J&B Drilling Company has recently acquired a lease to drill for natural gas in the remote region of southwest Louisana and southeast Texas. The area has long been known for oil and gas production, and the company is optismistic about the prospects of the lease. The lease contract has a three-year life and allows J&B to begin exploration at any time up until the end of the three-year term.

J&B's engineers have estimated the volume of the natural gas they hope to extract from the leasehold and have placed a value of $25 million on it, on the condition that explorations begin immediately. The cost of developing the property is extimated to be $25 million (regardless of when the property is developed over the next three years). Based on the historical volatilities in the returns of the similar investments and other relevant information, J&B's analysts have estimated that the value of the investment opportunity will evolve over the next three years. The risk-free rate of interest is currently 5%, and the risk-natural probability of an uptick in the value of the investment is estimated to be 46.26%.

Build a binomial tree, using the starting value of $25m and a volatility for the project value of 12% to calculate the up and down factors (u and d) and the risk-neutral probabilities (p and 1-p), and then construct a tree in Excel. You can assume that the time horizon of the project is three years and that the risk-free rate is 5%, but assume that there is no convenience yield. Once you have built your model, answer these questions:

a) What are the up and down factors for the change in project value in each period? What are the risk-neutral probabilities?

b) Would you invest in this project if there was no flexibility (i.e., develop now or never)? What would the NPV be in that case?

c) What is the value of the project with flexibility (i.e., with the option to develop the lease)?

d) Use the Black-Scholes equation to calculate the value of the option to develop. How does this compare to your answer from part c)? What assumption are you making about the option to develop when you use this approach? Is this a valid assumption in this case?

e) Suppose that there is a 4% lease penalty that accrues to the property owners for every year that you decide to delay (i.e., essentially a 4% convenience yield that reduces the value of the property). Update your binomial model and calculate the value with the flexibility to develop. (Hint: use the same approach you used for part a) but adjust the values for u and d by subtracting the convenience yield from the volatility, and the value for p by subtracting the convenience yield from r. What is the project value in this case? When would you exercise the option to develop in this case?