Forecasting Probability Distributions of Financial Returns with Deep Neural Networks

TL;DR

This paper demonstrates that deep neural networks, specifically CNNs and LSTMs, can effectively forecast probability distributions of financial returns, offering a competitive alternative to traditional econometric models for risk assessment.

Contribution

It introduces a novel application of deep neural networks with custom loss functions to directly forecast distribution parameters of financial returns.

Findings

LSTM with skewed Student's t distribution performs best across evaluation metrics.

Deep neural networks provide accurate distributional forecasts for financial data.

Models perform competitively with classical GARCH models for Value-at-Risk estimation.

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.