A Semi-Lagrangian method for the weather options of mean-reverting Brownian motion with jump–diffusion

The weather options were created to enable companies to hedge against climate risks. However, the valuation of weather options is complex, since the underlying temperature process has no negotiable price. This paper presents a new weather option pricing model, which is governed by a stochastic underlying temperature following a mean-reverting Browning motion with jump–diffusion under the assumption of mean-self-financing. Consequently, a two-dimensional partial integro-differential equation (PIDE) is derived to value the weather-based options.The numerical techniques applied in this paper are based on a Semi-Lagrangian method (SLM), in which at each time-step a set of one dimensional partial integro-differential equations are solved and the solution of each PIDE is updated using Semi-Lagrangian time-stepping. In addition, monotonicity, stability, and the convergence results of the discrete schemes are also derived in this paper. Lastly, this paper values a series of European HDD-based weather put options using SLM, and highlights the key details of the numerical implementation.