Question

Practice Question 1)

Jet Black is an international conglomerate with a petroleum division and is currently competing in an auction to win the right to drill for crude oil on a large of land in one year. The current market price of crude oil is $60 per barrel, and the land is believed to contains 495,000 barrels of oils. If found, the oil would cost $35 million to extract. Treasury bill that mature in one year yield a continuously compounded interest rate of 4 percent, and the standard deviation of the returns on the price of crude oil is 50 percent. Use one step binomial model to calculate the maximum bid that the company should be willing to make at the auction. Show steps please

Answer 1

Price of crude = $60 per barrel | Expected barrels of oils = 495,000 | Cost for extraction = $35,000,000

Standard Deviation of Price = 50% | Risk-free rate = 4%

Considering the auction an option where Exercise price is 35 million.

Using standard deviation, we can calculate the Up-state and Down-state price of crude in one year, which would be the one-step of our binomial model.

Up-state Price = Current Price * (1+50%) = 60 * 1.5

Up-state price = $ 90

Downstate Price = Current Price * (1-50%) = 60 * 0.5

Downstate Price = $30

Using the Upstate and Downstate price, we can calculate the risk-neutral probabilities of each state.

In risk-neutral environment, The forward price of crude should equal expected price of crude in one year. Let p be the probability of Up-state and (1-p) be the probability of down-state.

Forward Price of Crude = p * Up-state price + (1-p)* Down-state price

Forward Price = Spot price * e^RT

=> 60 * e^4% = p * 90 + (1-p)*30

=> 62.4486 = 90p + 30 - 30p

=> 32.4486 = 60p

=> p = 32.4486 / 60

Probability of up-state = 0.54081 or 54.081%

Probability of Down-state = 1 - p = 1 - 0.54081 = 0.45918 or 45.918%

Now we will calculate Payoffs in each state with 35 million being the Exercise price. As this matches with a call option, therefore, Payoff equals maximum of difference between Revenue - cost and 0. As you would not exercise and spend 35 million if crude is not found on the land.

Payoff at Upstate = Max(Crude Price * Number of barrels - Exercise Price, 0)

Payoff at Up-state = Max(90*495000 - 35,000,000, 0)

Payoff at Up-state = Max(44550000 - 35000000,0)

Payoff at Up-state = $ 9,550,000

Payoff at Down-state = Max(30*495000 - 35000000, 0)

Payoff at Down-state = Max(14850000 - 35000000, 0)

Payoff at Down-state = $0

Now we will calculate the value of the Option by discounting Expected payoff to Year 0 using continuously compounded risk-free rate.

Value of the Option = (p * Payoff at Upstate + (1-p) * Payoff at Downstate)*e^-RT

Value of the Option = (0.54081 * 9,550,000 + 0.45918 * 0) * e^-4%

Value of the Option = 5,164,742.89 * e^-4%

Value of the Option or Maximum Bid = $ 4,962,230.43

Hence, the maximum bid that the company should be willing to make is $ 4,962,230.43.

Below is the one-step binomial model: