Forecasting Probability Distributions of Financial Returns with Deep Neural Networks

Paper: arXiv 2508.18921

Authors: Jakub Michańków

Abstract

This study evaluates deep neural networks for forecasting probability distributions of financial returns. 1D convolutional neural networks (CNN) and Long Short-Term Memory (LSTM) architectures are used to forecast parameters of three probability distributions: Normal, Student’s t, and skewed Student’s t. Using custom negative log-likelihood loss functions, distribution parameters are optimized directly. The models are tested on six major equity indices (S&P 500, BOVESPA, DAX, WIG, Nikkei 225, and KOSPI) using probabilistic evaluation metrics including Log Predictive Score (LPS), Continuous Ranked Probability Score (CRPS), and Probability Integral Transform (PIT). Results show that deep learning models provide accurate distributional forecasts and perform competitively with classical GARCH models for Value-at-Risk estimation. The LSTM with skewed Student’s t distribution performs best across multiple evaluation criteria, capturing both heavy tails and asymmetry in financial returns. This work shows that deep neural networks are viable alternatives to traditional econometric models for financial risk assessment and portfolio management.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous empirical evaluation of deep learning architectures (CNN and LSTM) combined with parametric probability distribution loss functions (including skewed Student’s t) for financial return forecasting across six global indices. It features thorough metric evaluations (LPS, CRPS, PIT) and traditional risk validation metrics (VaR, Expected Shortfall tests), striking a strong balance between mathematical formulation and extensive empirical backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Forecast Financial Return Distributions using Deep Learning] --> B[Data Inputs: 6 Equity Indices<br/>S&P 500, BOVESPA, DAX, WIG, Nikkei 225, KOSPI]
    B --> C[Model Architectures: 1D CNN & LSTM]
    C --> D[Optimization: Custom Negative Log-Likelihood Loss<br/>Normal, Student's t, Skewed Student's t]
    D --> E[Evaluation: LPS, CRPS, PIT & VaR Backtesting]
    E --> F[Key Outcomes: DL Models Match/Exceed GARCH<br/>LSTM + Skewed Student's t Performs Best]