arXiv subjects

John A. Gubner

Publications and source records attributed to John A. Gubner.

5 recordsLinked to original sources

arXiv2019

On Positive Solutions of a Delay Equation Arising When Trading in Financial Markets

We consider a discrete-time, linear state equation with delay which arises as a model for a trader's account value when buying and selling a risky asset in a financial market. The state equation includes a nonnegative feedback gain $α$ and a sequence $v(k)$ which models asset returns which are within known bounds but otherwise arbitrary. We introduce two thresholds, $α_-$ and $α_+$, depending on these bounds, and prove that for $α< α_-$, state positivity is guaranteed for all time and all asset-return sequences; i.e., bankruptcy is ruled out and positive solutions of the state equation are continuable indefinitely. On the other hand, for $α> α_+$, we show that there is always a sequence of asset returns for which the state fails to be positive for all time; i.e., along this sequence, bankruptcy is certain and the solution of the state equation ceases to be meaningful after some finite time. Finally, this paper also includes a conjecture which says that for the "gap" interval $α_- \leq α\leq α_+,$ state positivity is also guaranteed for all time. Support for the conjecture, both theoretical and computational, is provided.

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arXiv2019

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

Stock trading based on Kelly's celebrated Expected Logarithmic Growth (ELG) criterion, a well-known prescription for optimal resource allocation, has received considerable attention in the literature. Using ELG as the performance metric, we compare the impact of trade execution delay on the relative performance of high-frequency trading versus buy and hold. While it is intuitively obvious and straightforward to prove that in the presence of sufficiently high transaction costs, buy and hold is the better strategy, is it possible that with no transaction costs, buy and hold can still be the better strategy? When there is no delay in trade execution, we prove a theorem saying that the answer is ``no.'' However, when there is delay in trade execution, we present simulation results using a binary lattice stock model to show that the answer can be ``yes.'' This is seen to be true whether self-financing is imposed or not.

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arXiv2018

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

In this paper, motivated by the celebrated work of Kelly, we consider the problem of portfolio weight selection to maximize expected logarithmic growth. Going beyond existing literature, our focal point here is the rebalancing frequency which we include as an additional parameter in our analysis. The problem is first set in a control-theoretic framework, and then, the main question we address is as follows: In the absence of transaction costs, does high-frequency trading always lead to the best performance? Related to this is our prior work on betting, also in the Kelly context, which examines the impact of making a wager and letting it ride. Our results on betting frequency can be interpreted in the context of weight selection for a two-asset portfolio consisting of one risky asset and one riskless asset. With regard to the question above, our prior results indicate that it is often the case that there are no performance benefits associated with high-frequency trading. In the present paper, we generalize the analysis to portfolios with multiple risky assets. We show that if there is an asset satisfying a new condition which we call dominance, then an optimal portfolio consists of this asset alone; i.e., the trader has "all eggs in one basket" and performance becomes a constant function of rebalancing frequency. Said another way, the problem of rebalancing is rendered moot. The paper also includes simulations which address practical considerations associated with real stock prices and the dominant asset condition.

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arXiv2018

At What Frequency Should the Kelly Bettor Bet?

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 gn*, change with n? Does the high-frequency 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) in {-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, gn* 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.

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arXiv2017

Kelly Betting Can Be Too Conservative

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 counterpart Xhat. Subsequently, we derive and compare the resulting Kelly bets for both X and Xhat 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 for Xhat. 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.

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