Monotone functional regression with hybrid depth weighting and block-conformal prediction bands for forward realized variance paths

We forecast forward realized variance (FRV) paths, defined as cumulative future daily variance proxy curves over a finite trading horizon, using a leakage-disciplined functional framework for multiday risk assessment. The framework combines multiresponse ridge regression, hybrid depth weighting, horizon-weighted blocked cross-validation, and isotonic post-projection to preserve the monotone structure of FRV paths. Uncertainty is summarized through upper one-sided block-calibrated conformal bands, interpreted as empirical risk envelopes under temporal dependence rather than exact distribution-free guarantees. In a fixed panel design for four liquid exchange-traded funds, GDX, GDXJ, XLE, and UUP, over the period 2010–2025, the proposed model reduces long-horizon mean squared error relative to rolling historical FRV by approximately 31.8%, 20.4%, 36.5%, and 28.0%, respectively, over $h=20:30$. Comparisons with heterogeneous autoregressive (HAR) ridge and functional principal component autoregressive (FPCA-AR) benchmarks are asset-dependent. The proposed model is most favorable for GDX and remains close to HAR ridge for GDXJ, whereas HAR ridge and FPCA-AR remain competitive for XLE and UUP. Coverage is conservative or close to nominal at $\alpha =0.05$ but more heterogeneous at $\alpha =0.10$. Robustness checks support a cautious interpretation of the method as a shape-aware enhancement of rolling FRV forecasting.