Orthogonal wavelet method for multi-stage expansion and contraction options under stochastic volatility

Abstract

Multi-stage expansion and contraction options are real options enabling an investment project to be scaled up or down in response to market conditions at predetermined future dates. We examine an investment project focused on producing a specific commodity, with the project value dependent on the market price of this commodity. We then study the value of options to either increase or decrease production at specific future dates based on predetermined factors and costs. Under the assumption that the commodity price follows a geometric Brownian motion and the volatility is stochastic, multiple partial differential equations represent the valuation model for these options. This paper aims to establish two new pricing models for multi-stage expansion and contraction options: one where variance follows a geometric Brownian motion and another governed by the Cox–Ingersoll–Ross process. Another aim is to propose and analyze an efficient wavelet-based numerical method for these models. The method employs the Galerkin method with a recently constructed orthogonal cubic spline wavelet basis and the Crank-Nicolson scheme enhanced by Richardson extrapolation. We establish the existence and uniqueness of the solution, provide error estimates for the proposed method, and derive bounds for condition numbers of the resulting matrices arising from discretization. The method is applied to options related to iron-ore mining investment projects to verify the relevance of the method and show its benefits, which are a high-order convergence rate, well-conditioned discretization matrices, and an efficient solution of the resulting system of equations using a small number of iterations.