Practice Question 1) Jet Black

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 * 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 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: