# American Institute of Mathematical Sciences

May  2020, 16(3): 1349-1368. doi: 10.3934/jimo.2019006

## Pricing options on investment project contraction and ownership transfer using a finite volume scheme and an interior penalty method

 1 Department of Mathematics & Statistics, Curtin University, GPO Box U1987, WA 6845, Australia 2 School of Mathematical & Software Sciences, Sichuan Normal University, Sichuan, China, 610000 3 Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics Tianjin 300222, China

Received  February 2018 Revised  September 2018 Published  May 2020 Early access  March 2019

In this work we develop partial differential equation (PDE) based computational models for pricing real options to contract the production or to transfer part/all of the ownership of a project when the underlying asset price of the project satisfies a geometric Brownian motion. The developed models are similar to the Black-Scholes equation for valuing conventional European put options or the partial differential linear complementarity problem (LCP) for pricing American put options. A finite volume method is used for the discretization of the PDE models and a penalty approach is applied to the discretized LCP. We show that the coefficient matrix of the discretized systems is a positive-definite $M$-matrix which guarantees that the solution from the penalty equation converges to that of the discretized LCP. Numerical experiments, performed to demonstrate the usefulness of our methods, show that our models and numerical methods are able to produce financially meaningful numerical results for the two non-trivial test problems.

Citation: Nan Li, Song Wang, Shuhua Zhang. Pricing options on investment project contraction and ownership transfer using a finite volume scheme and an interior penalty method. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1349-1368. doi: 10.3934/jimo.2019006
##### 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. Amram and N. Kulatilaka, Disciplined decisions: Aligning strategy with the financial markets, Harvard Business Review, 77 (1999), 95-104. [3] M. Andalaft-Chacur, M. Montaz Ali and J. Jorge Gonzalez Salazar, Real options pricing by the finite element method, Computers and Mathematics with Applications, 61 (2011), 2863-2873.  doi: 10.1016/j.camwa.2011.03.070. [4] L. Angermann and S. Wang, Convergence of a fitted finite volume method for the penalized Black-Scholes equation governing European and American Option pricing, Numer. Math., 106 (2007), 1-40.  doi: 10.1007/s00211-006-0057-7. [5] J. Ankudinova and M. Ehrhardt, On the numerical solution of nonlinear Black-Scholes equations, Computers and Mathematics with Applications, 56 (2008), 799-812.  doi: 10.1016/j.camwa.2008.02.005. [6] A. Bensoussan and J. L. Lions, Applications of Variational Inequalities in Stochastic Control, Studies in Mathematics and its Applications, 12. North-Holland Publishing Co., Amsterdam-New York, 1982. [7] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062. [8] R. A. Brualdi and S. Mellendorf, Regions in the complex plane containing the eigenvalues of a matrix, Amer. Math. Monthly, 101 (1994), 975-985.  doi: 10.1080/00029890.1994.12004577. [9] 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. [10] M. J. Brennan and E. S. Schwartz, Evaluating natural resource investments, The Journal of Business, 58 (1985), 135-157.  doi: 10.1086/296288. [11] W. Chen and S. Wang, A penalty method for a fractional order parabolic variational inequality governing American put option valuation, Computers & Mathematics with Applications, 67 (2014), 77-90.  doi: 10.1016/j.camwa.2013.10.007. [12] C. H. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optimization and Application, 5 (1996), 97-138.  doi: 10.1007/BF00249052. [13] G. Cortazar, E. S. Schwartz and J. Casassus, Optimal exploration investments under price and geological-technical uncertainty: A real options model, R and D Management, 31 (2001), 181–189. [14] G. Courtadon, A more accurate finite difference approximation for the valuation of options, J. Financial Economics Quant. Anal., 17 (1982), 697-703.  doi: 10.2307/2330857. [15] A. N. Daryina, A. F. Izmailov and M. V. Solodov, A class of active-set Newton methods for mixed complementarity problems, SIAM Journal on Optimization, 15 (2004), 409-429.  doi: 10.1137/S105262340343590X. [16] A. K. Dixit and R. S. Pindyck, Investment Under Uncertainty, Princeton University Press, Princeton, N.J., 1994. [17] D. J. Duffy, Finite Difference Methods in Financial Engineering – A Partial Differential Equation Approach, John Wiley & Sons Ltd, 2006. doi: 10.1002/9781118673447. [18] F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems, Vol. I, Springer Series in Operations Research, Springer-Verlag, New York, 2003. [19] M. C. Ferris and J. S. Pang, Engineering and economic applications of complementarity problems, SIAM Rev., 39 (1997), 669-713.  doi: 10.1137/S0036144595285963. [20] A. Forsgren, P. E. Gill and M. H. Wright, Interior methods for nonlinear optimization, SIAM Rev., 44 (2002), 525-597.  doi: 10.1137/S0036144502414942. [21] M. A. 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. [22] C. C. Huang and S. Wang, A power penalty approach to a nonlinear complementary problem, Operations Research Letters, 38 (2010), 72-76.  doi: 10.1016/j.orl.2009.09.009. [23] J. C. Hull, Options, Futures And Other Derivatives (9th Edition), Pearson Education, Harlow, 2014. [24] S. Jaimungal, M. O. de Souza and J. P. Zubelli, Real option pricing with mean-reverting investment and project value, The European Journal of Finance, 19 (2013), 625-644.  doi: 10.1080/1351847X.2011.601660. [25] C. Kanzow, Global optimization techniques for mixed complementarity problems, J. Glob. Optim., 16 (2000), 1-21.  doi: 10.1023/A:1008331803982. [26] D. C. Lesmana and S. Wang, Penalty approach to a nonlinear obstacle problem governing American put option valuation under transaction costs, Applied Mathematics & Computation, 251 (2015), 318–330. doi: 10.1016/j.amc.2014.11.060. [27] D. C. Lesmana and S. Wang, A numerical scheme for pricing American options with transaction costs under a jump diffusion process, J. Ind. Manag. Optim., 13 (2017), 1793–1813. doi: 10.3934/jimo.2017019. [28] N. Li and S. Wang, Pricing options on investment project expansions under commodity price uncertainty, J. Ind. Manag. Optim., 15 (2019), 261–273. doi: 10.3934/jimo.2018042. [29] W. Li and S. Wang, Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme, J. Ind. Manag. Optim., 9 (2013), 365–398. doi: 10.3934/jimo.2013.9.365. [30] D. Li and M. Fukushima, Smoothing Newton and Quasi-Newton Methods for Mixed Complementarity Problems, Computational Optimization and Applications, 17 (2000), 203–230. doi: 10.1023/A:1026502415830. [31] A. Moel and P. Tufano, When are real options exercised? An empirical study of mine closings, Review of Financial Studies, 15 (2002), 35-64. [32] N. Moyen, M. Slade and R. Uppal, Valuing risk and flexibility – a comparison of methods, Resources Policy, 22 (1996), 63-74. [33] S. C. Myers, Finance theory and financial strategy, Interfaces, 14 (1984), 126-137.  doi: 10.1287/inte.14.1.126. [34] B. F. Nielsen, O. Skavhaug and A. Tveito., Penalty and front-fixing methods for the numerical solution of American option problems, J. Comp. Fin., 5 (2002), 69-97.  doi: 10.21314/JCF.2002.084. [35] F. A. Potra and Y. Ye, Interior-point methods for nonlinear complementarity problems, Journal of Optimization Theory & Applications, 88 (1996), 617-642.  doi: 10.1007/BF02192201. [36] J. Savolainen, Real options in metal mining project valuation: Review of literature, Resources Policy, 50 (2016), 49-65.  doi: 10.1016/j.resourpol.2016.08.007. [37] R. Seydel, Tools for Computational Finance, Springer Verlag, London, 2017. doi: 10.1007/978-1-4471-7338-0. [38] L. Trigeorgis, Real Options, Princeton Series in Applied Mathematics, The MIT Press, Princeton, NJ, 1996. [39] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Engelwood Cliffs, NJ, 1962. [40] 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. [41] S. Wang, An interior penalty method for a large-scale finite-dimensional nonlinear double obstacle problem, Applied Mathematical Modelling, 58 (2018), 217-228.  doi: 10.1016/j.apm.2017.07.038. [42] S. Wang, X. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, Journal of Optimization Theory & Applications, 129 (2006), 227–254. doi: 10.1007/s10957-006-9062-3. [43] 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), 1190–1208. doi: 10.1002/num.21941. [44] S. Wang and K. Zhang, An interior penalty method for a finite-dimensional linear complementarity problem in financial engineering, Optimization Letters, 12 (2018), 1161-1178.  doi: 10.1007/s11590-016-1050-4. [45] P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, 1993. [46] K. Zhang and S. Wang, Convergence property of an interior penalty approach to pricing American option, J. Ind. Manag. Optim., 7 (2011), 435-447.  doi: 10.3934/jimo.2011.7.435. [47] S. Zhang, X. Wang and H. Li, Modeling and computation of water management by real options, J. Ind. Manag. Optim., 14 (2018), 81-103.  doi: 10.3934/jimo.2017038.

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##### 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. Amram and N. Kulatilaka, Disciplined decisions: Aligning strategy with the financial markets, Harvard Business Review, 77 (1999), 95-104. [3] M. Andalaft-Chacur, M. Montaz Ali and J. Jorge Gonzalez Salazar, Real options pricing by the finite element method, Computers and Mathematics with Applications, 61 (2011), 2863-2873.  doi: 10.1016/j.camwa.2011.03.070. [4] L. Angermann and S. Wang, Convergence of a fitted finite volume method for the penalized Black-Scholes equation governing European and American Option pricing, Numer. Math., 106 (2007), 1-40.  doi: 10.1007/s00211-006-0057-7. [5] J. Ankudinova and M. Ehrhardt, On the numerical solution of nonlinear Black-Scholes equations, Computers and Mathematics with Applications, 56 (2008), 799-812.  doi: 10.1016/j.camwa.2008.02.005. [6] A. Bensoussan and J. L. Lions, Applications of Variational Inequalities in Stochastic Control, Studies in Mathematics and its Applications, 12. North-Holland Publishing Co., Amsterdam-New York, 1982. [7] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062. [8] R. A. Brualdi and S. Mellendorf, Regions in the complex plane containing the eigenvalues of a matrix, Amer. Math. Monthly, 101 (1994), 975-985.  doi: 10.1080/00029890.1994.12004577. [9] 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. [10] M. J. Brennan and E. S. Schwartz, Evaluating natural resource investments, The Journal of Business, 58 (1985), 135-157.  doi: 10.1086/296288. [11] W. Chen and S. Wang, A penalty method for a fractional order parabolic variational inequality governing American put option valuation, Computers & Mathematics with Applications, 67 (2014), 77-90.  doi: 10.1016/j.camwa.2013.10.007. [12] C. H. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optimization and Application, 5 (1996), 97-138.  doi: 10.1007/BF00249052. [13] G. Cortazar, E. S. Schwartz and J. Casassus, Optimal exploration investments under price and geological-technical uncertainty: A real options model, R and D Management, 31 (2001), 181–189. [14] G. Courtadon, A more accurate finite difference approximation for the valuation of options, J. Financial Economics Quant. Anal., 17 (1982), 697-703.  doi: 10.2307/2330857. [15] A. N. Daryina, A. F. Izmailov and M. V. Solodov, A class of active-set Newton methods for mixed complementarity problems, SIAM Journal on Optimization, 15 (2004), 409-429.  doi: 10.1137/S105262340343590X. [16] A. K. Dixit and R. S. Pindyck, Investment Under Uncertainty, Princeton University Press, Princeton, N.J., 1994. [17] D. J. Duffy, Finite Difference Methods in Financial Engineering – A Partial Differential Equation Approach, John Wiley & Sons Ltd, 2006. doi: 10.1002/9781118673447. [18] F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems, Vol. I, Springer Series in Operations Research, Springer-Verlag, New York, 2003. [19] M. C. Ferris and J. S. Pang, Engineering and economic applications of complementarity problems, SIAM Rev., 39 (1997), 669-713.  doi: 10.1137/S0036144595285963. [20] A. Forsgren, P. E. Gill and M. H. Wright, Interior methods for nonlinear optimization, SIAM Rev., 44 (2002), 525-597.  doi: 10.1137/S0036144502414942. [21] M. A. 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. [22] C. C. Huang and S. Wang, A power penalty approach to a nonlinear complementary problem, Operations Research Letters, 38 (2010), 72-76.  doi: 10.1016/j.orl.2009.09.009. [23] J. C. Hull, Options, Futures And Other Derivatives (9th Edition), Pearson Education, Harlow, 2014. [24] S. Jaimungal, M. O. de Souza and J. P. Zubelli, Real option pricing with mean-reverting investment and project value, The European Journal of Finance, 19 (2013), 625-644.  doi: 10.1080/1351847X.2011.601660. [25] C. Kanzow, Global optimization techniques for mixed complementarity problems, J. Glob. Optim., 16 (2000), 1-21.  doi: 10.1023/A:1008331803982. [26] D. C. Lesmana and S. Wang, Penalty approach to a nonlinear obstacle problem governing American put option valuation under transaction costs, Applied Mathematics & Computation, 251 (2015), 318–330. doi: 10.1016/j.amc.2014.11.060. [27] D. C. Lesmana and S. Wang, A numerical scheme for pricing American options with transaction costs under a jump diffusion process, J. Ind. Manag. Optim., 13 (2017), 1793–1813. doi: 10.3934/jimo.2017019. [28] N. Li and S. Wang, Pricing options on investment project expansions under commodity price uncertainty, J. Ind. Manag. Optim., 15 (2019), 261–273. doi: 10.3934/jimo.2018042. [29] W. Li and S. Wang, Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme, J. Ind. Manag. Optim., 9 (2013), 365–398. doi: 10.3934/jimo.2013.9.365. [30] D. Li and M. Fukushima, Smoothing Newton and Quasi-Newton Methods for Mixed Complementarity Problems, Computational Optimization and Applications, 17 (2000), 203–230. doi: 10.1023/A:1026502415830. [31] A. Moel and P. Tufano, When are real options exercised? An empirical study of mine closings, Review of Financial Studies, 15 (2002), 35-64. [32] N. Moyen, M. Slade and R. Uppal, Valuing risk and flexibility – a comparison of methods, Resources Policy, 22 (1996), 63-74. [33] S. C. Myers, Finance theory and financial strategy, Interfaces, 14 (1984), 126-137.  doi: 10.1287/inte.14.1.126. [34] B. F. Nielsen, O. Skavhaug and A. Tveito., Penalty and front-fixing methods for the numerical solution of American option problems, J. Comp. Fin., 5 (2002), 69-97.  doi: 10.21314/JCF.2002.084. [35] F. A. Potra and Y. Ye, Interior-point methods for nonlinear complementarity problems, Journal of Optimization Theory & Applications, 88 (1996), 617-642.  doi: 10.1007/BF02192201. [36] J. Savolainen, Real options in metal mining project valuation: Review of literature, Resources Policy, 50 (2016), 49-65.  doi: 10.1016/j.resourpol.2016.08.007. [37] R. Seydel, Tools for Computational Finance, Springer Verlag, London, 2017. doi: 10.1007/978-1-4471-7338-0. [38] L. Trigeorgis, Real Options, Princeton Series in Applied Mathematics, The MIT Press, Princeton, NJ, 1996. [39] R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Engelwood Cliffs, NJ, 1962. [40] 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. [41] S. Wang, An interior penalty method for a large-scale finite-dimensional nonlinear double obstacle problem, Applied Mathematical Modelling, 58 (2018), 217-228.  doi: 10.1016/j.apm.2017.07.038. [42] S. Wang, X. Q. Yang and K. L. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, Journal of Optimization Theory & Applications, 129 (2006), 227–254. doi: 10.1007/s10957-006-9062-3. [43] 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), 1190–1208. doi: 10.1002/num.21941. [44] S. Wang and K. Zhang, An interior penalty method for a finite-dimensional linear complementarity problem in financial engineering, Optimization Letters, 12 (2018), 1161-1178.  doi: 10.1007/s11590-016-1050-4. [45] P. Wilmott, J. Dewynne and S. Howison, Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, 1993. [46] K. Zhang and S. Wang, Convergence property of an interior penalty approach to pricing American option, J. Ind. Manag. Optim., 7 (2011), 435-447.  doi: 10.3934/jimo.2011.7.435. [47] S. Zhang, X. Wang and H. Li, Modeling and computation of water management by real options, J. Ind. Manag. Optim., 14 (2018), 81-103.  doi: 10.3934/jimo.2017038.
The values of European and American contracting options when $\kappa = 0.5$ for Test 1
The difference between the values of the European and American contracting options for Test 1
The value of the option to abandon for Test 2
Computed $W - W^*$ and Greeks of $W$ when $\lambda = 0.5$ for Test 2
The American option value and its optimal exercise curve for $\lambda = 0.5$ for Test 2
Project, production and market data and functions used in Test 1
 $Q = 10^4$ million tons $B = 30\%$ per annum $c_0 = {\rm{US}}$ ＄25 $c(t) = c_0\times e^{0.005t}$ $R = 5\%$ per annum $r = 0.06$ per annum $C = {\rm{US}}$ ＄ $5 \times 10^2$ million $T = 1$ year $\sigma = 30\%$ $\delta = 0.02$ {{q}_{0}}=\left\{ \begin{array}{*{35}{l}} \begin{align} &0.01Q\times {{e}^{0.007t}},\ \ t\le T_{1}^{*}, \\ &0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t\in (T_{1}^{*},{{T}^{*}}) \\ \end{align} \\ \end{array} \right. {{q}_{1}}=\left\{ \begin{align} &0.01Q\times {{e}^{0.007t}},\ \ \ \ \ t Q = 10^4$million tons$B = 30\%$per annum$c_0 = {\rm{US}} $＄25$c(t) = c_0\times e^{0.005t}  R = 5\%$per annum$r = 0.06 $per annum$C = {\rm{US}}$＄$5 \times 10^2$million$T = 1$year$\sigma = 30\% \delta = 0.02{{q}_{0}}=\left\{ \begin{array}{*{35}{l}} \begin{align} &0.01Q\times {{e}^{0.007t}},\ \ t\le T_{1}^{*}, \\ &0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t\in (T_{1}^{*},{{T}^{*}}) \\ \end{align} \\ \end{array} \right.{{q}_{1}}=\left\{ \begin{align} &0.01Q\times {{e}^{0.007t}},\ \ \ \ \ t
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