Uncertainty Quantification

Index-Tracking Portfolio Construction and Rebalancing under Bayesian Sparse Modelling and Uncertainty Quantification ArXiv ID: 2512.22109 “View on arXiv” Authors: Dimitrios Roxanas Abstract We study the construction and rebalancing of sparse index-tracking portfolios from an operational research perspective, with explicit emphasis on uncertainty quantification and implementability. The decision variables are portfolio weights constrained to sum to one; the aims are to track a reference index closely while controlling the number of names and the turnover induced by rebalancing. We cast index tracking as a high-dimensional linear regression of index returns on constituent returns, and employ a sparsity-inducing Laplace prior on the weights. A single global shrinkage parameter controls the trade-off between tracking error and sparsity, and is calibrated by an empirical-Bayes stochastic approximation scheme. Conditional on this calibration, we approximate the posterior distribution of the portfolio weights using proximal Langevin-type Markov chain Monte Carlo algorithms tailored to the budget constraint. This yields posterior uncertainty on tracking error, portfolio composition and prospective rebalancing moves. Building on these posterior samples, we propose rules for rebalancing that gate trades through magnitude-based thresholds and posterior activation probabilities, thereby trading off expected tracking error against turnover and portfolio size. A case study on tracking the S&P~500 index is carried out to showcase how our tools shape the decision process from portfolio construction to rebalancing. ...

Trading with the Devil: Risk and Return in Foundation Model Strategies ArXiv ID: 2510.17165 “View on arXiv” Authors: Jinrui Zhang Abstract Foundation models - already transformative in domains such as natural language processing - are now starting to emerge for time-series tasks in finance. While these pretrained architectures promise versatile predictive signals, little is known about how they shape the risk profiles of the trading strategies built atop them, leaving practitioners reluctant to commit serious capital. In this paper, we propose an extension to the Capital Asset Pricing Model (CAPM) that disentangles the systematic risk introduced by a shared foundation model - potentially capable of generating alpha if the underlying model is genuinely predictive - from the idiosyncratic risk attributable to custom fine-tuning, which typically accrues no systematic premium. To enable a practical estimation of these separate risks, we align this decomposition with the concepts of uncertainty disentanglement, casting systematic risk as epistemic uncertainty (rooted in the pretrained model) and idiosyncratic risk as aleatory uncertainty (introduced during custom adaptations). Under the Aleatory Collapse Assumption, we illustrate how Monte Carlo dropout - among other methods in the uncertainty-quantization toolkit - can directly measure the epistemic risk, thereby mapping trading strategies to a more transparent risk-return plane. Our experiments show that isolating these distinct risk factors yields deeper insights into the performance limits of foundation-model-based strategies, their model degradation over time, and potential avenues for targeted refinements. Taken together, our results highlight both the promise and the pitfalls of deploying large pretrained models in competitive financial markets. ...

(Non-Parametric) Bootstrap Robust Optimization for Portfolios and Trading Strategies ArXiv ID: 2510.12725 “View on arXiv” Authors: Daniel Cunha Oliveira, Grover Guzman, Nick Firoozye Abstract Robust optimization provides a principled framework for decision-making under uncertainty, with broad applications in finance, engineering, and operations research. In portfolio optimization, uncertainty in expected returns and covariances demands methods that mitigate estimation error, parameter instability, and model misspecification. Traditional approaches, including parametric, bootstrap-based, and Bayesian methods, enhance stability by relying on confidence intervals or probabilistic priors but often impose restrictive assumptions. This study introduces a non-parametric bootstrap framework for robust optimization in financial decision-making. By resampling empirical data, the framework constructs flexible, data-driven confidence intervals without assuming specific distributional forms, thus capturing uncertainty in statistical estimates, model parameters, and utility functions. Treating utility as a random variable enables percentile-based optimization, naturally suited for risk-sensitive and worst-case decision-making. The approach aligns with recent advances in robust optimization, reinforcement learning, and risk-aware control, offering a unified perspective on robustness and generalization. Empirically, the framework mitigates overfitting and selection bias in trading strategy optimization and improves generalization in portfolio allocation. Results across portfolio and time-series momentum experiments demonstrate that the proposed method delivers smoother, more stable out-of-sample performance, offering a practical, distribution-free alternative to traditional robust optimization methods. ...

FinZero: Launching Multi-modal Financial Time Series Forecast with Large Reasoning Model ArXiv ID: 2509.08742 “View on arXiv” Authors: Yanlong Wang, Jian Xu, Fei Ma, Hongkang Zhang, Hang Yu, Tiantian Gao, Yu Wang, Haochen You, Shao-Lun Huang, Danny Dongning Sun, Xiao-Ping Zhang Abstract Financial time series forecasting is both highly significant and challenging. Previous approaches typically standardized time series data before feeding it into forecasting models, but this encoding process inherently leads to a loss of important information. Moreover, past time series models generally require fixed numbers of variables or lookback window lengths, which further limits the scalability of time series forecasting. Besides, the interpretability and the uncertainty in forecasting remain areas requiring further research, as these factors directly impact the reliability and practical value of predictions. To address these issues, we first construct a diverse financial image-text dataset (FVLDB) and develop the Uncertainty-adjusted Group Relative Policy Optimization (UARPO) method to enable the model not only output predictions but also analyze the uncertainty of those predictions. We then proposed FinZero, a multimodal pre-trained model finetuned by UARPO to perform reasoning, prediction, and analytical understanding on the FVLDB financial time series. Extensive experiments validate that FinZero exhibits strong adaptability and scalability. After fine-tuning with UARPO, FinZero achieves an approximate 13.48% improvement in prediction accuracy over GPT-4o in the high-confidence group, demonstrating the effectiveness of reinforcement learning fine-tuning in multimodal large model, including in financial time series forecasting tasks. ...

An Interval Type-2 Version of Bayes Theorem Derived from Interval Probability Range Estimates Provided by Subject Matter Experts ArXiv ID: 2509.08834 “View on arXiv” Authors: John T. Rickard, William A. Dembski, James Rickards Abstract Bayesian inference is widely used in many different fields to test hypotheses against observations. In most such applications, an assumption is made of precise input values to produce a precise output value. However, this is unrealistic for real-world applications. Often the best available information from subject matter experts (SMEs) in a given field is interval range estimates of the input probabilities involved in Bayes Theorem. This paper provides two key contributions to extend Bayes Theorem to an interval type-2 (IT2) version. First, we develop an IT2 version of Bayes Theorem that uses a novel and conservative method to avoid potential inconsistencies in the input IT2 MFs that otherwise might produce invalid output results. We then describe a novel and flexible algorithm for encoding SME-provided intervals into IT2 fuzzy membership functions (MFs), which we can use to specify the input probabilities in Bayes Theorem. Our algorithm generalizes and extends previous work on this problem that primarily addressed the encoding of intervals into word MFs for Computing with Words applications. ...

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. ...

Quantile deep learning models for multi-step ahead time series prediction ArXiv ID: 2411.15674 “View on arXiv” Authors: Unknown Abstract Uncertainty quantification is crucial in time series prediction, and quantile regression offers a valuable mechanism for uncertainty quantification which is useful for extreme value forecasting. Although deep learning models have been prominent in multi-step ahead prediction, the development and evaluation of quantile deep learning models have been limited. We present a novel quantile regression deep learning framework for multi-step time series prediction. In this way, we elevate the capabilities of deep learning models by incorporating quantile regression, thus providing a more nuanced understanding of predictive values. We provide an implementation of prominent deep learning models for multi-step ahead time series prediction and evaluate their performance under high volatility and extreme conditions. We include multivariate and univariate modelling, strategies and provide a comparison with conventional deep learning models from the literature. Our models are tested on two cryptocurrencies: Bitcoin and Ethereum, using daily close-price data and selected benchmark time series datasets. The results show that integrating a quantile loss function with deep learning provides additional predictions for selected quantiles without a loss in the prediction accuracy when compared to the literature. Our quantile model has the ability to handle volatility more effectively and provides additional information for decision-making and uncertainty quantification through the use of quantiles when compared to conventional deep learning models. ...

Quantile Regression using Random Forest Proximities ArXiv ID: 2408.02355 “View on arXiv” Authors: Unknown Abstract Due to the dynamic nature of financial markets, maintaining models that produce precise predictions over time is difficult. Often the goal isn’t just point prediction but determining uncertainty. Quantifying uncertainty, especially the aleatoric uncertainty due to the unpredictable nature of market drivers, helps investors understand varying risk levels. Recently, quantile regression forests (QRF) have emerged as a promising solution: Unlike most basic quantile regression methods that need separate models for each quantile, quantile regression forests estimate the entire conditional distribution of the target variable with a single model, while retaining all the salient features of a typical random forest. We introduce a novel approach to compute quantile regressions from random forests that leverages the proximity (i.e., distance metric) learned by the model and infers the conditional distribution of the target variable. We evaluate the proposed methodology using publicly available datasets and then apply it towards the problem of forecasting the average daily volume of corporate bonds. We show that using quantile regression using Random Forest proximities demonstrates superior performance in approximating conditional target distributions and prediction intervals to the original version of QRF. We also demonstrate that the proposed framework is significantly more computationally efficient than traditional approaches to quantile regressions. ...

Enhancing Mean-Reverting Time Series Prediction with Gaussian Processes: Functional and Augmented Data Structures in Financial Forecasting ArXiv ID: 2403.00796 “View on arXiv” Authors: Unknown Abstract In this paper, we explore the application of Gaussian Processes (GPs) for predicting mean-reverting time series with an underlying structure, using relatively unexplored functional and augmented data structures. While many conventional forecasting methods concentrate on the short-term dynamics of time series data, GPs offer the potential to forecast not just the average prediction but the entire probability distribution over a future trajectory. This is particularly beneficial in financial contexts, where accurate predictions alone may not suffice if incorrect volatility assessments lead to capital losses. Moreover, in trade selection, GPs allow for the forecasting of multiple Sharpe ratios adjusted for transaction costs, aiding in decision-making. The functional data representation utilized in this study enables longer-term predictions by leveraging information from previous years, even as the forecast moves away from the current year’s training data. Additionally, the augmented representation enriches the training set by incorporating multiple targets for future points in time, facilitating long-term predictions. Our implementation closely aligns with the methodology outlined in, which assessed effectiveness on commodity futures. However, our testing methodology differs. Instead of real data, we employ simulated data with similar characteristics. We construct a testing environment to evaluate both data representations and models under conditions of increasing noise, fat tails, and inappropriate kernels-conditions commonly encountered in practice. By simulating data, we can compare our forecast distribution over time against a full simulation of the actual distribution of our test set, thereby reducing the inherent uncertainty in testing time series models on real data. We enable feature prediction through augmentation and employ sub-sampling to ensure the feasibility of GPs. ...

Quantifying neural network uncertainty under volatility clustering ArXiv ID: 2402.14476 “View on arXiv” Authors: Unknown Abstract Time-series with volatility clustering pose a unique challenge to uncertainty quantification (UQ) for returns forecasts. Methods for UQ such as Deep Evidential regression offer a simple way of quantifying return forecast uncertainty without the costs of a full Bayesian treatment. However, the Normal-Inverse-Gamma (NIG) prior adopted by Deep Evidential regression is prone to miscalibration as the NIG prior is assigned to latent mean and variance parameters in a hierarchical structure. Moreover, it also overparameterizes the marginal data distribution. These limitations may affect the accurate delineation of epistemic (model) and aleatoric (data) uncertainties. We propose a Scale Mixture Distribution as a simpler alternative which can provide favorable complexity-accuracy trade-off and assign separate subnetworks to each model parameter. To illustrate the performance of our proposed method, we apply it to two sets of financial time-series exhibiting volatility clustering: cryptocurrencies and U.S. equities and test the performance in some ablation studies. ...