Probabilistic Forecasting

ProbFM: Probabilistic Time Series Foundation Model with Uncertainty Decomposition ArXiv ID: 2601.10591 “View on arXiv” Authors: Arundeep Chinta, Lucas Vinh Tran, Jay Katukuri Abstract Time Series Foundation Models (TSFMs) have emerged as a promising approach for zero-shot financial forecasting, demonstrating strong transferability and data efficiency gains. However, their adoption in financial applications is hindered by fundamental limitations in uncertainty quantification: current approaches either rely on restrictive distributional assumptions, conflate different sources of uncertainty, or lack principled calibration mechanisms. While recent TSFMs employ sophisticated techniques such as mixture models, Student’s t-distributions, or conformal prediction, they fail to address the core challenge of providing theoretically-grounded uncertainty decomposition. For the very first time, we present a novel transformer-based probabilistic framework, ProbFM (probabilistic foundation model), that leverages Deep Evidential Regression (DER) to provide principled uncertainty quantification with explicit epistemic-aleatoric decomposition. Unlike existing approaches that pre-specify distributional forms or require sampling-based inference, ProbFM learns optimal uncertainty representations through higher-order evidence learning while maintaining single-pass computational efficiency. To rigorously evaluate the core DER uncertainty quantification approach independent of architectural complexity, we conduct an extensive controlled comparison study using a consistent LSTM architecture across five probabilistic methods: DER, Gaussian NLL, Student’s-t NLL, Quantile Loss, and Conformal Prediction. Evaluation on cryptocurrency return forecasting demonstrates that DER maintains competitive forecasting accuracy while providing explicit epistemic-aleatoric uncertainty decomposition. This work establishes both an extensible framework for principled uncertainty quantification in foundation models and empirical evidence for DER’s effectiveness in financial applications. ...

Random processes for long-term market simulations ArXiv ID: 2511.18125 “View on arXiv” Authors: Gilles Zumbach Abstract For long term investments, model portfolios are defined at the level of indexes, a setup known as Strategic Asset Allocation (SAA). The possible outcomes at a scale of a few decades can be obtained by Monte Carlo simulations, resulting in a probability density for the possible portfolio values at the investment horizon. Such studies are critical for long term wealth plannings, for example in the financial component of social insurances or in accumulated capital for retirement. The quality of the results depends on two inputs: the process used for the simulations and its parameters. The base model is a constant drift, a constant covariance and normal innovations, as pioneered by Bachelier. Beyond this model, this document presents in details a multivariate process that incorporate the most recent advances in the models for financial time series. This includes the negative correlations of the returns at a scale of a few years, the heteroskedasticity (i.e. the volatility’ dynamics), and the fat tails and asymmetry for the distributions of returns. For the parameters, the quantitative outcomes depend critically on the estimate for the drift, because this is a non random contribution acting at each time step. Replacing the point forecast by a probabilistic forecast allows us to analyze the impact of the drift values, and then to incorporate this uncertainty in the Monte Carlo simulations. ...

The Evolution of Probabilistic Price Forecasting Techniques: A Review of the Day-Ahead, Intra-Day, and Balancing Markets ArXiv ID: 2511.05523 “View on arXiv” Authors: Ciaran O’Connor, Mohamed Bahloul, Steven Prestwich, Andrea Visentin Abstract Electricity price forecasting has become a critical tool for decision-making in energy markets, particularly as the increasing penetration of renewable energy introduces greater volatility and uncertainty. Historically, research in this field has been dominated by point forecasting methods, which provide single-value predictions but fail to quantify uncertainty. However, as power markets evolve due to renewable integration, smart grids, and regulatory changes, the need for probabilistic forecasting has become more pronounced, offering a more comprehensive approach to risk assessment and market participation. This paper presents a review of probabilistic forecasting methods, tracing their evolution from Bayesian and distribution based approaches, through quantile regression techniques, to recent developments in conformal prediction. Particular emphasis is placed on advancements in probabilistic forecasting, including validity-focused methods which address key limitations in uncertainty estimation. Additionally, this review extends beyond the Day-Ahead Market to include the Intra-Day and Balancing Markets, where forecasting challenges are intensified by higher temporal granularity and real-time operational constraints. We examine state of the art methodologies, key evaluation metrics, and ongoing challenges, such as forecast validity, model selection, and the absence of standardised benchmarks, providing researchers and practitioners with a comprehensive and timely resource for navigating the complexities of modern electricity markets. ...

Orderbook Feature Learning and Asymmetric Generalization in Intraday Electricity Markets ArXiv ID: 2510.12685 “View on arXiv” Authors: Runyao Yu, Ruochen Wu, Yongsheng Han, Jochen L. Cremer Abstract Accurate probabilistic forecasting of intraday electricity prices is critical for market participants to inform trading decisions. Existing studies rely on specific domain features, such as Volume-Weighted Average Price (VWAP) and the last price. However, the rich information in the orderbook remains underexplored. Furthermore, these approaches are often developed within a single country and product type, making it unclear whether the approaches are generalizable. In this paper, we extract 384 features from the orderbook and identify a set of powerful features via feature selection. Based on selected features, we present a comprehensive benchmark using classical statistical models, tree-based ensembles, and deep learning models across two countries (Germany and Austria) and two product types (60-min and 15-min). We further perform a systematic generalization study across countries and product types, from which we reveal an asymmetric generalization phenomenon. ...

Probabilistic Forecasting Cryptocurrencies Volatility: From Point to Quantile Forecasts ArXiv ID: 2508.15922 “View on arXiv” Authors: Grzegorz Dudek, Witold Orzeszko, Piotr Fiszeder Abstract Cryptocurrency markets are characterized by extreme volatility, making accurate forecasts essential for effective risk management and informed trading strategies. Traditional deterministic (point) forecasting methods are inadequate for capturing the full spectrum of potential volatility outcomes, underscoring the importance of probabilistic approaches. To address this limitation, this paper introduces probabilistic forecasting methods that leverage point forecasts from a wide range of base models, including statistical (HAR, GARCH, ARFIMA) and machine learning (e.g. LASSO, SVR, MLP, Random Forest, LSTM) algorithms, to estimate conditional quantiles of cryptocurrency realized variance. To the best of our knowledge, this is the first study in the literature to propose and systematically evaluate probabilistic forecasts of variance in cryptocurrency markets based on predictions derived from multiple base models. Our empirical results for Bitcoin demonstrate that the Quantile Estimation through Residual Simulation (QRS) method, particularly when applied to linear base models operating on log-transformed realized volatility data, consistently outperforms more sophisticated alternatives. Additionally, we highlight the robustness of the probabilistic stacking framework, providing comprehensive insights into uncertainty and risk inherent in cryptocurrency volatility forecasting. This research fills a significant gap in the literature, contributing practical probabilistic forecasting methodologies tailored specifically to cryptocurrency markets. ...

Isotonic Quantile Regression Averaging for uncertainty quantification of electricity price forecasts ArXiv ID: 2507.15079 “View on arXiv” Authors: Arkadiusz Lipiecki, Bartosz Uniejewski Abstract Quantifying the uncertainty of forecasting models is essential to assess and mitigate the risks associated with data-driven decisions, especially in volatile domains such as electricity markets. Machine learning methods can provide highly accurate electricity price forecasts, critical for informing the decisions of market participants. However, these models often lack uncertainty estimates, which limits the ability of decision makers to avoid unnecessary risks. In this paper, we propose a novel method for generating probabilistic forecasts from ensembles of point forecasts, called Isotonic Quantile Regression Averaging (iQRA). Building on the established framework of Quantile Regression Averaging (QRA), we introduce stochastic order constraints to improve forecast accuracy, reliability, and computational costs. In an extensive forecasting study of the German day-ahead electricity market, we show that iQRA consistently outperforms state-of-the-art postprocessing methods in terms of both reliability and sharpness. It produces well-calibrated prediction intervals across multiple confidence levels, providing superior reliability to all benchmark methods, particularly coverage-based conformal prediction. In addition, isotonic regularization decreases the complexity of the quantile regression problem and offers a hyperparameter-free approach to variable selection. ...

Hedging with Sparse Reward Reinforcement Learning ArXiv ID: 2503.04218 “View on arXiv” Authors: Unknown Abstract Derivatives, as a critical class of financial instruments, isolate and trade the price attributes of risk assets such as stocks, commodities, and indices, aiding risk management and enhancing market efficiency. However, traditional hedging models, constrained by assumptions such as continuous trading and zero transaction costs, fail to satisfy risk control requirements in complex and uncertain real-world markets. With advances in computing technology and deep learning, data-driven trading strategies are becoming increasingly prevalent. This thesis proposes a derivatives hedging framework integrating deep learning and reinforcement learning. The framework comprises a probabilistic forecasting model and a hedging agent, enabling market probability prediction, derivative pricing, and hedging. Specifically, we design a spatiotemporal attention-based probabilistic financial time series forecasting Transformer to address the scarcity of derivatives hedging data. A low-rank attention mechanism compresses high-dimensional assets into a low-dimensional latent space, capturing nonlinear asset relationships. The Transformer models sequential dependencies within this latent space, improving market probability forecasts and constructing an online training environment for downstream hedging tasks. Additionally, we incorporate generalized geometric Brownian motion to develop a risk-neutral pricing approach for derivatives. We model derivatives hedging as a reinforcement learning problem with sparse rewards and propose a behavior cloning-based recurrent proximal policy optimization (BC-RPPO) algorithm. This pretraining-finetuning framework significantly enhances the hedging agent’s performance. Numerical experiments in the U.S. and Chinese financial markets demonstrate our method’s superiority over traditional approaches. ...

OrderFusion: Encoding Orderbook for End-to-End Probabilistic Intraday Electricity Price Forecasting ArXiv ID: 2502.06830 “View on arXiv” Authors: Unknown Abstract Probabilistic intraday electricity price forecasting is becoming increasingly important with the growth of renewable generation and the rise in demand-side engagement. Their uncertainties have increased the trading risks closer to delivery and the subsequent imbalance settlement costs. As a consequence, intraday trading has emerged to mitigate these risks. Unlike auction markets, intraday trading in many jurisdictions is characterized by the continuous posting of buy and sell orders on power exchange platforms. This dynamic orderbook microstructure of price formation presents special challenges for price forecasting. Conventional methods represent the orderbook via domain features aggregated from buy and sell trades, or by treating it as a multivariate time series, but such representations neglect the full buy-sell interaction structure of the orderbook. This research therefore develops a new order fusion methodology, which is an end-to-end and parameter-efficient probabilistic forecasting model that learns a full interaction-aware representation of the buy-sell dynamics. Furthermore, as quantile crossing is often a problem in probabilistic forecasting, this approach hierarchically estimates the quantiles with non-crossing constraints. Extensive experiments on the market price indices across high-liquidity (German) and low-liquidity (Austrian) markets demonstrate consistent improvements over conventional baselines, and ablation studies highlight the contributions of the main modeling components. The methodology is available at: https://runyao-yu.github.io/OrderFusion/. ...

IVE: Enhanced Probabilistic Forecasting of Intraday Volume Ratio with Transformers ArXiv ID: 2411.10956 “View on arXiv” Authors: Unknown Abstract This paper presents a new approach to volume ratio prediction in financial markets, specifically targeting the execution of Volume-Weighted Average Price (VWAP) strategies. Recognizing the importance of accurate volume profile forecasting, our research leverages the Transformer architecture to predict intraday volume ratio at a one-minute scale. We diverge from prior models that use log-transformed volume or turnover rates, instead opting for a prediction model that accounts for the intraday volume ratio’s high variability, stabilized via log-normal transformation. Our input data incorporates not only the statistical properties of volume but also external volume-related features, absolute time information, and stock-specific characteristics to enhance prediction accuracy. The model structure includes an encoder-decoder Transformer architecture with a distribution head for greedy sampling, optimizing performance on high-liquidity stocks across both Korean and American markets. We extend the capabilities of our model beyond point prediction by introducing probabilistic forecasting that captures the mean and standard deviation of volume ratios, enabling the anticipation of significant intraday volume spikes. Furthermore, an agent with a simple trading logic demonstrates the practical application of our model through live trading tests in the Korean market, outperforming VWAP benchmarks over a period of two and a half months. Our findings underscore the potential of Transformer-based probabilistic models for volume ratio prediction and pave the way for future research advancements in this domain. ...

Forecasting Company Fundamentals ArXiv ID: 2411.05791 “View on arXiv” Authors: Unknown Abstract Company fundamentals are key to assessing companies’ financial and overall success and stability. Forecasting them is important in multiple fields, including investing and econometrics. While statistical and contemporary machine learning methods have been applied to many time series tasks, there is a lack of comparison of these approaches on this particularly challenging data regime. To this end, we try to bridge this gap and thoroughly evaluate the theoretical properties and practical performance of 24 deterministic and probabilistic company fundamentals forecasting models on real company data. We observe that deep learning models provide superior forecasting performance to classical models, in particular when considering uncertainty estimation. To validate the findings, we compare them to human analyst expectations and find that their accuracy is comparable to the automatic forecasts. We further show how these high-quality forecasts can benefit automated stock allocation. We close by presenting possible ways of integrating domain experts to further improve performance and increase reliability. ...