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On Robustness of Double Linear Policy with Time-Varying Weights

Xin-Yu Wang, Chung-Han Hsieh

TL;DR

The paper addresses the robustness of a double linear trading policy when weights are time-varying, proving that the robust positive expectation property persists in a discrete-time setting. It introduces an elementary symmetric-polynomial framework to derive closed-form expressions for the expected cumulative gain-loss $\overline{\mathcal{G}}$ and its variance, and establishes a key result that $\overline{\mathcal{G}}(k,\mu)>0$ for all $k>1$ with $\alpha=\tfrac{1}{2}$ and at least two positive weights, for any nonzero $\mu$. The authors provide explicit formulas for $\overline{R}_+(k)$ and $\overline{R}_-(k)$ and the gain-loss variance in terms of the time-varying weights, and validate the theory via Monte Carlo simulations across diverse weighting schemes, including moving-average based designs. The results demonstrate practical integration with standard technical analysis (e.g., moving averages) and offer a pathway to robust, time-adaptive trading strategies with potential extensions to multi-asset settings and scenarios with transaction costs.

Abstract

In this paper, we extend the existing double linear policy by incorporating time-varying weights instead of constant weights and study a certain robustness property, called robust positive expectation (RPE), in a discrete-time setting. We prove that the RPE property holds by employing a novel elementary symmetric polynomials characterization approach and derive an explicit expression for both the expected cumulative gain-loss function and its variance. To validate our theory, we perform extensive Monte Carlo simulations using various weighting functions. Furthermore, we demonstrate how this policy can be effectively incorporated with standard technical analysis techniques, using the moving average as a trading signal.

On Robustness of Double Linear Policy with Time-Varying Weights

TL;DR

The paper addresses the robustness of a double linear trading policy when weights are time-varying, proving that the robust positive expectation property persists in a discrete-time setting. It introduces an elementary symmetric-polynomial framework to derive closed-form expressions for the expected cumulative gain-loss and its variance, and establishes a key result that for all with and at least two positive weights, for any nonzero . The authors provide explicit formulas for and and the gain-loss variance in terms of the time-varying weights, and validate the theory via Monte Carlo simulations across diverse weighting schemes, including moving-average based designs. The results demonstrate practical integration with standard technical analysis (e.g., moving averages) and offer a pathway to robust, time-adaptive trading strategies with potential extensions to multi-asset settings and scenarios with transaction costs.

Abstract

In this paper, we extend the existing double linear policy by incorporating time-varying weights instead of constant weights and study a certain robustness property, called robust positive expectation (RPE), in a discrete-time setting. We prove that the RPE property holds by employing a novel elementary symmetric polynomials characterization approach and derive an explicit expression for both the expected cumulative gain-loss function and its variance. To validate our theory, we perform extensive Monte Carlo simulations using various weighting functions. Furthermore, we demonstrate how this policy can be effectively incorporated with standard technical analysis techniques, using the moving average as a trading signal.
Paper Structure (9 sections, 4 theorems, 24 equations, 4 figures, 2 tables)

This paper contains 9 sections, 4 theorems, 24 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contributions of the Paper
  3. Problem Formulation
  4. Double Linear Policy with Time-Varying Weights
  5. Survivability Considerations
  6. Robust Positive Expectation Problem
  7. Gain-Loss Analysis
  8. Illustrative Examples
  9. Concluding Remarks

Key Result

Lemma 3.2

Fix $m \geq 1.$ Let $\alpha \in (0,1)$ and $w(k) \in \mathcal{W}$ for all $k$, $\overline{R}_+(k)$ and $\overline{R}_-(k)$ for stage $k = 2m+1$ satisfies On the other hand, for $k =2m$, $\overline{R}_+(k)$ and $\overline{R}_-(k)$ satisfies

Figures (4)

  • Figure 1: Weighting Functions (Left) and Expected Gain-Loss for $\mu^* \in (-1, 1)$.
  • Figure 2: Twitter Minutely Prices from May 4, 2022 to May 19, 2022.
  • Figure 3: Cumulative Gain-Loss Using Twitter Minute-by-Minute Data.
  • Figure 4: Cumulative Gain-Loss Using Various MA Indicators.

Theorems & Definitions (15)

  • Definition 2.1: Robust Positive Expectation
  • Example 3.1: Elementary Symmetric Polynomials
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4: RPE with Time-Varying Weights
  • proof
  • Remark 3.5
  • Lemma 3.6: Variance of the Gain-Loss Function
  • ...and 5 more