Numerical solutions of an option pricing rainfall weather derivatives model

Abstract

Weather derivatives are financial products used to cover non catastrophic weather events with a weather index as the underlying asset. In this work, we derive a rainfall weather derivative price model, based in the assumption that the rainfall dynamics follows a Ornstein-Uhlenbeck process. To calculate the price of the option we arrive at a two dimensional time dependent partial differential equation, where the spatial variables are the rainfall index and the total rainfall. Appropriate boundary conditions are suggested and they differ from the boundaries presented in literature in similar contexts. To compute the approximate solutions of the partial differential equation, we propose an explicit numerical method in order to deal efficiently with the different choices of the coefficients involved in the equation, that depend on the rainfall defice (or excess) and on the precipitation (amount of rain). Being an explicit numerical method, it will be conditionally stable and we discuss the stability region of the numerical method and its order of convergence. In the end we examine two test cases where the parameters of the model presented are estimated based on precipitation data.