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August  2019, 24(8): 3503-3523. doi: 10.3934/dcdsb.2018254

## Numerical methods for PDE models related to pricing and expected lifetime of an extraction project under uncertainty

 1 Department of Mathematics, University of A Coruña and CITIC, Campus Elviña s/n, 15071 - A Coruña, Spain 2 Department of Mathematics, University of A Coruña, CITIC and ITMATI, Campus Elviña s/n, 15071 - A Coruña, Spain

* Corresponding author: Carlos Vázquez

Received  March 2018 Revised  April 2018 Published  August 2018

Fund Project: This article has been funded by Spanish MINECO (Projects MTM2013-47800-C2-1-P and MTM2016-76497-R) and Xunta de Galicia (Grant GRC2014/044), including FEDER funds.

Numerical techniques for solving some mathematical models related to a mining extraction project under uncertainty are proposed. The mine valuation is formulated as a complementarity problem associated to a degenerate second order partial differential equation (PDE), which incorporates the option to abandon the project. The probability of completion and the expected lifetime of the project are the respective solutions of problems governed by similar degenerated PDE operators. In all models, the underlying stochastic factors are the commodity price and the remaining resource. After justifying the required boundary conditions on the computational bounded domain, the proposed numerical techniques mainly consist of a Crank-Nicolson characteristics method for the time discretization to cope with the convection dominating setting and Lagrange finite elements for the discretization in the commodity and resource variables, with the additional use of an augmented Lagrangian active set method for the complementarity problem. Some numerical examples are discussed to illustrate the performance of the methods and models.

Citation: María Suárez-Taboada, Carlos Vázquez. Numerical methods for PDE models related to pricing and expected lifetime of an extraction project under uncertainty. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3503-3523. doi: 10.3934/dcdsb.2018254
##### References:
 [1] M. Bercovier, O. Pironneau and V. Sastri, Finite elements and characteristics for some parabolic-hyperbolic problems, Applied Mathematical Modelling, 7 (1983), 89-96.  doi: 10.1016/0307-904X(83)90118-X.  Google Scholar [2] A. Bermúdez, M. R. Nogueiras and C. Vázquez, Numerical analysis of convection-diffusion-reaction problems with higher order characteristics finite elements. Part Ⅱ: Fully discretized scheme and quadrature formulas, SIAM Journal Numerical Analysis, 44 (2006), 1854-1876.  doi: 10.1137/040615109.  Google Scholar [3] A. Bermúdez, M. R. Nogueiras and C. Vázquez, Numerical solution of variational inequalities for pricing Asian options by higher order Lagrange-Galerkin methods, Applied Numerical Mathematics, 56 (2006), 1256-1270.  doi: 10.1016/j.apnum.2006.03.026.  Google Scholar [4] A. Bermúdez, M. R. Nogueiras and C. Vázquez, Comparison of two algorithms to solve a fixed-strike Amerasian options pricing problem, in Free Boundary Problems, International Series in Numerical Mathematics, 154 (eds. I. N. Figueiredo, J. F. Rodrigues and L. Santos), Birkhäuser, (2007), 95-106. doi: 10.1007/978-3-7643-7719-9_10.  Google Scholar [5] M. J. Brennan and E. S. Schwartz, Evaluating natural resources investments, Journal of Business, 58 (1985), 135-157.  doi: 10.1086/296288.  Google Scholar [6] F. Black and M. Scholes, The pricing of option and corporate liabilities, Journal Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar [7] D. Castillo, A. M. Ferreiro, J. A. García-Rodríguez and C. Vázquez, Numerical methods to solve PDE models for pricing business companies in different regimes and implementation in GPUs, Applied Mathematics and Computation, 219 (2013), 11233-11257.  doi: 10.1016/j.amc.2013.05.032.  Google Scholar [8] Z. Cheng and P. A. Forsyth, A semi-Lagrangian approach for natural gas storage, SIAM Journal on Scientific Computing, 30 (2007), 339-368.  doi: 10.1137/060672911.  Google Scholar [9] Y. D'Halluin, P. A. Forsyth and G. Labahn, A semi-Lagrangian approach for American Asian options under jump diffusion, SIAM Journal on Scientific Computing, 27 (2005), 315-345.  doi: 10.1137/030602630.  Google Scholar [10] A. K. Dixit and R. S. Pindyck, Investment Under Uncertainty, Princeton University Press, Princeton, NJ, 1994. Google Scholar [11] J. Douglas and T. F. Russell Jr, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM Journal on Numerical Analysis, 19 (1982), 871-885.  doi: 10.1137/0719063.  Google Scholar [12] G. W. Evatt, P. V. Johnson, P. W. Duck and S. D. Howell, Mine valuations in the presence of a Stochastic ore-grade, Int. Assoc. Eng., 3 (2010), 1811-1816.   Google Scholar [13] G. W. Evatt, P. V. Johnson, P. W. Duck, S. D. Howell and J. Moriarty, The expected lifetime of an extraction project, Proceedings of the Royal Society, 467 (2011), 244-263.  doi: 10.1098/rspa.2010.0247.  Google Scholar [14] G. W. Evatt, P. V. Johnson, P. W. Duck and S. D. Howell, Optimal costless extraction rate changes from a non-renewable resource, European Journal of Applied Mathematics, 25 (2014), 681-705.  doi: 10.1017/S0956792514000229.  Google Scholar [15] G. Fichera, On a Unified theory of boundary value problems for elliptic-parabolic equations of second order in boundary value problems, University of Wisconsin Press, 1960. Google Scholar [16] R. Kangro and R. Nicolaides, Far field boundary conditions for Black-Scholes equations, SIAM Journal Numerical Analysis, 38 (2000), 1357-1368.  doi: 10.1137/S0036142999355921.  Google Scholar [17] T. Kärkkäinen, K. Kunisch and P. Tarvainen, Augmented Lagrangian active set methods for obstacle problems, SIAM Journal Numerical Analysis, 38 (2000), 1357-1368.  doi: 10.1023/B:JOTA.0000006687.57272.b6.  Google Scholar [18] B. ∅ksendal, Stochastic Differential Equations, 5$^{th}$ edition, Springer, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar [19] O. A. Oleinik and E. V. Radkevic, Second Order Equations with Nonnegative Characterisitc Form, A. M. S. and Plenum Press, Providence, 1973.  Google Scholar [20] A. Pascucci, PDE and Martingale Methods in Option Pricing, Bocconi & Springer Series, Springer-Verlag, New York, 2011. doi: 10.1007/978-88-470-1781-8.  Google Scholar [21] A. Pascucci, M. Suárez-Taboada and C. Vázquez, Mathematical analysis and numerical methods for a PDE model of a stock loan pricing problem, Journal of Mathematical Analysis and Applications, 403 (2013), 38-53.  doi: 10.1016/j.jmaa.2013.02.007.  Google Scholar

show all references

##### References:
 [1] M. Bercovier, O. Pironneau and V. Sastri, Finite elements and characteristics for some parabolic-hyperbolic problems, Applied Mathematical Modelling, 7 (1983), 89-96.  doi: 10.1016/0307-904X(83)90118-X.  Google Scholar [2] A. Bermúdez, M. R. Nogueiras and C. Vázquez, Numerical analysis of convection-diffusion-reaction problems with higher order characteristics finite elements. Part Ⅱ: Fully discretized scheme and quadrature formulas, SIAM Journal Numerical Analysis, 44 (2006), 1854-1876.  doi: 10.1137/040615109.  Google Scholar [3] A. Bermúdez, M. R. Nogueiras and C. Vázquez, Numerical solution of variational inequalities for pricing Asian options by higher order Lagrange-Galerkin methods, Applied Numerical Mathematics, 56 (2006), 1256-1270.  doi: 10.1016/j.apnum.2006.03.026.  Google Scholar [4] A. Bermúdez, M. R. Nogueiras and C. Vázquez, Comparison of two algorithms to solve a fixed-strike Amerasian options pricing problem, in Free Boundary Problems, International Series in Numerical Mathematics, 154 (eds. I. N. Figueiredo, J. F. Rodrigues and L. Santos), Birkhäuser, (2007), 95-106. doi: 10.1007/978-3-7643-7719-9_10.  Google Scholar [5] M. J. Brennan and E. S. Schwartz, Evaluating natural resources investments, Journal of Business, 58 (1985), 135-157.  doi: 10.1086/296288.  Google Scholar [6] F. Black and M. Scholes, The pricing of option and corporate liabilities, Journal Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.  Google Scholar [7] D. Castillo, A. M. Ferreiro, J. A. García-Rodríguez and C. Vázquez, Numerical methods to solve PDE models for pricing business companies in different regimes and implementation in GPUs, Applied Mathematics and Computation, 219 (2013), 11233-11257.  doi: 10.1016/j.amc.2013.05.032.  Google Scholar [8] Z. Cheng and P. A. Forsyth, A semi-Lagrangian approach for natural gas storage, SIAM Journal on Scientific Computing, 30 (2007), 339-368.  doi: 10.1137/060672911.  Google Scholar [9] Y. D'Halluin, P. A. Forsyth and G. Labahn, A semi-Lagrangian approach for American Asian options under jump diffusion, SIAM Journal on Scientific Computing, 27 (2005), 315-345.  doi: 10.1137/030602630.  Google Scholar [10] A. K. Dixit and R. S. Pindyck, Investment Under Uncertainty, Princeton University Press, Princeton, NJ, 1994. Google Scholar [11] J. Douglas and T. F. Russell Jr, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM Journal on Numerical Analysis, 19 (1982), 871-885.  doi: 10.1137/0719063.  Google Scholar [12] G. W. Evatt, P. V. Johnson, P. W. Duck and S. D. Howell, Mine valuations in the presence of a Stochastic ore-grade, Int. Assoc. Eng., 3 (2010), 1811-1816.   Google Scholar [13] G. W. Evatt, P. V. Johnson, P. W. Duck, S. D. Howell and J. Moriarty, The expected lifetime of an extraction project, Proceedings of the Royal Society, 467 (2011), 244-263.  doi: 10.1098/rspa.2010.0247.  Google Scholar [14] G. W. Evatt, P. V. Johnson, P. W. Duck and S. D. Howell, Optimal costless extraction rate changes from a non-renewable resource, European Journal of Applied Mathematics, 25 (2014), 681-705.  doi: 10.1017/S0956792514000229.  Google Scholar [15] G. Fichera, On a Unified theory of boundary value problems for elliptic-parabolic equations of second order in boundary value problems, University of Wisconsin Press, 1960. Google Scholar [16] R. Kangro and R. Nicolaides, Far field boundary conditions for Black-Scholes equations, SIAM Journal Numerical Analysis, 38 (2000), 1357-1368.  doi: 10.1137/S0036142999355921.  Google Scholar [17] T. Kärkkäinen, K. Kunisch and P. Tarvainen, Augmented Lagrangian active set methods for obstacle problems, SIAM Journal Numerical Analysis, 38 (2000), 1357-1368.  doi: 10.1023/B:JOTA.0000006687.57272.b6.  Google Scholar [18] B. ∅ksendal, Stochastic Differential Equations, 5$^{th}$ edition, Springer, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar [19] O. A. Oleinik and E. V. Radkevic, Second Order Equations with Nonnegative Characterisitc Form, A. M. S. and Plenum Press, Providence, 1973.  Google Scholar [20] A. Pascucci, PDE and Martingale Methods in Option Pricing, Bocconi & Springer Series, Springer-Verlag, New York, 2011. doi: 10.1007/978-88-470-1781-8.  Google Scholar [21] A. Pascucci, M. Suárez-Taboada and C. Vázquez, Mathematical analysis and numerical methods for a PDE model of a stock loan pricing problem, Journal of Mathematical Analysis and Applications, 403 (2013), 38-53.  doi: 10.1016/j.jmaa.2013.02.007.  Google Scholar
Computed mine value at time $t = 0$ for the real case
Abandonnment region (black) and non abandonment region (white) at time $t = 0$ in the computational domain (left) and its zoom in the domain $[0, 1.5] \times [0, 2.5]$ (right)
Probability of project completion (left) and expected lifetime (right) at time $t = 0$
Probability of completion with respect to time to expiry for $S = 0.8, \, 1$ and $1.2$, with fixed $Q = 0.5$ (left) and $Q = 10$ (right)
Data of the quadrangular finite element meshes
 Mesh 8 Mesh 16 Mesh 32 Mesh 64 Number of nodes 289 1089 4225 16641 Nomber of elements 64 256 1024 4096
 Mesh 8 Mesh 16 Mesh 32 Mesh 64 Number of nodes 289 1089 4225 16641 Nomber of elements 64 256 1024 4096
Parameter values for the academic test with analytical solution and the real mine
 Academic test Real mine Extraction costs ($\epsilon_M$) 1 1 ＄ $tonne^{-1}$ Processing costs ($\epsilon_P$) 4 4 ＄ $tonne^{-1}$ Interest rate ($r$) 0.1 10 $\%$ $yr^{-1}$ Dividend yield ($\delta$) 0.1 10 $\%$ $yr^{-1}$ Volatility ($\sigma$) 0.3 30 $\%$ $yr^{-\frac{1}{2}}$ Maximum duration extraction ($T$) 1 14 $yr$ $q$ 1 1 $G$ 9.74 9.74 g $tonne^{-1}$
 Academic test Real mine Extraction costs ($\epsilon_M$) 1 1 ＄ $tonne^{-1}$ Processing costs ($\epsilon_P$) 4 4 ＄ $tonne^{-1}$ Interest rate ($r$) 0.1 10 $\%$ $yr^{-1}$ Dividend yield ($\delta$) 0.1 10 $\%$ $yr^{-1}$ Volatility ($\sigma$) 0.3 30 $\%$ $yr^{-\frac{1}{2}}$ Maximum duration extraction ($T$) 1 14 $yr$ $q$ 1 1 $G$ 9.74 9.74 g $tonne^{-1}$
Relative errors in $l^{\infty}((0, T);l^2(\Omega))$ discrete norm between the exact and numerical solutions for the academic test
 $\Delta \tau= 10^{-1}$ $\Delta \tau= 10^{-2}$ $\Delta \tau= 10^{-3}$ $\Delta \tau= 10^{-4}$ Mesh 8 $4.3913 \times 10^{-3}$ $5.4307\times 10^{-3}$ $2.8440\times 10^{-5}$ $2.8942\times 10^{-5}$ Mesh 16 $5.4574 \times 10^{-3}$ $5.4133\times 10^{-5}$ $4.9435\times 10^{-6}$ $4.4366\times 10^{-6}$ Mesh 32 $7.8917 \times 10^{-3}$ $7.9003\times 10^{-5}$ $2.5526\times 10^{-6}$ $3.0282\times 10^{-7}$ Mesh 64 $8.9779 \times 10^{-3}$ $9.0633\times 10^{-5}$ $6.5549\times 10^{-7}$ $1.0258\times 10^{-7}$
 $\Delta \tau= 10^{-1}$ $\Delta \tau= 10^{-2}$ $\Delta \tau= 10^{-3}$ $\Delta \tau= 10^{-4}$ Mesh 8 $4.3913 \times 10^{-3}$ $5.4307\times 10^{-3}$ $2.8440\times 10^{-5}$ $2.8942\times 10^{-5}$ Mesh 16 $5.4574 \times 10^{-3}$ $5.4133\times 10^{-5}$ $4.9435\times 10^{-6}$ $4.4366\times 10^{-6}$ Mesh 32 $7.8917 \times 10^{-3}$ $7.9003\times 10^{-5}$ $2.5526\times 10^{-6}$ $3.0282\times 10^{-7}$ Mesh 64 $8.9779 \times 10^{-3}$ $9.0633\times 10^{-5}$ $6.5549\times 10^{-7}$ $1.0258\times 10^{-7}$
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