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# 1) A 1-year European put option with exercise price of \$45 on an underlying stock with price of \$25 trades at a premium of \$4. Suppose the 1-year...

1)   A 1-year European put option with exercise price of \$45 on an underlying stock with price of \$25 trades at a premium of \$4.75. Suppose the 1-year interest rate is 4%.

a.    Calculate the intrinsic value of this option. (2 pts)

b.    Does the option's premium violate a lower bound condition? Explain or illustrate briefly (2 pts)

Part 2: Option Valuation (25 points)

1)   Saber & Co. stock currently trades at a price of \$38 per share. You are trying to value a European call option with exercise price of \$40 per share on Armstrong stock that expires in 12 months. You have modeled Saber's stock price path such that the stock can go up 25% during each six month period or it can go down 25% during each six month period. The annual interest rate is 8%. Given the assumptions, calculate a fair value of this option. (8 pts)

2)   The output below shows the valuation of a two-period European put option in which the exercise price is \$40, the underlying price is \$42, price can go up by 25% or down by 25% per period, and the periodic risk-free rate is 2%.

Given the information shown, please calculate the value of a corresponding American put option. (5 pts)

3)   The output below shows the valuation of a two-period European call option with exercise price of \$40 given the assumptions shown below. The "Eu" cells indicate the value of the option at different points in time, given the assumptions.

Please use the information above to calculate deltas (i.e., the hedge ratios) at t=0 and at t=1 (assuming the stock price moves up to \$54.60). The delta for t=1 (assuming the stock price is down) has been provided for you. (3 pts)

4)   Please use the delta at t=0 and the delta at t=1 (if the stock is down) to explain the design of a risk-free trading strategy at t=0 (assuming that one part of the strategy involves selling 1,000 call options on the stock described in Question 3) and how the strategy is altered at t=1 (if the stock goes down to \$33.60). (NOTE: I am not asking you to "prove" that the strategy is risk-free). (3 pts)

5)   Armor Corp. stock trades at a price of \$18.78 per share currently. The annualized volatility of Armor's's stock is 40%. The continuously-compounded risk-free rate is 5.01% annually. You are considering a 9-month European call option on Armor's stock with an exercise price of \$20. The following output shows the Black-Scholes valuation of this option.

Please demonstrate (i.e., show your work) as to how the call option value of \$2.38 is obtained. Please show all steps, including estimation of N(d1) and N(d2). You have been provided with a statistical table of the CDF of the standard normal distribution. NOTE: If you round d1 and/or d2 in order to find N(d1) and/or N(d2), your answer may be different by up to \$0.03 from the correct answer. (4 pts)

6)   Please explain as to what information "Vega" provides in the above output? Be specific. (2 pts)

Part 3: Hedging with Futures (30 points)

1)   Suppose a producer anticipates buying 112,000 pounds of wheat on November, 2016. The October 2016 wheat futures contract's last trading day is November 30, 2016, and it currently trades at 21.34 cents per pound (NOTE: one wheat futures contract has an underlying of 11,000 pounds of wheat). Suppose that the spot price of wheat is currently 11.84 cents per pound. Please assume that the producer hedges against its price risk as of the date associated with the prices above, and answer the following questions.

a.    Please identify the direction of the producers unhedged exposure to wheat prices. (2 pts)

b.    What trade (involving a futures contract) will the producer pursue to hedge against its wheat's price exposure? (2 pts)

c.    Please show the profit/(loss) expected on the hedge? (3 pts)

2)   The spot price of Wheat during the time the hedge was started was \$2.89 per gallon. When the hedge was started, the basis between fuel spot and crude oil futures was \$0.556 per gallon. During the time frame when the hedge was liquidated, the average basis was approximately \$0.47 per gallon. Given this information, did the hedge lock in a price above or below (or equal to) \$2.89 per gallon? Briefly explain your answer. (4 pts)

3)   On June 30, the manager of the oil company's refinery notes that the spot price of Chicago gasoline is \$2.00 per gallon. At the same time, the price of August gasoline futures contracts is \$1.97 per gallon. The manager anticipates selling 4.2 million gallons of gasoline in the Chicago area during July, and sells 100 gasoline futures contracts at \$1.97 on June 30 to hedge its gasoline price exposure. (NOTE: each gasoline futures contract calls for delivery of 42,000 gallons of gasoline in New York Harbor). Answer the following questions.

a.    As of June 30, what is the refining manager's best estimate of its selling price of gasoline in July? (2 points)

b.    During July, the refinery purchased August gasoline futures contracts at a weighted average price of \$2.05 per gallon. During July, the weighted average spot price of Chicago gasoline is \$2.10 per gallon. What is the refiner's effective sales price of gasoline during July (after accounting for hedging gains or losses) given that the refiner sells gasoline in Chicago at prevailing spot prices? (4 points)

4)   Recall that In & Out Fueling needs to buy oil futures contracts. Suppose that In & Out Fueling "chooses a hedge ratio greater than the minimum variance hedge ratio."

a.    What risk is In & Out Fueling taking by this decision? (3 pts)

b.    Use the framework of the minimum variance hedge ratio equation to suggest a possible reason why In & Out may decide to employ a higher hedge ratio "than the minimum variance hedge ratio." (2 pts)

Part 4: Swaps (10 points)

1)    Aniket Industries has just issued a 4-year \$5100 million note with floating interest rate payments based on LIBOR made annually. Aniket has approached a derivatives dealer, Russel Solutions, to price a plain-vanilla interest rate swap to serve as a hedge on the interest payments of its debt issuance. The term structure of LIBOR over the next three years is shown below:

1-year LIBOR: 2%

2-year LIBOR: 2.5%

3-year LIBOR: 3%

a.    Calculate a fair fixed-rate for a 3-year plain-vanilla interest rate swap with annual settlement dates (assuming a floating rate of LIBOR). (8 pts)

b.    What is Aniket trying to achieve by entering into the swap (circle the best answer below)? (2 pts)

i.              The lowest interest rate possible on its debt

ii.             A fixed rate of interest on its debt

iii.           A floating rate of interest on its debt

iv.           That its debt value stays constant for three years

Part 5: Forward Pricing (10 points)

1)   Suppose that the price of a share of XYZ Co. stock is \$40 per share today. A futures contract on XYZ stock that expires in exactly 6 months is currently trading at \$41. Suppose that the annual interest rate is 5%. XYZ stock is expected to pay two quarterly dividends of \$0.40 per share (the first in exactly 3 months and the next in exactly 6 months). If you are looking to buy XYZ, would it be better to buy the shares now, or buy the futures contract? (5 pts)

2)   The output below shows a portion of gold futures prices (in US dollars per ounce) on April 29, 2016.

a.    Is the gold market in contango or backwardation? (2 pts)

b.    If interest rates are close to zero, what cost of carry factor (in words) accounts for the difference in futures prices over time (and about how large is this cost)? (3 pts)

Part 6: Risk Takers Cases (15 points)

1)   Ryan was not betting on the outright direction of natural gas price (i.e., the fund held many offsetting long and short positions). Given this statement, explain two fundamental factors that caused Ryan to lose most of the fund's value in the natural gas futures market during August - September 2006. (5 pts)

2)   In November of 1993,  (A&G) entered into an interest rate swap contract that ended up resulting in a loss of approximately \$160 million.

a.    What basic option strategy did A&G employ that resulted in this loss? (2.5 pts)

b.    What was A&G trying to achieve through the basic option strategy? (2.5 pts)

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