# 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  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 & Management Optimization, 2020, 16 (3) : 1349-1368. doi: 10.3934/jimo.2019006
##### References:

show all references

##### References:
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
 [1] Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095 [2] Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021018 [3] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351 [4] Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250 [5] Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial & Management Optimization, 2021, 17 (1) : 133-149. doi: 10.3934/jimo.2019103 [6] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 [7] Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 [8] Hui Gao, Jian Lv, Xiaoliang Wang, Liping Pang. An alternating linearization bundle method for a class of nonconvex optimization problem with inexact information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 805-825. doi: 10.3934/jimo.2019135 [9] Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021006 [10] Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 [11] Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127 [12] Qiang Fu, Xin Guo, Sun Young Jeon, Eric N. Reither, Emma Zang, Kenneth C. Land. The uses and abuses of an age-period-cohort method: On the linear algebra and statistical properties of intrinsic and related estimators. Mathematical Foundations of Computing, 2020  doi: 10.3934/mfc.2021001 [13] Matúš Tibenský, Angela Handlovičová. Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1181-1195. doi: 10.3934/dcdss.2020226 [14] Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075 [15] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [16] Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391 [17] Lan Luo, Zhe Zhang, Yong Yin. Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems. Journal of Industrial & Management Optimization, 2021, 17 (2) : 779-803. doi: 10.3934/jimo.2019134 [18] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [19] Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013 [20] Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020180

2019 Impact Factor: 1.366