Deep Neural Operator Learning for Probabilistic Models ArXiv ID: 2511.07235 “View on arXiv” Authors: Erhan Bayraktar, Qi Feng, Zecheng Zhang, Zhaoyu Zhang Abstract We propose a deep neural-operator framework for a general class of probability models. Under global Lipschitz conditions on the operator over the entire Euclidean space-and for a broad class of probabilistic models-we establish a universal approximation theorem with explicit network-size bounds for the proposed architecture. The underlying stochastic processes are required only to satisfy integrability and general tail-probability conditions. We verify these assumptions for both European and American option-pricing problems within the forward-backward SDE (FBSDE) framework, which in turn covers a broad class of operators arising from parabolic PDEs, with or without free boundaries. Finally, we present a numerical example for a basket of American options, demonstrating that the learned model produces optimal stopping boundaries for new strike prices without retraining. ...

Diffolio: A Diffusion Model for Multivariate Probabilistic Financial Time-Series Forecasting and Portfolio Construction ArXiv ID: 2511.07014 “View on arXiv” Authors: So-Yoon Cho, Jin-Young Kim, Kayoung Ban, Hyeng Keun Koo, Hyun-Gyoon Kim Abstract Probabilistic forecasting is crucial in multivariate financial time-series for constructing efficient portfolios that account for complex cross-sectional dependencies. In this paper, we propose Diffolio, a diffusion model designed for multivariate financial time-series forecasting and portfolio construction. Diffolio employs a denoising network with a hierarchical attention architecture, comprising both asset-level and market-level layers. Furthermore, to better reflect cross-sectional correlations, we introduce a correlation-guided regularizer informed by a stable estimate of the target correlation matrix. This structure effectively extracts salient features not only from historical returns but also from asset-specific and systematic covariates, significantly enhancing the performance of forecasts and portfolios. Experimental results on the daily excess returns of 12 industry portfolios show that Diffolio outperforms various probabilistic forecasting baselines in multivariate forecasting accuracy and portfolio performance. Moreover, in portfolio experiments, portfolios constructed from Diffolio’s forecasts show consistently robust performance, thereby outperforming those from benchmarks by achieving higher Sharpe ratios for the mean-variance tangency portfolio and higher certainty equivalents for the growth-optimal portfolio. These results demonstrate the superiority of our proposed Diffolio in terms of not only statistical accuracy but also economic significance. ...

Forecasting implied volatility surface with generative diffusion models ArXiv ID: 2511.07571 “View on arXiv” Authors: Chen Jin, Ankush Agarwal Abstract We introduce a conditional Denoising Diffusion Probabilistic Model (DDPM) for generating arbitrage-free implied volatility (IV) surfaces, offering a more stable and accurate alternative to existing GAN-based approaches. To capture the path-dependent nature of volatility dynamics, our model is conditioned on a rich set of market variables, including exponential weighted moving averages (EWMAs) of historical surfaces, returns and squared returns of underlying asset, and scalar risk indicators like VIX. Empirical results demonstrate our model significantly outperforms leading GAN-based models in capturing the stylized facts of IV dynamics. A key challenge is that historical data often contains small arbitrage opportunities in the earlier dataset for training, which conflicts with the goal of generating arbitrage-free surfaces. We address this by incorporating a standard arbitrage penalty into the loss function, but apply it using a novel, parameter-free weighting scheme based on the signal-to-noise ratio (SNR) that dynamically adjusts the penalty’s strength across the diffusion process. We also show a formal analysis of this trade-off and provide a proof of convergence showing that the penalty introduces a small, controllable bias that steers the model toward the manifold of arbitrage-free surfaces while ensuring the generated distribution remains close to the real-world data. ...

Machine-learning a family of solutions to an optimal pension investment problem ArXiv ID: 2511.07045 “View on arXiv” Authors: John Armstrong, Cristin Buescu, James Dalby, Rohan Hobbs Abstract We use a neural network to identify the optimal solution to a family of optimal investment problems, where the parameters determining an investor’s risk and consumption preferences are given as inputs to the neural network in addition to economic variables. This is used to develop a practical tool that can be used to explore how pension outcomes vary with preference parameters. We use a Black-Scholes economic model so that we may validate the accuracy of network using a classical and provably convergent numerical method developed using the duality approach. ...

A Risk-Neutral Neural Operator for Arbitrage-Free SPX-VIX Term Structures ArXiv ID: 2511.06451 “View on arXiv” Authors: Jian’an Zhang Abstract We propose ARBITER, a risk-neutral neural operator for learning joint SPX-VIX term structures under no-arbitrage constraints. ARBITER maps market states to an operator that outputs implied volatility and variance curves while enforcing static arbitrage (calendar, vertical, butterfly), Lipschitz bounds, and monotonicity. The model couples operator learning with constrained decoders and is trained with extragradient-style updates plus projection. We introduce evaluation metrics for derivatives term structures (NAS, CNAS, NI, Dual-Gap, Stability Rate) and show gains over Fourier Neural Operator, DeepONet, and state-space sequence models on historical SPX and VIX data. Ablation studies indicate that tying the SPX and VIX legs reduces Dual-Gap and improves NI, Lipschitz projection stabilizes calibration, and selective state updates improve long-horizon generalization. We provide identifiability and approximation results and describe practical recipes for arbitrage-free interpolation and extrapolation across maturities and strikes. ...

Bitcoin Forecasting with Classical Time Series Models on Prices and Volatility ArXiv ID: 2511.06224 “View on arXiv” Authors: Anmar Kareem, Alexander Aue Abstract This paper evaluates the performance of classical time series models in forecasting Bitcoin prices, focusing on ARIMA, SARIMA, GARCH, and EGARCH. Daily price data from 2010 to 2020 were analyzed, with models trained on the first 90 percent and tested on the final 10 percent. Forecast accuracy was assessed using MAE, RMSE, AIC, and BIC. The results show that ARIMA provided the strongest forecasts for short-run log-price dynamics, while EGARCH offered the best fit for volatility by capturing asymmetry in responses to shocks. These findings suggest that despite Bitcoin’s extreme volatility, classical time series models remain valuable for short-run forecasting. The study contributes to understanding cryptocurrency predictability and sets the stage for future work integrating machine learning and macroeconomic variables. ...

Push-response anomalies in high-frequency S&P 500 price series ArXiv ID: 2511.06177 “View on arXiv” Authors: Dmitrii Vlasiuk, Mikhail Smirnov Abstract We test the hypothesis that consecutive intraday price changes in the most liquid U.S. equity ETF (SPY) are conditionally nonrandom. Using NBBO event-time data for about 1,500 regular trading days, we form for every lag L ordered pairs of a backward price increment (“push”) and a forward price increment (“response”), standardize them, and estimate the expected responses on a fine grid of push magnitudes. The resulting lag-by-magnitude maps reveal a persistent structural shift: for short lags (1-5,000 ticks), expected responses cluster near zero across most push magnitudes, suggesting high short-term efficiency; beyond that range, pronounced tails emerge, indicating that larger historical pushes increasingly correlate with nonzero conditional responses. We also find that large negative pushes are followed by stronger positive responses than equally large positive pushes, consistent with asymmetric liquidity replenishment after sell-side shocks. Decomposition into symmetric and antisymmetric components and the associated dominance curves confirm that short-horizon efficiency is restored only partially. The evidence points to an intraday, lag-resolved anomaly that is invisible in unconditional returns and that can be used to define tradable pockets and risk controls. ...

Equilibrium Portfolio Selection under Utility-Variance Analysis of Log Returns in Incomplete Markets ArXiv ID: 2511.05861 “View on arXiv” Authors: Yue Cao, Zongxia Liang, Sheng Wang, Xiang Yu Abstract This paper investigates a time-inconsistent portfolio selection problem in the incomplete mar ket model, integrating expected utility maximization with risk control. The objective functional balances the expected utility and variance on log returns, giving rise to time inconsistency and motivating the search of a time-consistent equilibrium strategy. We characterize the equilibrium via a coupled quadratic backward stochastic differential equation (BSDE) system and establish the existence theory in two special cases: (i)the two Brownian motions driven the price dynamics and the factor process are independent with $ρ= 0$; (ii) the trading strategy is constrained to be bounded. For the general case with correlation coefficient $ρ\neq 0$, we introduce the notion of an approximate time-consistent equilibrium. Employing the solution structure from the equilibrium in the case $ρ= 0$, we can construct an approximate time-consistent equilibrium in the general case with an error of order $O(ρ^2)$. Numerical examples and financial insights are also presented based on deep learning algorithms. ...

Competitive optimal portfolio selection under mean-variance criterion ArXiv ID: 2511.05270 “View on arXiv” Authors: Guojiang Shao, Zuo Quan Xu, Qi Zhang Abstract We investigate a portfolio selection problem involving multi competitive agents, each exhibiting mean-variance preferences. Unlike classical models, each agent’s utility is determined by their relative wealth compared to the average wealth of all agents, introducing a competitive dynamic into the optimization framework. To address this game-theoretic problem, we first reformulate the mean-variance criterion as a constrained, non-homogeneous stochastic linear-quadratic control problem and derive the corresponding optimal feedback strategies. The existence of Nash equilibria is shown to depend on the well-posedness of a complex, coupled system of equations. Employing decoupling techniques, we reduce the well-posedness analysis to the solvability of a novel class of multi-dimensional linear backward stochastic differential equations (BSDEs). We solve a new type of nonlinear BSDEs (including the above linear one as a special case) using fixed-point theory. Depending on the interplay between market and competition parameters, three distinct scenarios arise: (i) the existence of a unique Nash equilibrium, (ii) the absence of any Nash equilibrium, and (iii) the existence of infinitely many Nash equilibria. These scenarios are rigorously characterized and discussed in detail. ...

Economic uncertainty and exchange rates linkage revisited: modelling tail dependence with high frequency data ArXiv ID: 2511.05315 “View on arXiv” Authors: Nourhaine Nefzi, Abir Abid Abstract The aim of this paper is to dig deeper into understanding the exchange rates and uncertainty dependence. Using the novel Baker et al. (2020)’s daily Twitter Uncertainty Index and BRICS exchange rates, we investigate their extreme tail dependence within an original time-varying copula framework. Our analysis makes several noteworthy results. Evidence for Indian, Russian and South African currencies indicates an elliptical copulas’ dominance implying neither asymmetric features nor extreme movements in their dependence structure with the global economic uncertainty. Importantly, Brazilian and Chinese currencies tail dependence is upward trending suggesting a safe-haven role in times of high global economic uncertainty including the recent COVID-19 pandemic. In such circumstances, these markets offer opportunities to significant gains through portfolio diversification. ...