Enhanced Sharpe Ratio Inference with Sensitivity Decay and Strategy Rotation

Overview

This repository extends the statistical framework established in "Sharpe Ratio Inference under GARCH Returns" (SSRN, 2026) by integrating a novel decision-theoretic layer: the Jessicka Formulation for sensitivity decay, edge rotation, and biological barriers in trading strategies.

While the original SSRN paper provides a rigorous method for calculating the asymptotic variance of the Sharpe ratio under heavy-tailed GARCH processes, it treats the strategy as static. This project addresses the critical gap: how to act when the theoretical variance is infinite or unstable? By modeling the trader's "edge" as a decaying biological response to repeated market exposure, we derive optimal rotation thresholds that maximize information gain and preserve capital in non-stationary regimes.

Key Contributions

1. Unified Framework: Blends stochastic volatility modeling (GARCH) with neural adaptation dynamics (Power-Law Decay).
2. Decision Rule: Derives a mathematically grounded rotation trigger $\theta$ based on optimal information gain theory ($0.3 \leq \theta \leq 0.6$).
3. Reproducibility: Provides a complete Jupyter notebook reproducing SSRN baseline results while demonstrating the out-of-sample superiority of the rotation-aware strategy.

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Theoretical Background

1. The Baseline: SSRN GARCH Inference

The foundational work, "A Closed-Form Solution for Sharpe Ratio Inference under GARCH Returns" by López de Prado, Porcu, Zoonekynd, and Engle (2026), establishes that for returns $r_t$ following a GARCH(1,1) process with standardized innovations having tail index $\kappa$:

$$ r_t = \sigma_t z_t, \quad \sigma_t^2 = \omega + \alpha r_{t-1}^2 + \beta \sigma_{t-1}^2 $$

where $z_t$ follows a distribution with power-law tails $P(|z| > x) \sim c x^{-\kappa}$.

Critical Insight: When $2 < \kappa < 4$, the fourth moment of returns is infinite ($E[r_t^4] = \infty$). Consequently, the standard asymptotic variance of the Sharpe ratio estimator diverges. The SSRN paper derives a corrected asymptotic variance $V_{GARCH}$ (Formula 15) dependent on $\kappa$:

$$ \sqrt{T}(\hat{SR} - SR) \xrightarrow{d} N(0, V_{GARCH}) $$

However, this inference assumes a static strategy. It tells us how uncertain our estimate is, but not when to stop using the strategy as market dynamics evolve.

2. The Extension: The Jessicka Formulation

The Sensitivity Decay Whitepaper posits that trading edge is not a fixed parameter but a dynamic state variable subject to habituation. Just as biological systems desensitize to repeated stimuli, a strategy's alpha decays with repeated exposure to the same market regime.

A. Power-Law Edge Decay

Instead of assuming constant expected return $\mu$, we model the effective edge $\mu_{eff}(n)$ after $n$ exposures (trades) as:

$$ \sigma(n) = (1 + \beta n)^{-\eta} $$ $$ \mu_{eff}(n) = \bar{\mu} \cdot \sigma(n) $$

Where:

• $\bar{\mu}$ is the ceiling edge (maximum initial alpha).
• $\eta$ is the decay exponent, linked directly to the tail index from the SSRN model: $$ \eta = 1 - \frac{2}{\kappa} $$ Rationale: Heavier tails (lower $\kappa$) imply faster structural instability and quicker edge decay.

B. Optimal Rotation Threshold

Drawing from optimal information gain theory (eLife, 2025), staying in a decaying strategy too long wastes energy (drawdown), while rotating too early incurs transaction costs and loses information. The optimal rotation threshold $\theta$ lies in the intermediate range:

$$ \text{Rotate if } \sigma(n) < \theta, \quad \theta \in [0.3, 0.6] $$

C. Overload Threshold & Position Sizing

To account for market-wide arousal (volatility clustering), we adjust the entry threshold $\tau_t$ and position size $w_t$:

$$ \tau_t = \tau_0 \left( 1 + \alpha_{load} \frac{\sigma_t}{\bar{\sigma}} \right) $$ $$ w_t \propto \sigma(n) $$

This ensures we trade smaller sizes as the edge decays and require stronger signals during high-volatility regimes.

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Methodology: Blending the Models

This project implements a two-stage simulation pipeline:

Stage 1: SSRN Baseline Reproduction

We simulate $N$ paths of GARCH(1,1) returns with $\kappa \in (2, 4)$.

1. Estimate parameters $(\omega, \alpha, \beta, \kappa)$ using functions.py.
2. Compute the sample variance of the Sharpe ratio across paths.
3. Verify convergence to the theoretical $V_{GARCH}$ from Formula 15.
4. Result: Confirms that standard inference fails without tail correction.

Stage 2: Jessicka Rotation Overlay

Using the same simulated paths:

1. Calibration: Estimate $\bar{\mu}$ and $\kappa$ from the first 100 trades (training window).
2. Decay Calculation: Compute $\eta = 1 - 2/\kappa$ and track $\sigma(n)$ in real-time.
3. Execution:
   • If $\sigma(n) &lt; \theta$: Rotate (stop strategy, reset P&L).
   • If $|r_t| &lt; \tau_t$: Skip (overload filter).
   • Else: Trade with size $w_t \propto \sigma(n)$.
4. Evaluation: Compare the final Sharpe ratio and Maximum Drawdown of the Rotating Strategy vs. Buy-and-Hold.

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Repository Structure

.
├── functions.py                  # Core GARCH utilities (SSRN)
├── Jessicka_Enhanced_Sharpe_Ratio.ipynb  # Main analysis notebook
├── Descriptive_Statistics.ipynb  # Original SSRN descriptive analysis
├── Simulation_with_skewness.ipynb # Original SSRN simulation baseline
├── README.md                     # This file
├── requirements.txt              # Dependencies
└── output/                       # Generated plots and summary stats
    ├── figure_1_ssrn_baseline.png
    ├── panel_b_sharpe_distribution.png
    ├── panel_c_decay_curve.png
    ├── panel_d_theta_sensitivity.png
    └── summary.txt

Installation & Usage

Prerequisites

Python 3.9+, with the following packages:

pip install numpy pandas matplotlib scipy arch statsmodels ray tqdm notebook papermill

Running the Analysis

1. Ensure functions.py is in the same directory.
2. Launch Jupyter:

   jupyter notebook Jessicka_Enhanced_Sharpe_Ratio.ipynb # has some minor issues
   streamlit run jessica_app.py # this one is better!

3. Run all cells. The notebook will:
   • Reproduce Figure 1 from the SSRN paper.
   • Simulate the rotation strategy.
   • Generate comparative panels (A-D).
   • Save high-resolution PNGs to the output/ folder.

Alternative: Batch Execution with Papermill

mkdir -p outputs
papermill Jessicka_Enhanced_Sharpe_Ratio.ipynb outputs/Jessicka_Enhanced_Sharpe_Ratio.ipynb

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Key Findings (Preliminary)

In regimes where $\kappa &lt; 4$ (infinite fourth moment):

1. Variance Reduction: The rotating strategy exhibits significantly lower variance in realized Sharpe ratios compared to buy-and-hold.
2. Drawdown Control: By exiting when $\sigma(n) &lt; 0.5$, the strategy avoids the "long tail" of decay where noise dominates signal.
3. Robustness: The power-law decay model fits empirical trade sequences better than exponential decay, aligning with neural adaptation literature.

Visual Summary

We provide a comprehensive infographic (before_after_infographic.png) that visualizes the reduction in estimator variance and the shift in performance distribution.

Figure: Comparison of Sharpe Ratio distributions and stability metrics between the static SSRN baseline and the adaptive Jessicka rotation strategy.

Pre-Analysis Plan (PAP) Compliance

This project adheres to the Double-Blind protocol outlined in the Whitepaper:

• Pre-registered Parameters: $\eta$ derivation, $\theta$ grid $[0.3, 0.5, 0.6]$, $\alpha_{load} = 0.5$.
• Data Splitting: Training (first 100 obs), Calibration (next 200), Holdout (remainder).
• No Look-Ahead: Decay states $\sigma(n)$ are computed strictly using past information.

Limitations & Future Work

• Estimation Error: The tail index $\kappa$ is noisy in finite samples; robust estimators (Hill estimator) are used but introduce variance.
• Regime Detection: Current implementation uses a simple exposure count $n$. Future versions will integrate Statistical Jump Models (JM) to detect regime changes $C_t$ explicitly, resetting $n$ only when the regime persists.
• Real Data Validation: While simulations match theoretical properties, validation on live hedge fund indices (e.g., HFR) is required to confirm the $\eta = 1 - 2/\kappa$ relationship.

References

1. López de Prado, M., Porcu, E., Zoonekynd, V., & Engle, R. F. (2026). A Closed-Form Solution for Sharpe Ratio Inference under GARCH Returns. SSRN. Link
2. Samokhvalov, E. (2025). From Arousal Decay to Edge Decay — A Unified Formulation for Markets. GitHub. Link
3. Nature Communications (2023). Power-law adaptation in primary visual cortex.
4. eLife (2025). Optimal information gain at intermediate habituation.

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License: MIT
Author: Eduard Samokhvalov
Date: 2025-02-19