# American Institute of Mathematical Sciences

January  2019, 15(1): 261-273. doi: 10.3934/jimo.2018042

## Pricing options on investment project expansions under commodity price uncertainty

 1 Department of Mathematics & Statistics, Curtin University, GPO Box U1987, WA 6845, Australia 2 School of Mathematical & Software Sciences, Sichuan Normal University, Sichuan 610000, China

Received  May 2017 Revised  October 2017 Published  April 2018

In this work we develop PDE-based mathematical models for valuing real options on investment project expansions when the underlying commodity price follows a geometric Brownian motion. The models developed are of a similar form as the Black-Scholes model for pricing conventional European call options. However, unlike the Black-Scholes' model, the payoff conditions of the current models are determined by a PDE system. An upwind finite difference scheme is used for solving the models. Numerical experiments have been performed using two examples of pricing project expansion options in the mining industry to demonstrate that our models are able to produce financially meaningful numerical results for the two non-trivial test problems.

Citation: Nan Li, Song Wang. Pricing options on investment project expansions under commodity price uncertainty. Journal of Industrial & Management Optimization, 2019, 15 (1) : 261-273. doi: 10.3934/jimo.2018042
##### References:

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##### References:
The computed option value for Test 1.
Computed option values at $t = 0$ for the different values of $\kappa$ and $\sigma$
Computed values of compound and normal options.
Computed option values and their differences for Test 2.
Project and market data used in Test 1.
 $Q = 10^4$ million tons $B = 30\%$ per annum $C_0 = {\rm US}\$$35 C(t) = C_0\times e^{0.005t} R = 5\% per annum r = 0.06 per annum K = {\rm US}\10^4 million T = 2 years \sigma = 30\% \delta = 0.02 q_0 = 0.01Q \times e^{0.007t} q_1 = \begin{cases} q_0&t < T \\ \kappa \times q_0&t \ge T \end{cases}  Q = 10^4 million tons B = 30\% per annum C_0 = {\rm US}\$$35$ C(t) = C_0\times e^{0.005t} R = 5\%$per annum$ r = 0.06$per annum$ K = {\rm US}\$10^4$ million $T = 2$ years $\sigma = 30\%$ $\delta = 0.02$ $q_0 = 0.01Q \times e^{0.007t}$ $q_1 = \begin{cases} q_0&t < T \\ \kappa \times q_0&t \ge T \end{cases}$
Project and market data used in Test 2.
 $T_1 = 2$ years $T_2 = 4$ years $K_1 = {\rm US}\$10^4$million$ K_2 = {\rm US}\$2 \times 10^4$ million $q_1 = \begin{cases} q_0&t < T_1 \\ 2 q_0&t \ge T_1 \end{cases}$ $q_2 = \begin{cases} q_1&t < T_2 \\ 2 q_1&t \ge T_2 \end{cases}$
 $T_1 = 2$ years $T_2 = 4$ years $K_1 = {\rm US}\$10^4$million$ K_2 = {\rm US}\$2 \times 10^4$ million $q_1 = \begin{cases} q_0&t < T_1 \\ 2 q_0&t \ge T_1 \end{cases}$ $q_2 = \begin{cases} q_1&t < T_2 \\ 2 q_1&t \ge T_2 \end{cases}$
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