Wavelet Method for Pricing Options on Two-Stage Expansion of Investment Project

The paper aims to derive a model in logarithmic prices for the valuation of two-stage expansion options, propose and study a wavelet-based numerical method for the model, and compare the performance of the method for different cubicspline wavelet bases. Under the assumption that the underlying commodity price follows a geometric Brownian motion, the model is represented by partial differential equations of the Black-Scholes type. However, the situation is more complicated than for the Black-Scholes model for financial options because expansion options do not have closed-form solutions, the model is represented by five equations, the payoff function is not given but is determined as the solution of a differential equation, and equipping equations with appropriate boundary conditions also is a delicate task. Since the model is represented by equations in logarithmic prices, the degeneracy of the original differential operator is removed and the variational formulation can be derived using the standard Sobolev spaces. The proposed method is based on the Crank-Nicolson scheme with Richardson extrapolation combined with the Galerkin method with cubic spline wavelet bases. Numerical experiments are conducted on the benchmark problem in the iron-ore mining industry. The results confirm a higher-order convergence rate with respect to both spatial and time variables and show that wavelet bases combined with diagonal preconditioning lead to uniformly bounded numbers of iterations needed to compute the sufficiently accurate solution. The method is implemented with different types of cubic spline wavelet bases, and it is shown that all tested bases give relevant results and that the L
²
-orthogonal wavelet basis is superior to others concerning the number of iterations.