# Optimal Investment Timing and Scale Choice of Overseas Oil Projects: A Real Option Approach
## 3 Model Description
### Optimal Investment Strategy  
Considering management flexibility of delaying investment and uncertainties of oil prices and the US dollar exchange rate, the choice of optimal investment timing and scale in overseas oil projects is based on a comparison of net present value and option value. In the highly uncertain environment, the investment decision is a matter of balancing the following [36]: Firstly, the interest from investment cost, \(I(\mathbf{s})\), will be saved if the investors do not invest. Secondly, investing in that moment means that the cash flow of the oil project is produced. Thirdly, investing in that moment means that if the oil price drops to a low level in the future, the investors will suffer the losses. The optimal investment strategy follows from two critical investment conditions that are in line with the above tradeoff [38; 44; 51].  
The first one is the \"value-matching\" condition, which says that the real option value equals the net present value of oil projects when investment occurs.  
\[\begin{gathered} F(\theta^{*})\quad=D\theta^{*\beta}=\mathrm{V} \left(\theta^{*},s_{optimal}(\theta^{*})\right)-TC(s_{optimal}(\theta^{*}))- I\left(s_{optimal}(\theta^{*})\right)\\ =\left[\left(\frac{1}{B\cdot OIP\cdot\gamma}\right)^{\frac{1}{ \gamma-1}}-B\cdot OIP\cdot\left(\frac{1}{B\cdot OIP\cdot\gamma}\right)^{\frac {\gamma}{\gamma-1}}\right]\cdot\left(\theta^{*}\xi-c\kappa\right)^{\frac{ \gamma}{\gamma-1}}\end{gathered} \tag{22}\]  
where \(F(\theta)\) denotes the option value when the unit revenue is \(\theta\).  
The second one is the \"smooth pasting\" condition, which entails that both functions have the same slope.  
\[\begin{array}{ll}F^{\prime}(\theta^{*})&=D\beta\theta^{*(\beta-1)}\\ &=\left[\left(\frac{1}{B\cdot OIP\cdot\gamma}\right)^{\frac{1}{\gamma-1}}-B \cdot OIP\cdot\left(\frac{1}{B\cdot OIP\cdot\gamma}\right)^{\frac{\gamma}{ \gamma-1}}\right]\cdot\frac{\gamma\xi}{\gamma-1}\cdot(\theta^{*}\xi-c\kappa)^ {\frac{1}{\gamma-1}}\end{array} \tag{23}\]  
Solving Equations (22) and (23), the following analytical solutions can be obtained:  
\[\theta^{*}=\frac{(\gamma-1)c\kappa\beta}{\xi[(\gamma-1)\beta-\gamma]} \tag{24}\]  
\[s^{*}=\left\{\frac{c\cdot\kappa}{B\cdot OIP\cdot[(\gamma-1)\beta-\gamma]} \right\}^{\frac{1}{\gamma-1}}(s_{optimal}\geq s^{*}) \tag{25}\]  
\[P^{*}=\frac{(\gamma-1)c\kappa\beta}{\eta\xi[(\gamma-1)\beta-\gamma]} \tag{26}\]  
where \(\theta^{*}\), \(P^{*}\), and \(s^{*}\) denote the critical unit revenue, critical oil price, and critical recovery factor, respectively.  
Considering the investment timing, the critical oil price is the investment threshold for triggering oilfield development. If the oil price equals the critical oil price, the optimal recovery factor at that time will equal the critical recovery factor. For international oil companies, if the oil price is below the critical oil price, it is never optimal to invest. If the oil price is higher than the critical oil price, it is optimal to invest immediately, and the optimal recovery factor, which determines the optimal investment scale and cost, can be selected by Equation (16) at that time. Above all, the oil project value can be obtained according to Equation (27):  
\[\begin{array}{ll}V=\left\{\begin{array}{ll}R\Big{(}\theta,s_{optimal}( \theta)\Big{)}-TC(s_{optimal}(\theta))-I\Big{(}s_{optimal}(\theta)\Big{)},P>P ^{*}\\ F(\theta),P\leq P^{*}\end{array}\right.\end{array} \tag{27}\]  
In summary, it is true that this model is in line with three important properties of overseas oil projects: irreversible capital investment, high uncertainty, and management flexibility.