Consider a futures contract for asset X maturing in one year.

QUESTION

Consider a futures contract for asset X maturing in one year.Assume that asset X has a spot price of $50 and that it pays a dividend of 2% (to be distributed a year from now) of the spot price today. Also, assume that the annual risk-free interest rate is 3%.(a) What is the fair value of the futures contract in the context of the no-arbitrage principle?(b) Suppose that the futures price in the market is $3 more than the price found in part (a). Describe an investment strategy that profits from this arbitrage opportunity. State explicitly what an investor would have to do today and one year from now.(c) Suppose the futures price in the market is $3 less than the price found in part (a). Describe an investment strategy that profits from this arbitrage opportunity. State explicitly what an investor would have to do today and one year from now.Now, instead of the risk-free interest rate of 3%, assume that the risk-free interest rate for lending is 2% and that the risk-free interest rate for borrowing is 4%.(d) If an investor is only allowed to lend money and not borrow, what range of futures prices prevents this investor from making an arbitrage profit?(e) If an investor is only allowed to borrow money and not lend, what range of futures prices prevents this investor from making an arbitrage profit?(f) Combine (d) and (e) to find the range of potential futures prices, which are consistent with a lack of arbitrage opportunities in the market.
(a) The fair value of the described futures contract is given by the expression (1+r-y)spot price of asset X, where r is the riskless interest rate and y is the dividend rate for asset X. In consequence, the fair value of the described futures contract is (1+0.03-0.02)50=50.50. (b) Now, the futures price is $53.50. In this case, one obtains a riskless profit from the so-called cash and carry trading strategy: Period 0: 1. Sell futures contract. 2. Borrow $50. 3. Use borrowed money to buy underlying asset for $50. Period 1: 1. Collect $1=0.02$50 because of the Xs dividend payment. 2. Use underlying asset to settle futures contract, receive $53.50. 3. Pay off loan with $51.50=1.03$50. The total profit from these transactions is $3. (c) Now, the futures price is $47.50. In this case, one obtains a riskless profit from the so-called reverse cash and carry trading strategy: Period 0: 1. Buy futures contract. 2. Short-sell the underlying asset, and receive $50. 3. Lend $50. Period 1: 4. Receive $51.50=1.03$50 from loan. 5. Receive the asset from futures contract, and pay $47.50. 6. Return the asset from futures contract, and pay $1=0.02$50 dividend to settle short-sale. The total arbitrage profit from these transactions is $3. (d) Qualitatively, the fact that the futures price is above the fair value price of the futures contract results in a riskless profit opportunity (not

uiring any of the investors own capital) via the cash and carry trading strategy. However, this strategy requires the ability to borrow money. As borrowing is prohibited, it is not possible to exploit futures prices that are too high. In contrast, the fact that the futures price falls below the fair value price of the futures contract results in a riskless profit opportunity (not requiring any of the investors own capital) via the reverse cash and carry trading strategy. This strategy requires the ability to lend money, which is permitted in this part. Therefore, futures prices that are too low can be exploited, whereas futures prices that are too high cannot be corrected. More precisely, the bound on futures prices in this part such that there are no arbitrage opportunities is F (1+ry)P = (1+0.020.02)50 = 50. Note that in this equation, because lending is used, r = 0.02. (e) Qualitatively, the fact that the futures price falls below the fair value price of the futures contract results in a riskless profit opportunity (not requiring any of the investors own capital) via the reverse cash and carry trading strategy. However, this strategy requires the ability to lend money. As lending is prohibited, it is not possible to exploit futures prices that are too low. In contrast, the fact that the futures price is above the fair value price of the futures contract results in a riskless profit opportunity (not requiring any of the investors own capital) via the cash and carry trading strategy. This strategy requires the ability to borrow money, which is permitted in this part. Therefore, futures prices that are too high can be exploited, whereas futures prices that are too low cannot be corrected. More precisely, the bound on futures prices in this part such that there are no arbitrage opportunities is F (1+ry)P = (1+0.040.02)50 = 51. Note that in this equation, because lending is used, r = 0.04. (f) Combining (d) and (e), the range of futures prices that is consistent with the no-arbitrage principle is 50 F 51.

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