Forecasting Probability Distributions of Financial Returns with Deep
  Neural Networks
Authors: name: Jakub Michańków

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.