Numerical Solution of a Mean-Reverting Uncertain Rainfall Model for Deficit Index Option Pricing: Empirical Evidence From Berlin–Tempelhof

Rainfall-deficit derivatives provide financial protection against precipitation shortfalls in agriculture, water management, and other weather-sensitive activities. Their valuation is complicated by the seasonality, local dependence, and nontradability of rainfall. An inadequate dispersion specification may produce intervals that are too narrow in highly variable months or unnecessarily wide in relatively stable months. We introduce a seasonal mean-reverting uncertain rainfall-index model with month-dependent uncertainty intensity. We derive an explicit integral representation of its α-paths, prove that the associated cumulative rainfall-deficit functional is decreasing in the belief level, and obtain its inverse uncertainty distribution. These results reduce the valuation of a put on the cumulative deficit index to a one-dimensional deterministic integral. We also derive the exact monthly transition of the centered process, which provides the basis for parameter estimation. We calibrate the model to monthly observations from Berlin–Tempelhof by estimating the seasonal mean through a Fourier representation and the transition parameters from the centered series. Predictive performance is evaluated over a 1996–2005 holdout period against a matched Gaussian Ornstein–Uhlenbeck benchmark. The two models have nearly identical monthly point-forecast accuracy. The uncertain model yields slightly lower monthly interval scores and quantile losses and performs marginally better for the 1-month June deficit contract. The Gaussian benchmark performs better for the April–June accumulation contract, for which the uncertain model produces substantially wider predictive distributions. The uncertain model therefore provides a competitive short-horizon specification, while its excessive multimonth dispersion limits its suitability for longer accumulation contracts.