Practice Question 1) Jet Black

Practice Question 1)

Jet Black is an international conglomerate with a petroleumdivision and is currently competing in an auction to win the rightto drill for crude oil on a large of land in one year. The currentmarket price of crude oil is $60 per barrel, and the land isbelieved to contains 495,000 barrels of oils. If found, the oilwould cost $35 million to extract. Treasury bill that mature in oneyear yield a continuously compounded interest rate of 4 percent,and the standard deviation of the returns on the price of crude oilis 50 percent. Use one step binomial model to calculate the maximumbid that the company should be willing to make at the auction. Showsteps please


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 35million.

Using standard deviation, we can calculate the Up-state andDown-state price of crude in one year, which would be the one-stepof 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 therisk-neutral probabilities of each state.

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

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

Forward Price = Spot price * eRT

=> 60 * e4% = 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 or45.918%

Now we will calculate Payoffs in each state with 35 millionbeing 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 crudeis 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 discountingExpected payoff to Year 0 using continuously compounded risk-freerate.

Value of the Option = (p * Payoff at Upstate + (1-p) * Payoff atDownstate)*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 bewilling to make is $ 4,962,230.43.

Below is the one-step binomial model:

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