arXiv subjects

Chung-Han Hsieh

Publications and source records attributed to Chung-Han Hsieh.

32 records · Page 2Linked to original sources

arXiv2022

On Solving Robust Log-Optimal Portfolio: A Supporting Hyperplane Approximation Approach

A {log-optimal} portfolio is any portfolio that maximizes the expected logarithmic growth (ELG) of an investor's wealth. This maximization problem typically assumes that the information of the true distribution of returns is known to the trader in advance. However, in practice, the return distributions are indeed {ambiguous}; i.e., the true distribution is unknown to the trader or it is partially known at best. To this end, a {distributional robust log-optimal portfolio problem} formulation arises naturally. While the problem formulation takes into account the ambiguity on return distributions, the problem needs not to be tractable in general. To address this, in this paper, we propose a {supporting hyperplane approximation} approach that allows us to reformulate a class of distributional robust log-optimal portfolio problems into a linear program, which can be solved very efficiently. Our framework is flexible enough to allow {transaction costs}, {leverage and shorting}, {survival trades}, and {diversification considerations}. In addition, given an acceptable approximation error, an efficient algorithm for rapidly calculating the optimal number of hyperplanes is provided. Some empirical studies using historical stock price data are also provided to support our theory.

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arXiv2021

On Asymptotic Log-Optimal Buy-and-Hold Strategy

In this paper, we consider a frequency-based portfolio optimization problem with $m \geq 2$ assets when the expected logarithmic growth (ELG) rate of wealth is used as the performance metric. With the aid of the notion called dominant asset, it is known that the optimal ELG level is achieved by investing all available funds on that asset. However, such an "all-in" strategy is arguably too risky to implement in practice. Motivated by this issue, we study the case where the portfolio weights are chosen in a rather ad-hoc manner and a buy-and-hold strategy is subsequently used. Then we show that, if the underlying portfolio contains a dominant asset, buy and hold on that specific asset is asymptotically log-optimal with a sublinear rate of convergence. This result also extends to the scenario where a trader either does not have a probabilistic model for the returns or does not trust a model obtained from historical data. To be more specific, we show that if a market contains a dominant asset, buy and hold a market portfolio involving nonzero weights for each asset is asymptotically log-optimal. Additionally, this paper also includes a conjecture regarding the property called high-frequency maximality. That is, in the absence of transaction costs, high-frequency rebalancing is unbeatable in the ELG sense. Support for the conjecture, involving a lemma for a weak version of the conjecture, is provided. This conjecture, if true, enables us to improve the log-optimality result obtained previously. Finally, a result that indicates a way regarding an issue about when should one to rebalance their portfolio if needed, is also provided. Examples, some involving simulations with historical data, are also provided along the way to illustrate the~theory.

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arXiv2020

On Control of Epidemics with Application to COVID-19

At the time of writing, the ongoing COVID-19 pandemic, caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), had already resulted in more than thirty-two million cases infected and more than one million deaths worldwide. Given the fact that the pandemic is still threatening health and safety, it is in the urgency to understand the COVID-19 contagion process and know how it might be controlled. With this motivation in mind, in this paper, we consider a version of a stochastic discrete-time Susceptible-Infected-Recovered-Death~(SIRD)-based epidemiological model with two uncertainties: The uncertain rate of infected cases which are undetected or asymptomatic, and the uncertain effectiveness rate of control. Our aim is to study the effect of an epidemic control policy on the uncertain model in a control-theoretic framework. We begin by providing the closed-form solutions of states in the modified SIRD-based model such as infected cases, susceptible cases, recovered cases, and deceased cases. Then, the corresponding expected states and the technical lower and upper bounds for those states are provided as well. Subsequently, we consider two epidemic control problems to be addressed: One is almost sure epidemic control problem and the other average epidemic control problem. Having defined the two problems, our main results are a set of sufficient conditions on a class of linear control policy which assures that the epidemic is "well-controlled"; i.e., both of the infected cases and deceased cases are upper bounded uniformly and the number of infected cases converges to zero asymptotically. Our numerical studies, using the historical COVID-19 contagion data in the United States, suggest that our appealingly simple model and control framework can provide a reasonable epidemic control performance compared to the ongoing pandemic situation.

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arXiv2020

On Feedback Control in Kelly Betting: An Approximation Approach

In this paper, we consider a simple discrete-time optimal betting problem using the celebrated Kelly criterion, which calls for maximization of the expected logarithmic growth of wealth. While the classical Kelly betting problem can be solved via standard concave programming technique, an alternative but attractive approach is to invoke a Taylor-based approximation, which recasts the problem into quadratic programming and obtain the closed-form approximate solution. The focal point of this paper is to fill some voids in the existing results by providing some interesting properties when such an approximate solution is used. Specifically, the best achievable betting performance, positivity of expected cumulative gain or loss and its associated variance, expected growth property, variance of logarithmic growth, and results related to the so-called survivability (no bankruptcy) are provided.

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arXiv2020

Generalization of Affine Feedback Stock Trading Results to Include Stop-Loss Orders

The takeoff point of this paper is to generalize the existing stock trading results for a class of affine feedback controller to include consideration of a stop-loss order. Using the geometric Brownian motion as the underlying stock price model, our main result is to provide a closed-form expression for the cumulative distribution function for the trading profit or loss. In addition, we show that the affine feedback controller with stop-loss order indeed generalizes the result without stop order in the sense of distribution function. Some simulations and illustrative examples are also provided as supporting evidence of the theory. Moreover, we provide some technical results aimed at addressing the issues about survivability, cash-financing considerations, long-only property, and lower bound of the expected gain or loss.

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arXiv2020

Necessary and Sufficient Conditions for Frequency-Based Kelly Optimal Portfolio

In this paper, we consider a discrete-time portfolio with $m \geq 2$ assets optimization problem which includes the rebalancing~frequency as an additional parameter in the maximization. The so-called Kelly Criterion is used as the performance metric; i.e., maximizing the expected logarithmic growth of a trader's account, and the portfolio obtained is called the frequency-based Kelly optimal portfolio. The focal point of this paper is to extend upon the results of our previous work to obtain various optimality characterizations on the portfolio. To be more specific, using Kelly's criterion in our frequency-based formulation, we first prove necessary and sufficient conditions for the frequency-based Kelly optimal portfolio. With the aid of these conditions, we then show several new optimality characterizations such as expected ratio optimality and asymptotic relative optimality, and a result which we call the Extended Dominant Asset Theorem. That is, we prove that the $i$th asset is dominant in the portfolio if and only if the Kelly optimal portfolio consists of that asset only. The word "extended" on the theorem comes from the fact that it was only a sufficiency result that was proved in our previous work. Hence, in this paper, we improve it to involve a proof of the necessity part. In addition, the trader's survivability issue (no bankruptcy consideration) is also studied in detail in our frequency-based trading framework. Finally, to bridge the theory and practice, we propose a simple trading algorithm using the notion called dominant asset condition to decide when should one triggers a trade. The corresponding trading performance using historical price data is reported as supporting evidence.

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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

On Inefficiency of Markowitz-Style Investment Strategies When Drawdown is Important

The focal point of this paper is the issue of "drawdown" which arises in recursive betting scenarios and related applications in the stock market. Roughly speaking, drawdown is understood to mean drops in wealth over time from peaks to subsequent lows. Motivated by the fact that this issue is of paramount concern to conservative investors, we dispense with the classical variance as the risk metric and work with drawdown and mean return as the risk-reward pair. In this setting, the main results in this paper address the so-called "efficiency" of linear time-invariant (LTI) investment feedback strategies which correspond to Markowitz-style schemes in the finance literature. Our analysis begins with the following principle which is widely used in finance: Given two investment opportunities, if one of them has higher risk and lower return, it will be deemed to be inefficient or strictly dominated and generally rejected in the marketplace. In this framework, with risk-reward pair as described above, our main result is that classical Markowitz-style strategies are inefficient. To establish this, we use a new investment strategy which involves a time-varying linear feedback block K(k), called the drawdown modulator. Using this instead of the original LTI feedback block K in the Markowitz scheme, the desired domination is obtained. As a bonus, it is also seen that the modulator assures a worst-case level of drawdown protection with probability one.

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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

On Drawdown-Modulated Feedback Control in Stock Trading

Control of drawdown, that is, the control of the drops in wealth over time from peaks to subsequent lows, is of great concern from a risk management perspective. With this motivation in mind, the focal point of this paper is to address the drawdown issue in a stock trading context. Although our analysis can be carried out without reference to control theory, to make the work accessible to this community, we use the language of feedback systems. The takeoff point for the results to follow, which we call the Drawdown Modulation Lemma, characterizes any investment which guarantees that the percentage drawdown is no greater than a prespecified level with probability one. With the aid of this lemma, we introduce a new scheme which we call the drawdown-modulated feedback control. To illustrate the power of the theory, we consider a drawdown-constrained version of the well-known Kelly Optimization Problem which involves maximizing the expected logarithmic growth of the trader's account value. As the drawdown parameter dmax in our new formulation tends to one, we recover existing results as a special case. This new theory leads to an optimal investment strategy whose application is illustrated via an example with historical stock-price data.

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arXiv2017

On Kelly Betting: Some Limitations

The focal point of this paper is the so-called Kelly Criterion, a prescription for optimal resource allocation among a set of gambles which are repeated over time. The criterion calls for maximization of the expected value of the logarithmic growth of wealth. While significant literature exists providing the rationale for such an optimization, this paper concentrates on the limitations of the Kelly-based theory. To this end, we fill a void in published results by providing specific examples quantifying what difficulties are encountered when Taylor-style approximations are used and when wealth drawdowns are considered. For the case of drawdown, we describe some research directions which we feel are promising for improvement of the theory.

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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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