A fitted finite volume method for real option valuation of risks in climate change

A large number of industries will experience climate change related damages with the climate change processes over the coming years. For example, the risks from sea level rising will be faced. In addition, there are a lot of uncertainties for the climate change in the future. Therefore, making decisions when to invest in the long term the sea level rising risk related projects is important and complex. The complexity of the decisions mainly lies in the evolving nature of the sea level rising risk, particularly due to the global climate change but also the future socio-economic development scenarios. In this paper, we first regard the sea level and the temperature as the underlying assets, and then develop a real option model to evaluate potential sea level rising risk management opportunities. In the case of American real options, we reformulate the problem to a linear parabolic variational inequality (VI) in two spatial dimensions and develop a power penalty method to solve it. It is shown that the nonlinear partial differential equation (PDE) is uniquely solvable and the solution of the PDE converges to that of the VI at the rate of order O(λ−k2). A so-called fitted finite volume method is proposed to solve the nonlinear PDE in both cases of European and American options, and the convergence of the fully discrete system of equations is obtained. Finally, some numerical experiments are performed to illustrate the theoretical results of this method.