Question 1 You live in a country where the government has fixed the price of wheat at S = $300 per metric ton. Assume that the fixed price is credible and is definitely not expected to change over the next month. Suppose you can take long and short positions in wheat with equal facility and costlessly (i.e., no transactions costs, no shorting costs, etc). Assume there are no holding costs or benefits associated with wheat. The one-month interest rate is r = 6% in continuously-compounded and annualized terms. Show that it is NOT possible to define an arbitrage-free one-month forward price for wheat in this setting. Explain intuitively why this is the case. Comment One way to show that no arbitrage-free forward price exists is to use the following steps: 1. The usual cost-of-carry formula suggests F = SerT as the candidate forward price. Using S = 300, r = 0.06, and T = 1/12, calculate F. Using the standard arguments developed in class, show that if the forward price is not equal to F, there is always an arbitrage. 2. Next, show that in this setting if the forward price is not equal to the fixed spot price S, there is always an arbitrage. 3. Since the forward price cannot simultaneously be equal to S and F = SerT , there is no arbitrage-free forward price possible.
Question 2 A company manufactures and supplies gold wire to customers. Delivery is made one month after the customer order is received. Company practice is to buy gold on the day the order is received. Let date 0 denote the date of order and date T denote the date of delivery. Customers are given a choice between date-of-order and date-of-delivery price. (The choice is made after the date of delivery.) Specifically, if G0 represents gold price on the date of order and GT represents gold price on the date of delivery, customers can choose between paying G0 + K and GT + K, where K is a fixed margin. For simplicity, suppose that the company has no other costs of production; the only cost is the cost G0 of buying the gold when the order is received. Thus, if the customer chooses date-of-order pricing, the companys profit on the order is (G0 + K) - G0 = K, while if the customer chooses date-of-delivery pricing, the companys profit on the order is (GT + K) - G0 = K - (G0 - GT ). Answer the following questions: 1. Is there directional risk in offering customers this pricing choice? Of what sort? That is, how does the company do if gold prices go up? If they go down? 2. Is there volatility risk? That is, does the company gain or lose if gold prices become more volatile? 3. In the language of derivatives, what kind of option on gold has the company implicitly sold its customers? Put differently, assuming there are standard call and put options available on gold, what kind of option would you suggest the company use to hedge the cash-flow risk you face?
Question 3 A target forward is a derivative product that is a variant on plain vanilla forward contracts. A vanilla forward contract has a linear payoff structure. For example, if the delivery price in the contract is K and the price of the underlying at maturity is ST , the payoff to a short forward position is just K - ST In a target forward, the slope of the payoff differs on either side of the delivery price K. For example, we may have the following payoff to a short position: ( K - ST , if K > ST 2(K - ST ), if K < st="" intuitively,="" target="" forwards="" incorporate="" a="" view="" on="" direction.="" in="" the="" above="" example,="" the="" loss="" from="" a="" price="" increase="" (k="">< st="" )="" increases="" twice="" as="" fast="" as="" the="" gain="" from="" a="" price="" decrease="" (k=""> ST ), implying that the holder of this position is betting against a price increase. Target forwards have caused large losses for several companies (such as Aracruz Cellulose, the Brazilian manufacturer of eucalyptus pulp). Show that a target forward of the sort above is just a combination of a short position in a vanilla forward and a short position in a vanilla option.
Question 4 Indicate which of the alternatives in the following multiple-choice questions is most correct. No explanation is required, but you may, if you wish, provide a brief explanation of your answer. 1. Apple stock (ticker: AAPL) is currently at $415/share, while Google stock (ticker: GOOG) is currently at $835/share. Consider the following options: • Option A: A one-month call option to buy Apple stock at a strike of $415. Cost: CA • Option G: A one-month call option to buy Google stock at a strike of $835. Cost: CG • Option AG: A one-month call option to buy a package of one share each of Apple and Google at a strike of $1,250. Cost:CAG. [Note: Under option AG, you either buy the whole package or nothing.] Which of the following alternatives is most correct? (a) CAG > CA + CG. (b) CAG