# A Real Option Pricing Decision of Construction Project under Group Bidding Environment
## 2 A Real Option Pricing Decision Model under Group Bidding Environments
### The Option Value Calculation  
Set \(P\) as the price made by the bidder and remain unchanged; \(t\) as a moment in the bidding preparation stage; \(C_{t}\) as estimated construction costs for the bidder in the bidding preparation stage. In the option period, the construction costs are an uncertain factor. According to the Black-Scholes option pricing model, the estimated construction cost \(C_{t}\) follows the geometric Brownian motion.  
\[dC_{t}=\mu C_{t}dt+\sigma C_{t}dN_{t} \tag{1}\]where \(dC_{t}\) is the change of estimated costs within \(dt\) time interval; \(\mu\) is the drift percentage; \(\sigma\) is estimated costs volatility; \(N_{t}\) is a Wiener process, also known as Brown motion; \(dN_{t}\) is the increments of the standard Wiener process; \(\mu\) and \(\sigma\) are constants, usually based on the past bidding data, to represent the expected future investment costs trends.  
Assuming that the moment when the bidder is selected and invited to sign the contract is \(T\), the bidder's profit at \(T\) is:  
\[\max([(P-C_{T});0]) \tag{2}\]  
If the expected construction costs are higher than the bidding price in the bidding document at \(T\), the winning bidder can choose not to sign the contract. Only when the expected profit is positive at \(T\), the winning bidder will choose to sign. Since the bidding price, \(P\), is fixed when the bidder submits the bidding documents, and the estimated cost, \(C_{t}\), is flexible during the project execution period, the option income at \(T\) is similar to that of the put option. The put option means that if the market price of the underlying future asset falls below the exercise price (bidding price), \(P\), the holders (the winning bidder) can make a profit by taking the initial price (price higher than the current market price), \(P\), as the final exercise price.  
The put option formula in the Black-Scholes model is as follows:  
\[P^{\prime}(S_{t},\ X)=Xe^{-r(T-t)}N(-d_{2}^{\prime})-S_{t}N(-d_{1}^{\prime}) \tag{3}\]  
\[-d_{2}^{\prime}=\frac{-ln\frac{S_{t}}{X}-\left(r-\frac{1}{2}\sigma^{2}\right) (T-t)}{\sigma\sqrt{T-t}}=\frac{ln\frac{S_{t}}{X}-\left(r-\frac{1}{2}\sigma^{2 }\right)T^{\prime}}{\sigma\sqrt{T^{\prime}}} \tag{4}\]  
\[-d_{1}^{\prime}=-d_{2}^{\prime}-\sigma\sqrt{T^{\prime}} \tag{5}\]  
where \(P^{\prime}\) is the put option value; \(S_{t}\) is the current price of the asset being traded; \(X\) is the option delivery price; \(r\) is the risk-free interest rate; \(T^{\prime}\) is the option period, with year as the unit; \(N(\cdot)\) is the cumulative probability distribution function of a normally distributed variable; \(\sigma^{2}\) is the annualized variance.  
In the field of construction project, \(P\)\(\equiv\)\(X\), \(C_{t}\)\(\equiv\)\(S_{t}\). The option value \(F\) of the contract signed at any time when \(t\) < \(T\) can be expressed as:  
\[F(P,\ C_{t})=Pe^{-r(T-t)}N(d_{1})-C_{t}N(d_{2}) \tag{6}\]  
\[d_{1}=\frac{\ln\frac{P}{C_{t}}-\left(r-\frac{1}{2}\sigma^{2}\right)(T-t)}{ \sigma\sqrt{T-t}} \tag{7}\]  
\[d_{2}=d_{1}-\sigma\sqrt{T-t} \tag{8}\]  
where \(T-t\) is the option period, that is, the period from making price decisions to winning the final bid, in the unit of the year.