Calibrated predictive regions for static hedging of piecewise-affine claims with option portfolios

Abstract We develop a modular forecast-to-construction framework for static option-portfolio design with piecewise-affine contingent claims. The forecasting layer is treated as a replaceable front-end: any statistical, econometric, machine-learning, expert-based, or scenario-based method may be used, provided that it produces a predictive region for the future underlying-price vector. For concreteness, the empirical illustration predicts the future log-return vector $$ Y_{t,\\Delta }:=\\log (S_{t+\\Delta }/S_t) =F(z_t)+\\xi _{t+\\Delta }, $$ where the logarithm and division are coordinatewise. A fitted member of a small sigmoidal family approximates $$F$$ , and absolute residuals on a chronologically later calibration block determine a finite-sample corrected empirical quantile. The resulting return region is mapped back to the positive price space by the exponential transformation. Training, calibration, and test blocks are separated by prediction-horizon embargoes, so that every response needed for fitting or calibration is observed before the next evaluation block begins. In the construction layer, the investor combines a static market payoff $$\\Pi $$ with a target claim payoff $$f$$ , allowing either sign according to whether the claim is held long or short, and studies $$H=\\Pi +\\eta f$$ . In one dimension, if $$f$$ is piecewise affine with finitely many branches, then $$H$$ is piecewise affine with finitely many breakpoints. Hence statewise constraints over intervals, finite unions of intervals, or scenario sets admit finite certification. In the multi-asset case, an analogous vertex certificate applies after the predictive region is refined into boxes on which the net payoff is affine. The resulting portfolio problems reduce to linear or mixed-integer linear programs. The numerical section gives an end-to-end frozen-data HTZ illustration. After leakage-free evaluation, a separate deployment fit produces the raw price interval $$[4.0258,8.3385]$$ , which is enlarged outward to $$[4.0,8.5]$$ . The construction selects option strikes from that boundary and produces a region-binding writer hedge with nonnegative profit on the certified interval and the global loss bound $$\\textrm{Profit}(x)\\ge -1$$ for every $$x\\ge 0$$ .