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

• * Corresponding author: Carlos Vázquez

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.

Mathematics Subject Classification: Primary: 91G80, 65M25; Secondary: 91G60, 65M22.

 Citation: • • Figure 1.  Computed mine value at time $t = 0$ for the real case

Figure 2.  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)

Figure 3.  Probability of project completion (left) and expected lifetime (right) at time $t = 0$

Figure 4.  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)

Table 1.  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

Table 2.  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 $tonne^{-1}$

Table 3.  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 $8.9779 \times 10^{-3}$ $9.0633\times 10^{-5}$ $6.5549\times 10^{-7}$ $1.0258\times 10^{-7}$
• Figures(4)

Tables(3)

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