Numerical methods for weather derivatives pricing

Nhangumbe, Clarinda

Numerical methods for weather derivatives pricing

Thesis / Dissertation

2025

Publisher

University of Cape Town

Department

Department of Mathematics and Applied Mathematics

Faculty

Faculty of Science

Abstract

Weather derivatives are financial products used to hedge non-catastrophic weather risks with a weather index as an underlying asset. Mispricing these contracts poses a significant risk due the nature of weather variable. On rainfall derivatives pricing, the rainfall process is considered to be a stochastic, consisting of two random variables: one representing frequency, which is a two state Markov Chain, and the other representing the rainfall amount. Generally, these variables are modelled separately. The frequency is modelled by the discrete models and the rain-fall amount by the continuous models. However, the debate on how to model the dynamics of rainfall amounts still open. The main objective of this thesis, is to price rainfall based derivatives using only monthly rainfall amount. The monthly rainfall amount are modeled by Ornstein-Uhlenbeck process. Then, applying the Feynman-Kac theorem we derive the partial differential equations that govern the price of an European derivative option. Since the partial deferential equation does not admit analytical solutions, we use the numerical methods to solve it. The explicit numerical methods that are special cases of finite-difference schemes and nonstandard finite difference combined with the operator splitting approaches, are proposed. The methods are effective on handling with convection dominant condition and preserve the positivity. The positivity and stability conditions are established and the numerical solutions are simulated. Furthermore, we propose the boundary conditions which have financial interpretation that are also compatible with the mathematical view points.