Conditional continuous quantile forecasting under rough and heavy-tailed market dynamics

We propose Conditional Continuous Quantile Forecasting (CCQF), a roughness-aware neural framework for conditional density forecasting in financial time series characterized by heavy tails, rough dynamics, and frequent regime shifts. Motivated by empirical evidence of volatility clustering, long-memory–like rough sample paths, and horizon-dependent uncertainty, CCQF combines a long short-term memory (LSTM) encoder with a Student-t output head to jointly estimate the conditional location, scale, and degrees of freedom of returns. This parametric structure enables flexible modeling of heavy-tailed behavior and rapidly varying heteroskedasticity.A key component of CCQF is a conditioning vector that incorporates a roughness indicator—implemented via a Hurst-based estimator or a volatility-of-volatility measure—together with the forecast horizon. CCQF is trained under a unified objective that integrates negative log-likelihood and multi-quantile pinball loss, yielding coherent predictive distributions across heterogeneous roughness regimes and time scales.We evaluate CCQF in a large-scale fractional Brownian motion (fBm) environment spanning a wide Hurst grid and multiple sampling resolutions, as well as in a rough fractional Ornstein–Uhlenbeck (fOU) setting. Real-world experiments using Bitcoin–USDT (BTCUSDT) one-minute returns further illustrate the challenges posed by extreme volatility and heavy-tailed innovations. Across both synthetic and empirical datasets, CCQF achieves superior likelihood-based accuracy, competitive multi-quantile performance, and well-calibrated yet sharp prediction intervals relative to the Monte Carlo simulation, CCQF-Gaussian variants, the Temporal Fusion Transformer (TFT), and GARCH-family baselines.These results demonstrate that explicitly conditioning on roughness and jointly optimizing likelihood and quantile objectives yields a robust and adaptable framework for probabilistic forecasting in modern high-volatility markets.