Price options on investment project expansion under commodity price and volatility uncertainties using a novel finite difference method

Highlights：

• We propose a novel 9-point finite difference scheme with a upwind technique is designed for solving the PDE system, as well that for determining the terminal (payoff) condition, established. We show that the coefficient matrix of the system from this discretization is an M-matrix and the numerical solution generated by the finite difference scheme converge to the exact one by proving that the scheme is consistent, monotone and stable.

• Extensive numerical experiments on the model and numerical methods using a model investment problem in an iron-ore industry have been performed. The numerical results show that our model and numerical methods for solving the model are able to produce numerical results which are financially meaningful.

摘要

•A real option is a contract which gives its holder the flexibility to expand the scale of an investment project or production. Real options are often used to hedge risks or capture opportunities in investments. In this paper, we establish a mathematical model for pricing a real option of expansion whose underlying asset price and its volatility/variance satisfy two separate stochastic equations. Based on Ito’s lemma and a hedging technique, we show that the option price satisfies a 2nd-order parabolic partial differential equation (PDE) in two spatial dimensions. We also derive the boundary and terminal conditions for the PDE and some of these conditions are also determined by PDEs.•We propose a novel 9-point finite difference scheme with a upwind technique is designed for solving the PDE system, as well that for determining the terminal (payoff) condition, established. We show that the coefficient matrix of the system from this discretization is an M-matrix and the numerical solution generated by the finite difference scheme converge to the exact one by proving that the scheme is consistent, monotone and stable.•Extensive numerical experiments on the model and numerical methods using a model investment problem in an iron-ore industry have been performed. The numerical results show that our model and numerical methods for solving the model are able to produce numerical results which are financially meaningful.