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

María Suárez-Taboada; Carlos Vázquez

doi:10.3934/dcdsb.2018254

Article Contents

2019, Volume 24, Issue 8: 3503-3523. Doi: 10.3934/dcdsb.2018254

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

María Suárez-Taboada^1, and
Carlos Vázquez^2, ,

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
* Corresponding author: Carlos Vázquez

Received: February 28, 2018

Revised: March 31, 2018

Early access: August 2018

Published: July 2019

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

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Abstract

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.

Keywords:

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

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

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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)

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

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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)

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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

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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 $tonne^{-1}$

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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	$8.9779 \times 10^{-3}$	$9.0633\times 10^{-5}$	$6.5549\times 10^{-7}$	$1.0258\times 10^{-7}$

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