Pseudo-Siamese Network for Asset Pricing (SNAP)

• The paper introduces a deep learning architecture that decomposes asset returns into interpretable components: deep alpha, deep beta, and factor risk premia.
• The methodology employs distinct, pseudo-Siamese subnetworks with LSTM modules and dropout regularization to capture nonlinear dynamics and long-term financial dependencies.
• Empirical results on U.S. equities show that SNAP outperforms traditional models with an out-of-sample Sharpe ratio >2 and improved predictive R², enabling effective mispricing and arbitrage detection.

A Pseudo-Siamese Network for Asset Pricing (SNAP) is a deep learning-based framework that decomposes asset returns into economically interpretable components—specifically, deep alpha (mispricing), deep beta (risk exposures), and deep factor risk premia—conditional on high-dimensional firm characteristics and macroeconomic states. Unlike classical Siamese networks with identical weight-tied branches, the pseudo–Siamese architecture organizes distinct but structurally similar subnetworks for each latent component and integrates long-term memory using LSTM modules. This enables memory-aware, dynamic modeling of financial interactions and anomalies. Empirical analysis on U.S. equities demonstrates that such architectures outperform traditional asset pricing models in both predictive accuracy and risk-adjusted portfolio performance, and facilitate the identification of arbitrage opportunities via mispricing error clustering (Liu, 5 Sep 2025).

1. Architectural Principles and Structure

SNAP is built on an end-to-end deep learning design where asset returns are modeled additively:

$$R_{i,t+1}^e = \alpha(z_{it}) + \beta(z_{it}) \cdot \lambda(\bar{z}_t, m_t) + \epsilon_{i,t+1}$$

• $\alpha(z_{it})$: Deep alpha branch. Learns nonlinear idiosyncratic mispricing from firm-specific characteristics vector $z_{it}$.
• $\beta(z_{it})$: Deep beta branch. Captures flexible, time-varying risk exposures, modeled as a nonlinear function of $z_{it}$.
• $\lambda(\bar{z}_t, m_t)$: Deep factor risk premia branch. Processes population-level characteristics $\bar{z}_t$ (cross-sectional averages) and macroeconomic states $m_t$.
• $\epsilon_{i,t+1}$: Unexplained error term.

Distinct branches emulate a pseudo–Siamese architecture: deep alpha and deep beta map the same input (firm characteristics) via different nonlinear functions, while the factor branch introduces non-shared inputs. Pseudo–Siamese delineation permits architectural flexibility (no strict weight sharing) while exploiting parallels in feature extraction and temporal memory.

Each branch utilizes LSTM cells to encode long-term dependencies from sequential financial data, addressing both short- and long-horizon market dynamics. Dropout is applied on non-recurrent connections to regularize model capacity. Optimization proceeds via Adam or similar adaptive algorithms.

2. Deep Learning Techniques for Conditional Asset Pricing

SNAP leverages modern deep learning components to capture the conditional nature and sequence dependence of asset pricing phenomena:

• LSTM Networks: Employed in each branch to aggregate historical firm characteristics and macroeconomic variables. Memory gates (input, forget, output) enable preservation and integration of temporal patterns—crucial for modeling asset price persistence, mean reversion, and regime shifts.
• Dropout Regularization: Mitigates overfitting in high-dimensional parameter spaces, enhancing out-of-sample generalization.
• End-to-End Differentiability: All branches are differentiable, allowing joint training and efficient propagation of gradients during backpropagation.

This multi-stream architecture subsumes both static and dynamic conditional beta modeling, moving beyond classical linear factor assumptions.

3. Empirical Performance and Sharpe Ratio Analysis

Applied to a comprehensive panel of U.S. monthly equity returns from 1970–2019, SNAP demonstrates superior predictive and economic performance:

Model	Out-of-Sample R²	Out-of-Sample Sharpe Ratio
SNAP (pseudo)	4%	>2
FF3/FF5	negative/small	near 0
LASSO/Elastic Net/Ridge	lower	lower
MLP	lower	lower

• Predictive R² is calculated for one-month ahead returns. SNAP maintains robust R² margins compared to traditional and regularized linear models.
• Sharpe Ratio: Zero-net investment portfolios trading decile spreads based on model predictions consistently show out-of-sample Sharpe ratios above 2.
• Performance Decay: Notably, SNAP retains predictive and risk-adjusted advantages in out-of-sample tests, whereas benchmarks experience substantial deterioration.

4. Deep Alpha, Mispricing Error Estimation, and Arbitrage

SNAP’s architecture facilitates direct extraction of deep alpha signals—nonlinear mispricing errors not captured by risk factors:

• Mispricing Estimation: Comparing residuals between the full (with $\alpha$) and masked ($\alpha=0$) models yields $\hat{\alpha}_{it}$.
• Statistical Testing: Mann–Whitney U and t-tests consistently reject the null $H_0: \alpha_{it}=0$ (p-values = 0), establishing the significance of deep mispricings.
• Arbitrage Portfolio Construction: Portfolio weights $w_t = \hat{\alpha}_t / N_t$ (where $N_t$ is the number of stocks) form arbitrage portfolios, showing statistically significant abnormal returns.
• Clustering: K-Means applied to $(\alpha, R)$ pairs groups stocks by mispricing patterns, informing sector or anomaly-driven strategy design.

5. Integration with Economic Theory and Model Interpretability

SNAP is designed to explicitly respect core risk factor decomposition from asset pricing theory:

• Economic Decomposition: Segregates predicted return into interpretable components: risk premium, conditional beta, and mispricing.
• Masked Model Testing: The architecture enables consistent testing of economic hypotheses (e.g., market efficiency via $\alpha=0$ constraint).
• Interpretability: The modular design clarifies the economic role of each subnet, in contrast to black-box approaches.
• Relevance for Investors: Clear identification of actionable mispricing and dynamic betas enhances managing market anomalies and constructing market-neutral portfolios.

6. Benchmark Comparisons and Robustness

SNAP is benchmarked against canonical models (Fama–French 3/5-factor), linear regularized regressions (LASSO, Ridge, Elastic Net), and standard deep networks (MLP). Distinctions:

• Modular Component Structure: SNAP’s separation between alpha, beta, and factor branches contrasts with the monolithic aggregation in FF models or MLPs.
• Higher Out-of-Sample Sharpe Ratios and Predictive R²: SNAP notably outperforms across these metrics. FF models often yield negative out-of-sample R² and Sharpe ratios near zero.
• Performance Consistency: Robust against exclusion of microcaps and alternative risk adjustments, due to dynamic LSTM memory and conditional structure.

7. Implications and Extensions

The pseudo–Siamese design of SNAP opens new methodological directions:

• Scalability: Modular branch structure is amenable to augmentation with richer macroeconomic indicators, alternative loss functions, or deeper/unified LSTM/convolutional layers.
• Interpretable Arbitrage: Deep alpha extraction and mispricing portfolio clustering enable data-driven anomaly detection and strategy backtesting.
• Robustness: Separation of systematic and idiosyncratic signals improves resistance to market regime shifts and enhances generalization.
• Research Frontiers: Extensions may involve hybrid architectures incorporating attention mechanisms, economic distillation for variable importance, or application to other asset classes.

A plausible implication is that further architectural refinements—such as deep recurrent features, augmented clustering, or integration with theory-driven losses—could improve both the predictive accuracy and actionable interpretability of deep learning models in empirical asset pricing and risk management.