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# Pricing options on investment project expansions under commodity price uncertainty

• 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.

Mathematics Subject Classification: Primary: 65M06; 91G20 Secondary: 91G60.

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• Figure 4.1.  The computed option value for Test 1.

Figure 4.2.  Computed option values at $t = 0$ for the different values of $\kappa$ and $\sigma$

Figure 4.3.  Computed values of compound and normal options.

Figure 4.4.  Computed option values and their differences for Test 2.

Table 4.1.  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}$Table 4.2. 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}\$
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