Question

The Karns Oil Company is deciding whether to drill for oil on a tract of land that the company owns. The company estimates that the project would cost $8 million today. Karns estimates that once drilled, the oil will generate positive net cash flows of $4 million a year at the end of each of the next 4 years. Although the company is fairly confident about its cash flow forecast, in 2 years it will have more information about the local geology and about the price of oil. Karns estimates that if it waits 2 years then the project would cost $9 million. Moreover, if it waits 2 years, then there is a 90% chance that the net cash flows would be $4.2 million a year for 4 years, and there is a 10% chance that they would be $2.2 million a year for 4 years. Assume that all cash flows are discounted at 10%. Use the Black-Scholes model to estimate the value of the option. Assume the variance of the project's rate of return is 1.11% and that the risk-free rate is 6%. Do not round intermediate calculations. Enter your answer in millions. For example, an answer of $1.2 million should be entered as 1.2, not 1,200,000. Round your answer to two decimal places.

Answer 1

Step 1: Estimate of P0Probability12345PV at Year 190%$4.2 $4.2 $4.2 $4.2 $13.31310%$2.2 $2.2 $2.2 $2.2 $6.974Example: 4.2/(1.1)^1 + 4.2/(1.1)^2 + 4.2/(1.1)^3 + 4.2/(1.1)^4 = $13.313Step 2: Find the expected PV at the current date, Year 0PV Year 0 PV Year 1$13,313 HighXXXXX$6.974 LowPV Year 0 = PV of Exp. PV Year 1 = [0.90*$13.313) + (0.10 * $6.974) PV Year 0 = 11.98170 + 0.69740PV Year 0 = $12.679**Based on the calculations P = $12.679Step 3: Estimating ?2 for the Black-Scholes modelEstimating with Direct Approach Model

PV Year 0 PV Year 1Return$13,313 High5% {(13.313 – 12.679)/12.679}$12.679$6.974 Low-45% {(6.974 – 12.679)/12.679}Expected Return and Variance of ReturnE(Ret.) = 0.9(0.05) + 0.10(-0.45) = 0.0%?2 = 0.9(0.05-0)2 + 0.1(-0.45-)2 = 0.02250 = 2.3%Estimating ?2 with the Indirect Approach?^2=(ln(CV?^2?+1))/tValue of the project for each scenario at the expiration dateHigh $13.313Low$6.974E(PV) = 0.90(13.313) + 0.10(6.974) = $12.679?PV = [0.90(13.313 – 12.679)2 + 0.10(6.974 – 12.679)2 ]1/2?PV = $1.90Expected Coefficient of Variation, CVPV (at the time the option expires)CVPV = 1.90/12.679 = 0.15Now we use the formula to estimate ?2 ?^2=(ln(0.15?^2?+1))/1?^2= 0.02225 = 2.23%Black-Scholes Inputs: P=$12.679, X=$8, rRF = 6%, t=1 year, ?2=0.02225V = $12.679[N(d1) - $8e-(0.06)(1) [N(d2)]d1 = ln(12.679/8)+[(0.06 + 0.02225/2)](1) / (0.02225)0.5 (1).05 = 0.436526

d2 = d1 – (0.02225)0.5 (1)0.05