Kelly betting can be too conservative

Abstract

Kelly betting is a prescription for optimal resource allocation among a set of gambles which are typically repeated in an independent and identically distributed manner. In this setting, there is a large body of literature which includes arguments that the theory often leads to bets which are "too aggressive" with respect to various risk metrics. To remedy this problem, many papers include prescriptions for scaling down the bet size. Such schemes are referred to as Fractional Kelly Betting. In this paper, we take the opposite tack. That is, we show that in many cases, the theoretical Kelly-based results may lead to bets which are "too conservative" rather than too aggressive. To make this argument, we consider a random vector X with its assumed probability distribution and draw m samples to obtain an empirically-derived counterpartX. Subsequently, we derive and compare the resulting Kelly bets for both X andX with consideration of sample size m as part of the analysis. This leads to identification of many cases which have the following salient feature: The resulting bet size using the true theoretical distribution for X is much smaller than that forX. If instead the bet is based on empirical data, "golden" opportunities are identified which are essentially rejected when the purely theoretical model is used. To formalize these ideas, we provide a result which we call the Restricted Betting Theorem. An extreme case of the theorem is obtained when X has unbounded support. In this situation, using X, the Kelly theory can lead to no betting at all.

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

“…It is also worth mentioning that the paradigm presented in this paper remains valid for a wide variety of other choices for the admissible feedback gain set K associated with leveraged investments with some components K i > 1. For example, in [16], this is achieved by imposing a "survival" constraint which disallows any trade that can potentially lead to V (k) < 0.…”

“…The appeal of this research to the control community is based on the fact that the Kelly-based rebalancing problem can be formulated as a stochastic control problem with a linear feedback and randomly varying inputs corresponding to the vector of stagewise returns X(k) on portfolio assets; see [15]- [16], [18]- [20], and [22] where a similar controltheoretic set-up is considered for finance problems in continuous time. To study the effect of rebalancing frequency for portfolio problems, let ∆t be the time between portfolio updates.…”

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.…”

Convert index trading to option strategies via LSTM architecture

“…The Kelly criterion can be used in situations such as gambling that can be played unlimited times and has a fixed probability of winning and odds [41], in the process of repeated betting to maximize the growth of his assets [20,30,34,42]. However, there is a gap between the financial market and the gambling [16], so many scholars have begun to explore how to use the Kelly formula in a more general situation [4,6,9,11,43].…”