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Dynamic Mode Decomposition for Financial Trading Strategies
Jordan Mann, J. Nathan Kutz
TL;DR
Financial markets are treated as complex, non-stationary dynamical systems for which conventional model specification is limited. The paper applies equation-free DMD to portfolio data, using low-rank modes and learned sampling windows to guide trading decisions. Across sectors, identified hot-spots support profitable strategies, while performance is not guaranteed when hot-spots are absent.
Problem
The paper addresses how to model complex, non-stationary financial dynamics without prescribing known governing laws and extract patterns relevant to trading.
Method
DMD approximates market dynamics from data using low-rank modes, eigenvalue-based temporal structure, adaptive windows, and a learning algorithm that identifies trading hot-spots.
Results
Hot-spots allowed the strategy to beat the S&P 5000 in the reported transport, home-construction, and retail cases, while home construction achieved 21.48% annualized over 10 years with the best input.
Takeaways & Limitations
DMD provides a data-driven framework for sector-specific buy, sell, or hold decisions by adapting sampling and prediction windows to detected market patterns.
Takeaways & Limitations
DMD fails when the data matrix lacks suitable low-rank structure, and performance is not guaranteed when no clear trading hot-spot is present.
Abstract
from arXiv · showhide
We demonstrate the application of an algorithmic trading strategy based upon the recently developed dynamic mode decomposition (DMD) on portfolios of financial data. The method is capable of characterizing complex dynamical systems, in this case financial market dynamics, in an equation-free manner by decomposing the state of the system into low-rank terms whose temporal coefficients in time are known. By extracting key temporal coherent structures (portfolios) in its sampling window, it provides a regression to a best fit linear dynamical system, allowing for a predictive assessment of the market dynamics and informing an investment strategy. The data-driven analytics capitalizes on stock market patterns, either real or perceived, to inform buy/sell/hold investment decisions. Critical to the method is an associated learning algorithm that optimizes the sampling and prediction windows of the algorithm by discovering trading hot-spots. The underlying mathematical structure of the algorithms is rooted in methods from nonlinear dynamical systems and shows that the decomposition is an effective mathematical tool for data-driven discovery of market patterns.
1. Introduction.
The introduction presents DMD as an equation-free, data-driven framework for modeling non-stationary financial dynamics and extracting actionable market patterns. It combines adaptive decomposition with learning-based window optimization to generate trading strategies across sectors.
- Motivation and approach: DMD reconstructs market dynamics directly from sampled data without prescribing an underlying dynamical model.It decomposes portfolio data into low-rank features with prescribed temporal dynamics.
- Adaptive trading design: A learning algorithm tracks slowly changing optimal sampling and prediction windows across different market sectors.The paper applies the method to technology, biotechnology, transportation, and banking markets.
- Methodological novelty: The method captures temporal and spatial structure simultaneously, unlike standalone time-series or PCA analyses.Its low-rank features can represent transient and non-stationary phenomena across multiple market timescales.
- Adaptive trading design: Adaptive sampling and prediction windows support distinct high-frequency, daily, and long-term trading strategies.An iterative refinement process optimizes the snapshot window for future-market prediction.
- Financial setting: DMD is positioned as a natural financial-modeling tool because markets lack known laws of nature and require statistical approaches.Its equation-free formulation provides short-time future estimates and coherent-structure assessments from sampled data.
2. Dynamic Mode Decomposition: An Equation-Free Architecture.
This section formulates DMD as a data-driven approximation of unknown nonlinear dynamics using measured snapshots and a fitted linear evolution. Its low-rank modes provide dimensionality reduction, time-scale decomposition, and short-term prediction for portfolio decisions.
- Equation-free formulation: The measured state x represents a portfolio of companies, while the underlying process may be nonlinear, time-dependent, and parameter-dependent.The measurement operator G(·) supplies observations of the evolving system.
- Equation-free formulation: DMD approximates unknown nonlinear dynamics from snapshot measurements and initial conditions rather than an explicitly specified governing model.The procedure constructs a proxy linear evolution from the available data.
- Dynamic modes: DMD decomposes snapshot data into dynamic modes whose eigenvalues and eigenfunctions encode growth rates and frequencies.The resulting linear model approximates nonlinear dynamics and connects modes to prescribed time scales.
- Financial interpretation: Positive real parts of DMD eigenvalues identify exponentially growing portfolio modes, whereas negative real parts identify exponentially decreasing modes.The paper uses these eigenvalue signs to inform investment strategies.
- Approximation and reduction: The approximation is optimal only over its sampling window, although it can still predict future states and separate dynamics across time scales.The number of retained modes can be much smaller than the system dimension, enabling dimensionality reduction.
3. The DMD Decomposition and Algorithm.
DMD constructs a low-rank, data-driven approximation of market dynamics from sampled portfolio data, then uses its modes and eigenvalue dynamics to project future states. The algorithm varies collection snapshots and sampling windows while relying on low-dimensional structure and least-squares fitting.
- Koopman representation: The method approximates nonlinear dynamics with a linear Koopman mapping between sampled states without explicitly linearizing the underlying system.The operator maps data from t_j to t_j+1, although the dynamics generating the observations may remain nonlinear.
- Best-fit decomposition: DMD fits the final data point as a least-squares combination of Krylov-space vectors, minimizing the residual with a pseudoinverse.This fit uses the first M−1 data points to construct a basis and represent x_M as accurately as possible.
- Low-rank reduction: An SVD reduces the data matrix to a rank-K approximation before constructing the reduced operator and solving for its eigenvalues and eigenvectors.The reduction makes computation tractable and lets DMD exploit low-dimensional structure in the data.
- Future-state prediction: DMD modes and eigenvalue dynamics generate projected future states for arbitrary future times from the initial-state coefficients.The coefficients are obtained with a Moore–Penrose pseudoinverse, and the real parts of the resulting growth rates influence portfolio growth or demise.
- Algorithm: The complete algorithm uses sampled data, reduced matrices, eigen-decomposition, pseudoinverse projection, and DMD modes to compute future states without requiring governing equations.The future prediction is obtained by evaluating the DMD representation at the desired time, provided the approximation remains valid.
- Data collection: DMD samples portfolio data at prescribed locations and times, forming matrices whose columns represent successive market states.Regularly spaced sampling is used in the basic algorithm, while the total number of snapshots can be varied for optimization.
4. Financial Trading with DMD.
The paper applies equation-free DMD to portfolio data, using low-rank modes and learned sampling and prediction windows to identify trading hot-spots. Back tests show that performance depends on sector-specific hot-spots, with strong results when those regions are present.
- Trading Algorithm and Training: The training algorithm evaluates combinations of sampling window m and prediction window ℓ to find regions with high prediction success, termed trading hot-spots.The search uses historical data and, in the described daily-trading setup, considers m=1–25 days and ℓ=1–10 days.
- Trading Algorithm and Training: Most hot-spots examined over 10 years used 8–10 days of prices to predict 4–5 days ahead, with success rates above 50% treated as promising.The resulting signals were used to enter positions daily across portfolio companies.
- Trading Algorithm and Training: DMD decomposes portfolio data into low-rank modes and uses their dynamics to predict short-term market states for trading decisions.The decomposition identifies coherent portfolio structures, eigenvalue growth modes, and company weightings from each sampling window.
- DMD Decomposition in Sectors: The decomposition reveals low-rank portfolio structure, with dominant modes representing major variance and the first mode capturing the average stock price over the sampling window.Eigenvalues distinguish growth, decay, and oscillatory behavior, while sector differences appear in their growth modes.
- Performance Evaluation: 21.48% annualized over 10 years was achieved by xDMD(11, 5) in the home construction example, alongside 19.22% for xDMD(11, 1) and 18.59% for xDMD(2, 5).The associated hot-spot had a success rate of about 52.5%.
5. Conclusions and Outlook.
The paper presents DMD as a data-driven decomposition of portfolio measurements into dynamic modes that approximate nonlinear dynamics through the Koopman operator. It identifies extensions for reducing data requirements and modeling external market drivers.
- DMD decomposes portfolio snapshots into dynamic modes and approximates the modes of the Koopman operator.
- Interpreting DMD eigenvalues as prescribed time-scale dynamics enables extraction of coherent structures from data.
- Outlook: Compressive sampling could reduce the number of market measurements when data acquisition is expensive or prohibitive.
- Outlook: DMD with control could distinguish underlying dynamics from external market drivers and produce input-output models for market sectors.