# Forecasting Probability Distributions of Financial Returns with Deep Neural Networks

arXiv: https://arxiv.org/abs/2508.18921 · HTML: https://mlsanity.com/paper/2508.18921/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 21 references that are in the map.
- First posted on arXiv 26 Aug 2025.

## Builds on (most cited first, 4 of 4)

- [DeepAR: Probabilistic Forecasting with Autoregressive Recurrent Networks](https://mlsanity.com/paper/1704.04110/) (2017)
- [Probabilistic Forecasting with Temporal Convolutional Neural Network](https://mlsanity.com/paper/1906.04397/) (2019)
- [Distributional neural networks for electricity price forecasting](https://mlsanity.com/paper/2207.02832/) (2022)
- [Probabilistic intraday electricity price forecasting using generative machine learning](https://mlsanity.com/paper/2506.00044/) (2025)

## Abstract (arXiv)

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.