# Optimal Investment Timing and Scale Choice of Overseas Oil Projects: A Real Option Approach
## 3 Model Description
### Theoretical Analysis of the Impact of Investment Scale on Investment Timing  
Changing the drift and volatility of unit revenue revises the optimal investment decisions of overseas oil projects. In highly uncertain environments, the fixed investment scale limits the flexibility of investment timing. The investment scale that can be selected freely allows international oil companies to make comparatively reasonable investment decisions on oil projects and increase the project value of management flexibility from the uncertainties. In this Section, we compare the critical oil price difference with and without the possibility of choosing the investment scale freely. We suppose that investors can choose optimal investment scale when they can choose oilfield recoverable factor freely. According to Section 3.3, the critical oil price, which determines the investment timing, can be rewritten as follows:  
\[P^{*}=\frac{(\gamma-1)c\kappa\beta}{\eta\xi[(\gamma-1)\beta-\gamma]}=\frac{ \beta c\kappa s^{*}+\beta I(s^{*})}{\eta(\beta-1)\xi s^{*}} \tag{28}\]  
where  
\[s^{*}=\left\{\frac{c\cdot\kappa}{B\cdot OIP\cdot[(\gamma-1)\beta-\gamma]} \right\}^{\frac{1}{\gamma-1}}\]Supposing that the oilfield recovery factor is fixed, the corresponding critical oil price (\(P_{0}\)) can be written as Equation (29):  
\[P_{0}=\frac{\beta c\kappa s_{0}+\beta I(s_{0})}{\eta(\beta-1)\xi s_{0}} \tag{29}\]  
The function \(G\) is set to compare the critical oil price difference between the fixed investment scale and the variable one. In this part, the fixed investment scale is optimal under the initial condition. \[ \begin{array}{ll}G&=P^{*}-P_{0}=\frac{\beta c\kappa s^{*}+\beta I(s^{*})}{ \eta(\beta-1)\xi s^{*}}-\frac{\beta c\kappa s_{0}+\beta I(s_{0})}{\eta(\beta- 1)\xi s_{0}}=\frac{\beta}{\eta(\beta-1)\xi}(\frac{I(s^{*})}{s^{*}}-\frac{I(s_{ 0})}{s_{0}})\\ &=\frac{\beta\cdot B\cdot OIP}{\eta(\beta-1)\xi}(s^{*\gamma-1}-s_{0}^{ \gamma-1})=(\frac{\beta\cdot B\cdot OIP}{\eta(\beta-1)\xi})\cdot(\frac{c\cdot \kappa}{B\cdot OIP\cdot[(\gamma-1)\beta-\gamma]}-s_{0}^{\gamma-1})\end{array}  \tag{30}\]  
Under the initial condition, \(G=0\) and \(s_{0}=s^{*}\) because the parameters of the drift and volatility of unit revenue are the same. Firstly, we focus on the drift rate and compare the critical oil price difference with and without the possibility of choosing the investment scale freely. The following is known by derivation of Equation (30):  
\[\frac{\partial G}{\partial u}=\frac{\partial(P^{*}-P_{0})}{\partial u}=(\frac {\beta\cdot B\cdot OIP}{\eta(\beta-1)\xi})\cdot(\frac{c\cdot\kappa}{B\cdot OIP \cdot[(\gamma-1)\beta-\gamma]}-s_{0}^{\gamma-1})^{{}^{\prime}}_{u}+(\frac{ \beta\cdot B\cdot OIP}{\eta(\beta-1)\xi})^{{}^{\prime}}_{u}(\frac{c\cdot \kappa}{B\cdot OIP\cdot[(\gamma-1)\beta-\gamma]}-s_{0}^{\gamma-1}) \tag{31}\]  
Under the initial condition, \(s_{0}=s^{*}\), we know the following:  
\[(\frac{\beta\cdot B\cdot OIP}{\eta(\beta-1)\xi})^{{}^{\prime}}_{u}\frac{c \cdot\kappa}{B\cdot OIP\cdot[(\gamma-1)\beta-\gamma]}-s_{0}^{\gamma-1}=0, \frac{\partial\kappa}{\partial u}>0,\frac{\partial\beta}{\partial u}<0\]  
Therefore,  
\[\frac{\partial G}{\partial u}>0 \tag{32}\]  
Secondly, we focus on the volatility rate and compare the critical oil price difference with and without the possibility of choosing the investment scale freely.