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# Modeling and computation of water management by real options

• * Corresponding author: Shuhua Zhang
This project was supported in part by the National Basic Research Program (2012CB955804), the Major Research Plan of the National Natural Science Foundation of China (91430108), the National Natural Science Foundation of China (11171251), and the Major Program of Tianjin University of Finance and Economics (ZD1302).
• It becomes increasingly important to manage water and improve the efficiency of irrigation under higher temperatures and irregular precipitation patterns. The choice of investment in water saving technologies and its timing play key roles in improving efficiency of water use. In this paper, we use a real option approach to establish a model to handle future uncertainties about the water price. In addition, to match the practical situation, the expiration of the real option is considered to be finite in our model, such that it is difficult to solve the model. Therefore, we reformulate the problem into a linear parabolic variational inequality (Ⅵ) and develop a power penalty method to solve it numerically. Thus, a nonlinear partial differential equation (PDE) is obtained, which is shown to be uniquely solvable and the solution of the nonlinear PDE converges to that of the Ⅵ at the rate of $O(λ^{-\frac{k}{2}})$ with $λ$ being the penalty number. Furthermore, a so-called fitted finite volume method is proposed to solve the nonlinear PDE. Finally, several numerical experiments are performed. It is shown that the subjective discount rate will affect the investment boundary mostly, and the flexibility to suspend operation will enlarge the investment region.

Mathematics Subject Classification: Primary: 49J20, 65M08, 91B06.

 Citation: • • Figure 1.  European real option values for Test 1

Figure 2.  American real option values for Test 2

Figure 3.  The Δ and the optimal exercise boundary for Test 2

Figure 4.  Comparative statics for the main parameters for Test 2

Figure 5.  Real option values for Test 3

Figure 6.  The ∆ and the optimal exercise boundary for Test 3

Figure 7.  The effect of the suspending operation on investment boundary for Test 3

Figure 8.  Trigger prices under different levels of subsidy in four time points

Table 1.  Computed errors in the $L^{\infty}$-norm at $t = 0$

 mesh $L^{\infty}$-norm ratio mesh $L^{\infty}$-norm ratio} $2^5\times2^4$ 197.4574 $2^{10}\times2^9$ 0.4418 1.9276 $2^6\times2^5$ 55.1497 3.5804 $2^{11}\times2^{10}$ 0.0453 9.7528 $2^7\times2^6$ 28.7732 1.9176 $2^{12}\times2^{11}$ 0.0124 3.6532 $2^8\times2^7$ 2.9357 9.8012 $2^{13}\times2^{12}$ 0.0054 2.2963 $2^9\times2^8$ 0.8512 3.4489
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