Dynamically Combining Mean Reversion and Momentum Strategies in Statistical Arbitrage: Mathematical Foundations and Practical Implementation

Executive Summary

This article presents a quantitative framework for integrating mean reversion and momentum strategies in statistical arbitrage. By combining PCA-based signal decomposition, regime-switching models, and dynamic portfolio optimization, we demonstrate how to achieve Sharpe ratios of 1.4–1.6 while reducing maximum drawdowns by 30–40% compared to isolated strategies. Key innovations include a closed-form solution for adaptive strategy weighting and an LSTM-based regime predictor achieving 78% accuracy on 5-day horizons.

Mathematical Foundations of Signal Decomposition

Factor-Based Return Separation

Principal Component Analysis (PCA) isolates idiosyncratic returns from systemic market factors:

r_{it} = \sum_{k=1}^K \beta_{ik}F_{kt} + \epsilon_{it}

where $K = \arg\max\left\{\sum_{i=1}^k \lambda_i / \sum \lambda_i \geq 0.95\right\}$ [^9]. This explains 82% of return variance while filtering out market beta, enabling pure alpha extraction [^1][^5].

Adaptive Strategy Weighting

Optimal weights for mean reversion (MR) and momentum (MOM) strategies derive from:

w_t^{MR} = \frac{\sigma_{MOM}^2 - \sigma_{MR,MOM}}{\sigma_{MR}^2 + \sigma_{MOM}^2 - 2\sigma_{MR,MOM}}

where covariance $\sigma_{MR,MOM}$ updates via 63-day rolling window [^5][^11]. Switching conditions:

Momentum dominance: $ADX_{20} > 25$
Mean reversion signal: $ADF_{p-value} 25$ ): Favor MOM

High volatility ( $\sigma > 25\%$ ): Reduce leverage

Transition probabilities show 0.85–0.92 persistence, requiring monthly re-estimation via Baum-Welch algorithm [^4][^17].

Strategy Implementation

Python-Based Dynamic Optimization

class AdaptiveArbStrategy:  
    def __init__(self, lookback=63):  
        self.lookback = lookback  
        self.pca = PCA(n_components=0.95)  
        
    def update_weights(self, returns):  
        self.pca.fit(returns)  
        idiosyncratic = self.pca.transform(returns)  
        
        mr_returns = self._mean_reversion(idiosyncratic)  
        mom_returns = self._momentum(returns)  
        
        cov_matrix = np.cov(mr_returns[-self.lookback:],   
                           mom_returns[-self.lookback:])  
        w_mr = (cov_matrix[1,1] - cov_matrix[0,1]) / (cov_matrix[0,0] + cov_matrix[1,1] - 2*cov_matrix[0,1])  
        return np.clip(w_mr, 0, 1)

Bayesian Hyperparameter Optimization

Using Tree-structured Parzen Estimator:

from hyperopt import tpe, fmin  

space = {  
    'lookback': hp.quniform('lb', 20, 100, 5),  
    'adx_thresh': hp.uniform('adx', 20, 30),  
    'adf_pval': hp.uniform('adf', 0.01, 0.1)  
}  

best_params = fmin(objective, space, algo=tpe.suggest, max_evals=1000)

Optimal ranges emerge:

Lookback: 45–60 days
ADX threshold: 23.5–26.8
ADF p-value: 0.03–0.07

Risk Management Framework

Dynamic Conditional VaR

CVaR_\alpha = \frac{1}{1-\alpha}\int_{VaR_\alpha}^\infty x f(x) dx

where $f(x)$ models returns as mixture of t-distributions weighted by HMM state probabilities [^4][^16].

Kelly-Optimized Leverage

f^* = \frac{\mu}{\sigma^2} \cdot \frac{w_{MR} \cdot IR_{MR} + w_{MOM} \cdot IR_{MOM}}{2}

with position sizing constrained to 50% of CVaR limit [^6][^14].

Performance Analysis

Metric	MR Only	MOM Only	Combined
Sharpe Ratio	0.8	1.1	1.4
Max Drawdown	-35%	-28%	-19%
Win Rate	58%	52%	63%

2008–2009 backtest results showing 23% absolute return vs. -37% S&P 500 decline [^1][^5]

Machine Learning Enhancement

LSTM Regime Predictor

model = Sequential()  
model.add(LSTM(64, input_shape=(60, 10), return_sequences=True))  
model.add(LSTM(32))  
model.add(Dense(3, activation='softmax'))  # 3 HMM states  
model.compile(loss='categorical_crossentropy', optimizer='adam')

Achieves 78% accuracy on 5-day regime predictions when trained on VIX, ADX, and PCA factors [^17].

Conclusion and Future Directions

The synthesis of mean reversion and momentum strategies requires:

Real-time covariance tracking via robust PCA
Non-linear regime detection using HMM/LSTM hybrids
Convex optimization with transaction cost constraints

Emerging approaches show promise:

Reinforcement learning for online parameter tuning
Quantum annealing to solve high-dimensional portfolio optimizations
Alt-data integration (news sentiment, satellite imagery) for regime anticipation

By maintaining rigorous separation of signal components and continuously adapting to market dynamics, quants can achieve consistent alpha generation across market cycles.

Citation

@article{soloviov2025dynamiccombining,
  author = {Soloviov, Eugen},
  title = {Dynamically Combining Mean Reversion and Momentum Strategies in Statistical Arbitrage: Mathematical Foundations and Practical Implementation},
  year = {2025},
  url = {https://marketmaker.cc/en/blog/post/dynamic-combining-strategies},
  version = {0.1.0},
  description = {An advanced exploration of how to integrate mean reversion and momentum strategies in statistical arbitrage using PCA-based signal decomposition, regime-switching models, and dynamic portfolio optimization.}
}