Browsing by Subject "Weather derivatives"

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    Lie Analysis for Partial Differential Equations in Finance
    (2019) Nhangumbe, Clarinda Vitorino; Fredericks, Ebrahim; Canhanga , Betuel
    Weather derivatives are financial tools used to manage the risks related to changes in the weather and are priced considering weather variables such as rainfall, temperature, humidity and wind as the underlying asset. Some recent researches suggest to model the amount of rainfall by considering the mean reverting processes. As an example, the Ornstein Uhlenbeck process was proposed by Allen [3] to model yearly rainfall and by Unami et al. [52] to model the irregularity of rainfall intensity as well as duration of dry spells. By using the Feynman-Kac theorem and the rainfall indexes we derive the partial differential equations (PDEs) that governs the price of an European option. We apply the Lie analysis theory to solve the PDEs, we provide the group classification and use it to find the invariant analytical solutions, particularly the ones compatible with the terminal conditions.
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    Numerical methods for weather derivatives pricing
    (2025) Nhangumbe, Clarinda; Fredericks, Ebrahim; Canhanga, Betuel
    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.