Complexity-Based Framework for Trading

• The framework leverages algorithmic information theory and lossless compression to quantify hidden structures in financial time series.
• It employs an iterative regularity erasing procedure by transforming returns into discrete symbols and measuring compressibility to detect subtle market patterns.
• Empirical findings indicate low residual compressibility after filtering stylized facts, supporting market efficiency and limiting exploitable trading edges.

A complexity-based framework for algorithmic trading is a class of models and methodologies that leverages the principles and measures of computational and statistical complexity to analyze, characterize, and potentially exploit regularities in financial time series. Unlike conventional approaches grounded in probability theory or closed-form statistical modeling, complexity-based frameworks apply tools from algorithmic information theory, dynamical systems, lossless compression, agent-based modeling, evolutionary computing, and multi-agent coordination to capture deep structural properties of financial data—often invisible to classical statistical tests—with the aim of providing new angles for prediction, characterization, or systematic trading.

1. Foundations: Algorithmic Information Theory and Kolmogorov Complexity

The core theoretical foundation of the classical complexity-based framework in finance is algorithmic information theory, and specifically Kolmogorov complexity. Formally, the Kolmogorov complexity $K(s)$ of a binary string $s$ is defined as the length of the shortest universal program that generates $s$. In the context of financial series, this is used to provide a non-probabilistic measure of the "randomness" or "regularity" present in observed market sequences.

A string $s$ of length $n$ is called algorithmically random if $K(s) \geq n - C$ for some constant $C$ and all $n$. In such a setting, no algorithm can compress $s$ to a length significantly shorter than $n$, implying that $s$ lacks exploitable regularities.

In practice, Kolmogorov complexity is uncomputable. Therefore, frameworks utilize lossless compression algorithms (e.g., Huffman encoding, Gzip, PAQ8o8) to empirically estimate compressibility, using the achieved compression ratio as a practical proxy for the string's inherent complexity (Brandouy et al., 2015).

2. Methodology: Regularity Erasing and Empirical Complexity Estimation

The methodology for measuring and exploiting complexity in financial time series is structured as an iterative "regularity erasing procedure" (REP):

1. Transformation to Returns: Raw price series are first transformed into returns (e.g., via logarithmic differences), which themselves act as a form of lossy compression by focusing on market changes.
2. Lossless Discretization: Returns, which are real-valued, are mapped into discrete symbols (e.g., quantizing to 256 bins for 8-bit representation) in a way that is lossless for the chosen compression algorithms. This process ensures that all intrinsic information is preserved up to the desired granularity.
3. Iterative REP and Compression: The discretized sequence is subjected to standard compression. If the series is compressible, it harbors regularities—structural patterns, volatility clustering, or regimes—that are "erased" in each iteration. Subsequent rounds seek to compress away known stylized facts, isolating residual, potentially novel structure.
4. Block-by-Block/Local REP: For well-known stylized phenomena such as volatility clustering, progressive or block-wise discretization is applied (e.g., using moving windows) to remove the impact of local variance effects, targeting structures undetectable by standard econometric/auto-correlation tests.

The amount of compression achieved in each round is used as an empirical measure of the intrinsic complexity left in the time series, with minimal compressibility indicating near-randomness and inefficiency (Brandouy et al., 2015).

3. Empirical Applications: Finance and Market Data

Application of this framework to real and synthetic financial series reveals several key results:

• Dow Jones Index Example: Uniformly discretized returns of the Dow Jones Index show weak compressibility (~0.82% with PAQ8o8) attributed largely to volatility clustering.
• Progressive Discretization: When volatility clustering is accounted for and regularized out, compressibility drops even further (~0.27%), signaling an extremely high Kolmogorov complexity.
• Detection of Hidden Regularities: Compression-based techniques sometimes detect structural features or patterns that are entirely invisible to typical statistical or econometric tests (unit-root, autocorrelation, BDS test), suggesting that algorithmic methods may be more sensitive to fine-grained or nonlinear information structures.

However, in mature markets, after stylized facts are removed, the residual complexity remains so high as to preclude robust trading edges, supporting a version of the Efficient Market Hypothesis at high levels of abstraction (Brandouy et al., 2015).

4. Implications for Algorithmic Trading

The direct implications for systematic trading are multifaceted:

• Signal Discovery: Compression rates act as indicators of residual information that could, in principle, be transformed into predictive signals or exploited as market inefficiencies.
• Risk Characterization: Complexity can correlate with "information risk"—regimes with lower complexity might signal more predictable (and thus riskier) structure.
• Limits of Predictability: In markets where REP leaves strings almost incompressible, any trading strategy seeking to exploit statistical or algorithmic regularities is constrained to at best ephemeral and weak effects.
• Non-Probabilistic Edge: Complexity-based probes require no strong distributional or stationarity assumptions and can be effective even in one-off (non-ergodic) market scenarios.

Despite these strengths, the practical yield for profitable trading is limited by the vanishing compressibility observed in highly-efficient electronic markets after standard regularities are filtered out (Brandouy et al., 2015).

5. Comparative Advantages and Limitations

A direct comparison with probabilistic and conventional machine learning methods highlights characteristic strengths and weaknesses:

Feature	Compression-Based	Probabilistic/ML
Distributional Assumptions	None	Often required
Detection of Hidden Structure	High (for some forms)	Limited by test scope
One-off Applicability	Yes	Rare
Quantification of Structure	Direct via compression ratio	Indirect (e.g., p-values)
Absolute Testability	No (depends on choice of universal language and constant c in the invariance theorem)	Sometimes yes (unbiased estimators)
Scalability	Limited (computation-intensive, algorithm-dependent)	Often scalable with high-performance implementations

The reliance on text-oriented compression for numeric data introduces calibration challenges. Not all compressible structures are detectable (e.g., patterns implicit in the digits of $\pi$ remain undetected by general compressors), and small residual effects from compression may not be robust enough for systematic exploitation in high-frequency contexts (Brandouy et al., 2015).

6. Extensions, Generalizations, and Future Directions

The complexity-based framework establishes a foundation for further research and method development:

• Development of Specialized Financial Compressors: Existing algorithms are not designed for the structural properties of financial time series. Compression schemes tailored to financial regime switching, high-frequency stylized facts, and microstructural features offer a promising direction.
• Complexity as Risk Metric: Measuring time-varying empirical Kolmogorov complexity could inform dynamic portfolio allocation or position sizing, offering a non-parametric alternative to volatility and drawdown metrics.
• Integration with Hybrid Modeling: Algorithmic complexity measures may be combined with agent-based simulations, deep learning, and evolutionary models for richer regime detection and regime-specific strategy adaptation.
• Non-Efficient and Frictional Markets: Secondary, illiquid, or structurally inefficient markets may harbor more exploitable algorithmic structure than liquid developed markets, warranting targeted empirical investigation.
• Systematic Comparison with Other Approaches: Direct empirical tests contrasting signal strengths found by complexity-based methods versus deep neural architectures or RL-based strategies remain an open area for research.

• "Estimating the Algorithmic Complexity of Stock Markets" (Brandouy et al., 2015).

In summary, the complexity-based framework for algorithmic trading interprets financial time series through the lens of non-probabilistic algorithmic complexity, operationalized via iterative regularity erasure and empirical lossless compression. Its principal contribution is the capacity to reveal hidden structural regularities and quantify the limits of predictability without prior distributional assumptions; however, empirical findings generally support the market's high complexity, implying that, barring the development of more sophisticated compressors or an application to structurally inefficient regimes, robust systematic trading edges will be correspondingly rare.