment level is given by a linear feedback I(k) = KV (k); e.g., see [8] and [12]- [14]. Hence, the long-only condition leads to the requirement K ≥ 0.…”

“…This paper is most closely related to more recent work such as [10]- [13] which provide results on the effect of rebalancing frequency on optimal trading performance. Specifically, in [12] and [13], it is shown that in a so-called idealized market with a stock satisfying a certain "sufficient attractiveness" condition, the buy and holder can match the performance of the high-frequency trader. Additionally, in [12], it was shown that when transaction costs are added into the mix, consistent with intuition, the buy and holder can strictly outperform the high-frequency trader.…”