# 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  January 2019 Early access  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 and Management Optimization, 2019, 15 (1) : 261-273. doi: 10.3934/jimo.2018042
##### References:
 [1] S. A. Abdel Sabour and R. Poulin, Valuing real capital investments using the least-squares Monte Carlo method, The Engineering Economist, 51 (2006), 141-160. [2] M. Bellalah, Irreversibility, Sunk costs and investment under uncertainty, R & D Management, 31 (2001), 115-126. [3] M. Bellalah, A reexamination of corporate risks under incomplete information, International Journal of Finance and Economics, 6 (2001), 41-58. [4] M. J. Brennan and E. S. Schwartz, Finite difference methods and jump processes arising in the pricing of contingent claims, Journal of Financial and Quantitative Analysis, 13 (1978), 461-474.  doi: 10.2307/2330152. [5] M. J. Brennan and E. S. Schwartz, Evaluating natural resource investments, The Journal of Business, 58 (1985), 135-157. [6] W. Chen and S. Wang, A finite difference method for pricing European and American options under a geometric Levy process, J. Ind. Mang. Optim., 11 (2015), 241-264. [7] G. Cortazar, E. S. Schwartz and J. Casassus, Optimal exploration investments under price and geological-technical uncertainty: A real options model, R & D Management, 31 (2001), 181-189. [8] L. Costa, A. Gabriel and S. B. Suslick, Estimating the volatility of mining projects considering price and operating cost uncertainties, Resources Policy, 31 (2006), 86-94.  doi: 10.1016/j.resourpol.2006.07.002. [9] A. K. Dixit and R. S. Pindyck, The Options approach to capital investment, The Economic Impact of Knowledge, (1995), 325-340.  doi: 10.1016/B978-0-7506-7009-8.50024-0. [10] S. E. Fadugba, F. H. Adefolaju and O. H. Akindutire, On the stability and accuracy of finite difference method for options pricing, Mathematical Theory and Modeling, 2 (2012), 101-108. [11] R. Geske, The valuation of compound options, Journal of Financial Economics, 7 (1979), 63-81. [12] M. Haque, E. Topal and E. Lilford, A numerical study for a mining project using real options valuation under commodity price uncertainty, Resources Policy, 39 (2014), 115-123.  doi: 10.1016/j.resourpol.2013.12.004. [13] Hartono, L. S. Jennings and S. Wang, Iterative upwind finite difference method with completed Richardson extrapolation for state-constrained Hamilton-Jacobi-Bellman equations, Pacific J. of Optim., 12 (2016), 379-397. [14] C. Hirsch, Numerical Computation of Internal and External Flows, John Wiley & Sons, 1990. [15] J. E. Hodder and H. E. Riggs, Pitfalls in evaluating risky projects, Harvard Business Reviews, 63 (1985), 128-135. [16] D. C. Lesmana and S. Wang, An upwind finite difference method for a nonlinear Black-Scholes equation governing European option valuation under transaction costs, Applied Mathematics and Computation, 219 (2013), 8811-8828.  doi: 10.1016/j.amc.2012.12.077. [17] D. C. Lesmana and S. Wang, A numerical scheme for pricing American options with transaction costs under a jump diffusion process, J. Ind. Mang. Optim., 13 (2017), 1793-1813. [18] W. Li and S. Wang, Penalty approach to the HJB Equation arising in European stock option pricing with proportional transaction costs, Journal of Optimization Theory and Applications, 143 (2009), 279-293.  doi: 10.1007/s10957-009-9559-7. [19] W. Li and S. Wang, Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme, J. Ind. Mang. Optim., 9 (2013), 365-389.  doi: 10.3934/jimo.2013.9.365. [20] W. Li and S. Wang, Pricing European options with proportional transaction costs and stochastic volatility using a penalty approach and a finite volume scheme, Computers and Mathematics with Applications, 73 (2017), 2454-2469.  doi: 10.1016/j.camwa.2017.03.024. [21] A. Moel and P. Tufano, When are real options exercised? An empirical study of mine closings, Review of Financial Studies, 15 (2002), 35-64. [22] N. Moyen, M. Slade and R. Uppal, Valuing risk and flexibility-a comparison of methods, Resources Policy, 22 (1996), 63-74. [23] S. C. Myers, Finance theory and financial strategy, Interfaces, 14 (1984), 126-137.  doi: 10.1287/inte.14.1.126. [24] L. Trigeorgis, The nature of option interactions and the valuation of investments with multiple real options, Journal of Financial and Quantitative Analysis, 28 (1993), 1-20.  doi: 10.2307/2331148. [25] L. Trigeorgis, Real Options, Princeton Series in Applied Mathematics, The MIT Press, Princeton, NJ, 1996. [26] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Engelwood Cliffs, NJ, 1962. [27] S. Wang, A novel fitted finite volume method for the Black-Scholes equation governing option pricing, IMA J. Numer. Anal., 24 (2004), 699-720.  doi: 10.1093/imanum/24.4.699. [28] S. Wang, L. S. Jennings and K. L. Teo, Numerical solution of Hamilton-Jacobi-Bellman equations by an upwind finite volume method, Journal of Global Optimization, 27 (2003), 177-192.  doi: 10.1023/A:1024980623095. [29] S. Wang, S. Zhang and Z. Fang, A superconvergent fitted finite volume Method for Black-Scholes equations governing European and American option valuation, Numerical Methods For Partial Differential Equations, 31 (2015), 181-208.  doi: 10.1002/num.21941. [30] P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, 1993.

show all references

##### References:
 [1] S. A. Abdel Sabour and R. Poulin, Valuing real capital investments using the least-squares Monte Carlo method, The Engineering Economist, 51 (2006), 141-160. [2] M. Bellalah, Irreversibility, Sunk costs and investment under uncertainty, R & D Management, 31 (2001), 115-126. [3] M. Bellalah, A reexamination of corporate risks under incomplete information, International Journal of Finance and Economics, 6 (2001), 41-58. [4] M. J. Brennan and E. S. Schwartz, Finite difference methods and jump processes arising in the pricing of contingent claims, Journal of Financial and Quantitative Analysis, 13 (1978), 461-474.  doi: 10.2307/2330152. [5] M. J. Brennan and E. S. Schwartz, Evaluating natural resource investments, The Journal of Business, 58 (1985), 135-157. [6] W. Chen and S. Wang, A finite difference method for pricing European and American options under a geometric Levy process, J. Ind. Mang. Optim., 11 (2015), 241-264. [7] G. Cortazar, E. S. Schwartz and J. Casassus, Optimal exploration investments under price and geological-technical uncertainty: A real options model, R & D Management, 31 (2001), 181-189. [8] L. Costa, A. Gabriel and S. B. Suslick, Estimating the volatility of mining projects considering price and operating cost uncertainties, Resources Policy, 31 (2006), 86-94.  doi: 10.1016/j.resourpol.2006.07.002. [9] A. K. Dixit and R. S. Pindyck, The Options approach to capital investment, The Economic Impact of Knowledge, (1995), 325-340.  doi: 10.1016/B978-0-7506-7009-8.50024-0. [10] S. E. Fadugba, F. H. Adefolaju and O. H. Akindutire, On the stability and accuracy of finite difference method for options pricing, Mathematical Theory and Modeling, 2 (2012), 101-108. [11] R. Geske, The valuation of compound options, Journal of Financial Economics, 7 (1979), 63-81. [12] M. Haque, E. Topal and E. Lilford, A numerical study for a mining project using real options valuation under commodity price uncertainty, Resources Policy, 39 (2014), 115-123.  doi: 10.1016/j.resourpol.2013.12.004. [13] Hartono, L. S. Jennings and S. Wang, Iterative upwind finite difference method with completed Richardson extrapolation for state-constrained Hamilton-Jacobi-Bellman equations, Pacific J. of Optim., 12 (2016), 379-397. [14] C. Hirsch, Numerical Computation of Internal and External Flows, John Wiley & Sons, 1990. [15] J. E. Hodder and H. E. Riggs, Pitfalls in evaluating risky projects, Harvard Business Reviews, 63 (1985), 128-135. [16] D. C. Lesmana and S. Wang, An upwind finite difference method for a nonlinear Black-Scholes equation governing European option valuation under transaction costs, Applied Mathematics and Computation, 219 (2013), 8811-8828.  doi: 10.1016/j.amc.2012.12.077. [17] D. C. Lesmana and S. Wang, A numerical scheme for pricing American options with transaction costs under a jump diffusion process, J. Ind. Mang. Optim., 13 (2017), 1793-1813. [18] W. Li and S. Wang, Penalty approach to the HJB Equation arising in European stock option pricing with proportional transaction costs, Journal of Optimization Theory and Applications, 143 (2009), 279-293.  doi: 10.1007/s10957-009-9559-7. [19] W. Li and S. Wang, Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme, J. Ind. Mang. Optim., 9 (2013), 365-389.  doi: 10.3934/jimo.2013.9.365. [20] W. Li and S. Wang, Pricing European options with proportional transaction costs and stochastic volatility using a penalty approach and a finite volume scheme, Computers and Mathematics with Applications, 73 (2017), 2454-2469.  doi: 10.1016/j.camwa.2017.03.024. [21] A. Moel and P. Tufano, When are real options exercised? An empirical study of mine closings, Review of Financial Studies, 15 (2002), 35-64. [22] N. Moyen, M. Slade and R. Uppal, Valuing risk and flexibility-a comparison of methods, Resources Policy, 22 (1996), 63-74. [23] S. C. Myers, Finance theory and financial strategy, Interfaces, 14 (1984), 126-137.  doi: 10.1287/inte.14.1.126. [24] L. Trigeorgis, The nature of option interactions and the valuation of investments with multiple real options, Journal of Financial and Quantitative Analysis, 28 (1993), 1-20.  doi: 10.2307/2331148. [25] L. Trigeorgis, Real Options, Princeton Series in Applied Mathematics, The MIT Press, Princeton, NJ, 1996. [26] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Engelwood Cliffs, NJ, 1962. [27] S. Wang, A novel fitted finite volume method for the Black-Scholes equation governing option pricing, IMA J. Numer. Anal., 24 (2004), 699-720.  doi: 10.1093/imanum/24.4.699. [28] S. Wang, L. S. Jennings and K. L. Teo, Numerical solution of Hamilton-Jacobi-Bellman equations by an upwind finite volume method, Journal of Global Optimization, 27 (2003), 177-192.  doi: 10.1023/A:1024980623095. [29] S. Wang, S. Zhang and Z. Fang, A superconvergent fitted finite volume Method for Black-Scholes equations governing European and American option valuation, Numerical Methods For Partial Differential Equations, 31 (2015), 181-208.  doi: 10.1002/num.21941. [30] P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, 1993.
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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