Option Pricing Unit 6
Expectations theory of forward rates.
. Prices of goods in different countries are equal when measured in terms of the same currency. For example, Dollar price of a product in the USA = peso price of the good in Mexico/number of pesos per dollar. It follows that the expected change in the spot rate equals the expected inflation differential
In March 1997 the exchange rate for the Indonesian rupiah was R2,419 = $1. Inflation in the year to March 1998 was about 30% in Indonesia and 2% in the United States. a. If purchasing power parity held, what should have been the nominal exchange rate in March 1998? b. The actual exchange rate in March 1998 (in the middle of the Asian currency crisis) was R8,325 = $1. What was the change in the real exchange rate?
2,419 X 1.3/1.02 = R3,083 = $1 b. Real value of rupiah fell by 3,083/8,325 - 1 = .63, or 63%
A company knows that it is due to receive a certain amount of a foreign currency in four months. What type of option contract is appropriate for hedging?
A long position in a four-month put option can provide insurance against the exchange rate falling below the strike price. It ensures that the foreign currency can be sold for at least the strike price.
Purchasing Power Parity (PPP)
Expected real interest rates in different countries are equal
The treasurer of a corporation is trying to choose between options and forward contracts to hedge the corporation's foreign exchange risk. Discuss the advantages and disadvantages of each.
Forward contracts lock in the exchange rate that will apply to a particular transaction in the future. Options provide insurance that the exchange rate will not be worse than some level. The advantage of a forward contract is that uncertainty is eliminated as far as possible. The disadvantage is that the outcome with hedging can be significantly worse than the outcome with no hedging. This disadvantage is not as marked with options. However, unlike forward contracts, options involve an up-front cost
Why is the expected loss from a default on a swap less than the expected loss from the default on a loan to the same counterparty with the same principal?
In an interest-rate swap a financial institution's exposure depends on the difference between a fixed-rate of interest and a floating-rate of interest. It has no exposure to the notional principal. In a loan the whole principal can be lost
What differences exist in the way prices are quoted in the foreign exchange futures market, the foreign exchange spot market, and the foreign exchange forward market?
In futures markets, prices are quoted as the number of US dollars per unit of foreign currency. Spot and forward rates are quoted in this way for the British pound, euro, Australian dollar, and New Zealand dollar. For other major currencies, spot and forward rates are quoted as the number of units of foreign currency per US dollar
What is the difference between the forward price and the value of a forward contract?
The forward price of an asset today is the price at which you would agree to buy or sell the asset at a future time. The value of a forward contract is zero when you first enter into it. As time passes the underlying asset price changes and the value of the contract may become positive or negative.
A U.S. company has committed to pay 10 million kronor to a Swedish company in one year. What is the cost (in present value) of covering this liability by buying kronor forward? The Swedish interest rate is .6%, and exchange rates are shown in Table 27.1 of the BMA text. Briefly explain.
Zero. Buying kronor in the forward market commits the U.S. company, but does not require an up-front expenditure.
Net convenience yield
The advantage from owning the commodity rather than the promise of future delivery less the cost of storing the commodity.
Alpha and Omega are U.S. corporations. Alpha has a plant in Hamburg that imports components from the United States, assembles them, and then sells the finished product in Germany. Omega is at the opposite extreme. It also has a plant in Hamburg, but it buys its raw material in Germany and exports its output back to the United States. How is each firm likely to be affected by a fall in the value of the euro? How could each firm hedge itself against exchange risk?
Alpha has revenues in euros and expenses in dollars. If the value of the euro falls, its profit will decrease. In the short run, Alpha could hedge this exchange risk by entering into a forward contract to sell euros for dollars. Omega has revenues in dollars and expenses in euros. If the value of the euro falls, its profit will increase. In the short run, Omega could hedge this exchange risk by entering into a forward contract to sell dollars for euros.
Explain why a bank is subject to credit risk when it enters into two offsetting swap contracts.
At the start of the swap, both contracts have a value of approximately zero. As time passes, it is likely that the swap values will change, so that one swap has a positive value to the bank and the other has a negative value to the bank. If the counterparty on the other side of the positive-value swap defaults, the bank still has to honor its contract with the other counterparty. It is liable to lose an amount equal to the positive value of the swap.
Companies A and B have been offered the following rates per annum on a $20 million five-year loan: Fixed Rate Floating Rate Company A 5.0% LIBOR+0.1% Company B 6.4% LIBOR+0.6% Company A requires a floating-rate loan; company B requires a fixed-rate loan. Design a swap that will net a bank, acting as intermediary, 0.1% per annum and that will appear equally attractive to both companies.
Company A has an apparent comparative advantage in fixed-rate markets but wants to borrow floating. Company B has an apparent comparative advantage in floating-rate markets but wants to borrow fixed. This provides the basis for the swap. There is a 1.4% per annum differential between the fixed rates offered to the two companies and a 0.5% per annum differential between the floating rates offered to the two companies. The total gain to all parties from the swap is therefore 1.4−0.5 = 0.9% per annum. Because the bank gets 0.1% per annum of this gain, the swap should make each of A and B 0.4% per annum better off. This means that it should lead to A borrowing at LIBOR −0.3% and to B borrowing at 6.0%. The appropriate arrangement is therefore as shown in the below figure. See picture on Question 10
Company X wishes to borrow U.S. dollars at a fixed rate of interest. Company Y wishes to borrow Japanese yen at a fixed rate of interest. The amounts required by the two companies are roughly the same at the current exchange rate. The companies have been quoted the following interest rates, which have been adjusted for the impact of taxes: Yen Dollars Company X 5.0% 9.6% Company Y 6.5% 10.0% Design a swap that will net a bank, acting as intermediary, 50 basis points per annum. Make the swap equally attractive to the two companies and ensure that all foreign exchange risk is assumed by the bank.
Company X has a comparative advantage in yen markets but wants to borrow dollars. Company Y has a comparative advantage in dollar markets but wants to borrow yen. This provides the basis for the swap. There is a 1.5% per annum differential between the yen rates and a 0.4% per annum differential between the dollar rates. The total gain to all parties from the swap is therefore 1.5−0.4=1.1% per annum. The bank requires 0.5% per annum, leaving 0.3% per annum for each of X and Y. The swap should lead to X borrowing dollars at 9.6−0.3=9.3% per annum and to Y borrowing yen at 6.5−0.3=6.2% per annum. The appropriate arrangement is therefore as shown in the below figure. All foreign exchange risk is borne by the bank. See Picture on Question 11
Explain the difference between the credit risk and the market risk in a financial contract.
Credit risk arises from the possibility of a default by the counterparty. Market risk arises from movements in market variables such as interest rates and exchange rates. A complication is that the credit risk in a swap is contingent on the values of market variables. A company's position in a swap has credit risk only when the value of the swap to the company is positive.
Calculate convenience yield for magnoosium scrap from the following information: Spot price: $2,550 per ton. Futures price: $2,408 for a one-year contract. Interest rate: 12%. Storage costs: $100 per year.
F = S(1 + rf + sc - cy). Therefore 2,408 = 2,550(1.12 + 100/2,550 − cy) cy = 548/2,550 = .215, or 21.5%.
Calculate the value of a six-month futures contract on a Treasury bond. You have the following information: Six-month interest rate: 10% per year, or 4.9% for six months. Spot price of bond: 95. The bond pays an 8% coupon, 4% every six months.
F = S(1 + rf − y) = 95(1 + .049 − 4/95) = 95.655.
In March 2012, 12 month futures on the Australian S&P/ASX 200 Index traded at 4,244. Spot was 4,276. The interest rate was 3.90%, and the dividend yield was about 5%. Were the futures fairly priced?
Ft = S0 (1 + rf − y)t = 4,276 × (1 + .039 - .05)1= 4,228.96. Because the futures calculation is lower than what the future and spot prices are, trading at the futures are slightly overpriced.
Penny Farthing, the treasurer of International Bicycles, Inc., has noticed that the interest rate in Japan is below the rates in most other countries. She is, therefore, suggesting that the company should make an issue of Japanese yen bonds. Does this make sense?
If international capital markets are competitive, the real cost of funds in Japan must be the same as the real cost of funds elsewhere. That is, the low Japanese yen interest rate is likely to reflect the relatively low expected rate of inflation in Japan and the expected appreciation of the Japanese yen. Note that the parity relationships imply that the difference in interest rates is equal to the expected change in the spot exchange rate. If the funds are to be used outside Japan, then Ms. Stone should consider whether to hedge against changes in the exchange rate and how much this hedging will cost.
On some catastrophe bonds, payments are reduced if the claims against the issuer exceed a specified sum. In other cases, payments are reduced only if claims against the entire industry exceed some sum. What are the advantages and disadvantages of the two structures? Which involves more basis risk? Which may create a problem of moral hazard?
If payments are reduced when claims against one issuer exceed a specified amount, the issuer is co-insured above some level, and some degree of ongoing viability is ensured in the event of a catastrophe. The disadvantage is that, knowing this, the insurance company may overcommit in this area in order to gain additional premiums, thereby creating a problem of moral hazard for the insurance company. If the payments are reduced based on claims against the entire industry, an ongoing and viable insurance market may be assured, but some firms may under commit and yet still enjoy the benefits of lower payments, thereby creating a problem of moral hazard for those firms. Basis risk will be highest in the first case due to the larger firm-specific risk.
Show that, if the futures price of a commodity is greater than the spot price during the delivery period, then there is an arbitrage opportunity. Does an arbitrage opportunity exist if the futures price is less than the spot price? Explain your answer
If the futures price is greater than the spot price during the delivery period, an arbitrageur buys the asset, shorts a futures contract, and makes delivery for an immediate profit. If the futures price is less than the spot price during the delivery period, there is no similar perfect arbitrage strategy. An arbitrageur can take a long futures position but cannot force immediate delivery of the asset. The decision on when delivery will be made is made by the party with the short position. Nevertheless companies interested in acquiring the asset may find it attractive to enter into a long futures contract and wait for delivery to be made.
A firm in the United States is due to receive payment of €1 million in eight years' time. It would like to protect itself against a decline in the value of the euro, but finds it difficult to get forward cover for such a long period. Is there any other way in which it can protect itself?
It can borrow the present value of €1 million, sell the Euros in the spot market, and invest the proceeds in an eight-year dollar loan.
You own a $1 million portfolio of aerospace stocks with a beta of 1.2. You are very enthusiastic about aerospace but uncertain about the prospects for the overall stock market. Explain how you could hedge out your market exposure by selling the market short. How much would you sell? How in practice would you go about "selling the market"?
Sell short $1.2 million of the market portfolio. In practice rather than "sell the market" you would sell futures on $1.2 million of the market index.
After a record harvest, grain silos are full to the brim. Are storage costs likely to be high or low? What does this imply for the net convenience yield?
Storage costs are likely to be high. Other things equal, firms will prefer to hold the future rather than the spot commodity, and net convenience yield will be low.
Companies may be affected by changes in the nominal exchange rate or in the real exchange rate. Explain how this can occur. Which changes are easiest to hedge against?
Suppose a firm has a known foreign currency income (e.g., a foreign currency receivable). Even if the law of one price holds, the firm is at risk if the overseas inflation rate is unexpectedly high and the value of the currency declines correspondingly. The firm can hedge this risk by selling the foreign currency forward or by borrowing foreign currency and selling it spot. Note, however, that this is a relative inflation risk, rather than a currency risk; e.g., if you were less certain about your domestic inflation rate, you might prefer to keep the funds in the foreign currency. If the firm owns a foreign real asset your worry is that changes in the exchange rate may not affect relative price changes. In other words, you are exposed to changes in the real exchange rate. You cannot so easily hedge against these changes unless, say, you can sell commodity futures to fix income in the foreign currency and then sell the currency
A US company knows it will have to pay 3 million euros in three months. The current exchange rate is 1.3500 dollars per euro. Discuss how forward and options contracts can be used by the company to hedge its exposure.
The company could enter into a forward contract obligating it to buy 3 million euros in three months for a fixed price (the forward price). The forward price will be close to but not exactly the same as the current spot price of 1.3500. An alternative would be to buy a call option giving the company the right but not the obligation to buy 3 million euros for a particular exchange rate (the strike price) in three months. The use of a forward contract locks in, at no cost, the exchange rate that will apply in three months. The use of a call option provides, at a cost, insurance against the exchange rate being higher than the strike price.
A US company expects to have to pay 1 million Canadian dollars in six months. Explain how the exchange rate risk can be hedged using (a) a forward contract and (b) an option.
The company could enter into a long forward contract to buy 1 million Canadian dollars in six months. This would have the effect of locking in an exchange rate equal to the current forward exchange rate. Alternatively the company could buy a call option giving it the right (but not the obligation) to purchase 1 million Canadian dollars at a certain exchange rate in six months. This would provide insurance against a strong Canadian dollar in six months while still allowing the company to benefit from a weak Canadian dollar at that time.
Suppose you are the treasurer of Lufthansa, the German international airline. How is company value likely to be affected by exchange rate changes? What policies would you adopt to reduce exchange rate risk?
Suppose, for example, that the real value of the euro declines relative to the dollar. Competition may not allow Lufthansa to raise trans-Atlantic fares in dollar terms. Thus, if dollar revenues are fixed, Lufthansa will earn fewer euros. This will be offset by the fact that Lufthansa's costs may be partly set in dollars, such as the cost of fuel and new aircraft. However, wages are fixed in euros. So the net effect will be a fall in euro profits from its trans-Atlantic business. However, this is not the whole story. For example, revenues may not be wholly in dollars. Also, if trans-Atlantic fares are unchanged in dollars, there may be extra traffic from German passengers who now find that the euro cost of travel has fallen. In addition, Lufthansa may be exposed to changes in the nominal exchange rate. For example, it may have bills for fuel that are awaiting payment. In this case, it would lose from a rise in the dollar. Note that Lufthansa is partly exposed to a commodity price risk (the price of fuel may rise in dollars) and partly to an exchange rate risk (the rise in fuel prices may not be offset by a fall in the value of the dollar). In some cases, the company can, to a great extent, fix the dollar cash flows, such as by buying oil futures. However, it still needs at least a rough-and-ready estimate of the hedge ratios, i.e., the percentage change in company value for each 1% change in the exchange rate. Lufthansa can then hedge in either the exchange markets (forwards, futures, or options) or the loan markets.
The holder of a commodities futures contract does not have to pay for storage costs, but foregoes convenience yield. T or F
TRUE
The forward price on the Swiss franc for delivery in 45 days is quoted as 1.1000. The futures price for a contract that will be delivered in 45 days is 0.9000. Explain these two quotes. Which is more favorable for an investor wanting to sell Swiss francs?
The 1.1000 forward quote is the number of Swiss francs per dollar. The 0.9000 futures quote is the number of dollars per Swiss franc. When quoted in the same way as the futures price the forward price is 1 / 1.1000 = 0.9091 The Swiss franc is therefore more valuable in the forward market than in the futures market. The forward market is therefore more attractive for an investor wanting to sell Swiss francs.
The price of gold is currently $1,400 per ounce. The forward price for delivery in one year is $1,500. An arbitrageur can borrow money at 4% per annum. What should the arbitrageur do? Assume that the cost of storing gold is zero and that gold provides no income.
The arbitrageur should borrow money to buy a certain number of ounces of gold today and short forward contracts on the same number of ounces of gold for delivery in one year. This means that gold is purchased for $1,400 per ounce and sold for $1,500 per ounce. Interest on the borrowed funds will be 0.04 × $1400 or $56 per ounce. A profit of $44 per ounce will therefore be made.
A bank finds that its assets are not matched with its liabilities. It is taking floating-rate deposits and making fixed-rate loans. How can swaps be used to offset the risk?
The bank is paying a floating-rate on the deposits and receiving a fixed-rate on the loans. It can offset its risk by entering into interest rate swaps (with other financial institutions or corporations) in which it contracts to pay fixed and receive floating.
A cattle farmer expects to have 120,000 pounds of live cattle to sell in three months. The live-cattle futures contract traded by the CME Group is for the delivery of 40,000 pounds of cattle. How can the farmer use the contract for hedging? From the farmer's viewpoint, what are the pros and cons of hedging?
The farmer can short 3 contracts that have 3 months to maturity. If the price of cattle falls, the gain on the futures contract will offset the loss on the sale of the cattle. If the price of cattle rises, the gain on the sale of the cattle will be offset by the loss on the futures contract. Using futures contracts to hedge has the advantage that the farmer can greatly reduce the uncertainty about the price that will be received. Its disadvantage is that the farmer no longer gains from favorable movements in cattle prices.
Interest rate parity.
The interest rate differential equals the forward premium or discount. (1 + Rx) / (1+ Rs) = (F x/s) / (S x / s) The percentage difference between the forward rate and today's spot rate is equal to the expected change in the spot rate. (F x / s) / (S x /s) = E(S x / s) /(S x / s)
Explain what happens when an investor shorts a certain share.
The investor's broker borrows the shares from another client's account and sells them in the usual way. To close out the position, the investor must purchase the shares. The broker then replaces them in the account of the client from whom they were borrowed. The party with the short position must remit to the broker dividends and other income paid on the shares. The broker transfers these funds to the account of the client from whom the shares were borrowed. Occasionally the broker runs out of places from which to borrow the shares. The investor is then short squeezed and has to close out the position immediately. A fee may be charged for borrowing shares.
It is July 2014. A mining company has just discovered a small deposit of gold. It will take six months to construct the mine. The gold will then be extracted on a more or less continuous basis for one year. Futures contracts on gold are available with delivery months every two months from August 2014 to December 2015. Each contract is for the delivery of 100 ounces. Discuss how the mining company might use futures markets for hedging
The mining company can estimate its production on a month by month basis. It can then short futures contracts to lock in the price received for the gold. For example, if a total of 3,000 ounces are expected to be produced in September 2015 and October 2015, the price received for this production can be hedged by shorting 30 October 2015 contracts.
A corporate treasurer tells you that he has just negotiated a five-year loan at a competitive fixed rate of interest of 5.2%. The treasurer explains that he achieved the 5.2% rate by borrowing at six-month LIBOR plus 150 basis points and swapping LIBOR for 3.7%. He goes on to say that this was possible because his company has a comparative advantage in the floating-rate market. What has the treasurer overlooked?
The rate is not truly fixed because, if the company's credit rating declines, it will not be able to roll over its floating rate borrowings at LIBOR plus 150 basis points. The effective fixed borrowing rate then increases. Suppose, for example, that the treasurer's spread over LIBOR increases from 150 basis points to 200 basis points. The borrowing rate increases from 5.2% to 5.7%
Risk management: A gold-mining firm is concerned about short-term volatility in its revenues. Gold currently sells for $1,600 an ounce, but the price is extremely volatile and could fall as low as $1,520 or rise as high as $1,680 in the next month. The company will bring 1,000 ounces to the market next month. What will be total revenues if the firm remains unhedged for gold prices of $1,520, $1,600, and $1,680 an ounce? The futures price of gold for delivery one month ahead is $1,610. What will be the firm's total revenues at each gold price if the firm enters into a one-month futures contract to deliver 1,000 ounces of gold? What will total revenues be if the firm buys a one-month put option to sell gold for $1,600 an ounce? The put option costs $110 per ounce.
To solve for unhedged revenue, multiply the price per ounce times the number of ounces purchased (1,000). If the firm enters into a futures contract at $1,610 the net total revenue will remain the same: $1,610 x 1,000 ounces = $1,610,000. The table below shows total revenue for each potential price when unhedged, hedged using futures, or hedged using options. (Titles across top) Gold Price Per Ounce Unhedged Revenue (a) Futures-Hedged Revenue (b) Options-Hedged Revenue (c) A B C $1,520 $1,520,000 $1,610,000 $1,490,000 $1,600 $1,600,000 $1,610,000 $1,490,000 $1,680 $1,680,000 $1,610,000 $1,570,000
Look at Table 27.1 of the BMA text. If the three-month interest rate on dollars is 0.2%, what do you think is the three-month interest rate on the Brazilian real? Explain what would happen if the rate were substantially above your figure.
We can utilize the interest rate parity theory: (1 + R real) / (1+ R$) = (F real / $) / (S real / $) 1 + R real/1.002 = 1.7781/1.7456 = .0207 = 2.07% If the three-month rand interest rate were substantially higher than 2.07%, then you could make an immediate arbitrage profit by buying rands, investing in a three-month rand deposit, and selling the proceeds forward.
Marshall Arts has just invested $1 million in long-term Treasury bonds. Marshall is concerned about increasing volatility in interest rates. He decides to hedge using bond futures contracts. a. Should he buy or sell such contracts? The treasurer of Zeta Corporation plans to issue bonds in three months. She is also concerned about interest rate volatility and wants to lock in the price at which her company could sell 5% coupon bonds. b. How would she use bond futures contracts to hedge?
a. Sell. If interest rates rise, the value of the long-term Treasury bonds will fall, a risk that can be offset by selling bond futures contracts. b. The treasurer faces the risk the interest rates will rise and can offset this risk by selling three-month bond futures.
sing the below table 27.1 from the BMA text, answer the following questions. a. How many Japanese yen do you get for your dollar? b. What is the one-month forward rate for yen? c. Is the yen at a forward discount or premium on the dollar? d. Use the one-year forward rate to calculate the annual percentage discount or premium on yen. e. If the one-year interest rate on dollars is 1.5% annually compounded, what do you think is the one-year interest rate on yen? f. According to the expectations theory, what is the expected spot rate for yen in three months' time? g. According to purchasing power parity theory, what then is the expected difference in the three-month rate of price inflation in the United States and Japan?
a. This is equal to the spot rate: 76.735. b. 76.7162 c. Yen is at premium (dollar is at discount). d. Premium = 76.735/76.282 - 1 = .005938, or .5938%. e. From interest rate parity, 76.282/76.735 = (1 + ryen)/ 1.015. Therefore ryen = .994097 × 1.015 - 1, ryen = .009008, or .9008%. f. 76.6595 yen = $1. g. If the real exchange rate is expected to be constant, expected difference in inflation is 76.6595/76.735 - 1 = -.00098; i.e., inflation in Japan over the three months is expected to be .098% less than in the United States.
