By virtually any measure, the foreign exchange market is enormous. In 2007,

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1 CHAPTER 8 Foreign Exchange Derivatives ROBERT W. KOLB Professor of Finance and Frank W. Considine Chair of Applied Ethics, Loyola University Chicago By virtually any measure, the foreign exchange market is enormous. In 2007, the daily turnover in foreign exchange markets averaged $3.2 trillion daily. This figure includes spot transactions in which one currency is exchanged for another for immediate delivery, and the spot exchange market accounts for slightly more than $1 trillion of this total (Bank for International Settlements [BIS], 2007, p. 4). But it also includes activity in foreign exchange (FX) derivatives markets. These derivative markets include forwards, futures, options, swaps, and a variety of other more esoteric instruments. This chapter briefly surveys the pricing principles for foreign exchange contracts and discusses the principal types of foreign exchange derivatives. BASIC PRICING PRINCIPLES Like many other financial instruments, the markets for foreign exchange have very few frictions and are characterized by a very high degree of transparency and low transaction costs. As a result, pricing of foreign exchange instruments in actual practice correspond closely to theoretical values, although there are some significant departures. The two most important pricing principles in foreign exchange are the purchasing power parity theorem and the interest rate parity theorem. Although these principles have distinct names, they both specify no-arbitrage conditions, and they have close analogues in other markets. Purchasing Power Parity Theorem The purchasing power parity theorem (PPPT) essentially claims that prices for identical goods trading in different countries with different currencies must have the same cost, except for factors due to market frictions. If this condition were not met, there would be arbitrage opportunities. For example, if a widget in England sold for a higher price, considering exchange rates, than a widget in the United This chapter draws on Robert W. Kolb and James A. Overdahl, Futures, Options and Swaps, 5th ed. (London: Blackwell Publishers, 2007). 115

2 116 Types of Financial Derivatives States, a trader could acquire the widget in England by paying pounds, transport the widget to the United States, sell it for dollars, and convert the dollars to more pounds than the widget cost in England. Thus, the price of a widget in the United States and England must be identical, except for market frictions. In our widget example, one clear kind of market friction is the cost of transporting the widget across the Atlantic. Other important kinds of transaction costs that arise in international trade would include tariffs, taxes, import quotas, and the like. As a result, in actual practice, the same goods can trade for substantially different prices in different countries. But the possibility of arbitrage keeps those pricing discrepancies to a level that reflects only market frictions. The Economist magazine frequently publishes its Big Mac Index, which compares the prices of McDonald s Big Mac sandwich in different currencies. In the summer of 2008, the U.S. price of a Big Mac was $3.57. At the same time, the equivalent price in China was $1.83, $4.57 in England, and $7.88 in Norway. These prices obviously represent enormous departures from the PPPT. With the very real difficulties in conducting arbitrage for Big Macs between countries, there are few market forces that can erase these huge differentials. By contrast, for goods that are easy to transport, that have less significant market frictions, those differentials between countries will be much lower and there will tend to be a single world price for such goods. Thus, for most commodities, there tends to be a single world price, yet this can be disturbed by transportation costs and government-imposed costs that frustrate the PPPT. Interest Rate Parity Theorem The interest rate parity theorem (IRPT) pertains to the differentials between spot exchange rates and forward exchange rates. The spot exchange rate is the rate of exchange between two currencies for immediate delivery, while the forward exchange rate is the rate at which one can contract today for the exchange of currencies at a specified future date. The IRPT asserts that the spot exchange rate and the forward exchange rate between two currencies, and the interest rates on the two currencies form a system of prices that must bear a certain relationship to one another to prevent arbitrage. The theorem can be understood best through an example. Assume that the spot exchange rate is $1.42 = 1, that the forward rate of exchange for delivery in one year is $1.35 = 1, that the one-year interest rate for dollars is 5 percent and the one-year interest rate on the euro is 6 percent. Faced with these exchange rates and interest rates, a European investor would borrow with 1 at the 6 percent interest rate, exchange the euro for $1.42, and invest the dollar proceeds in the United States at the 5 percent rate for one year, giving anticipated dollar proceeds in one year of $ Simultaneously, the investor would sell $ in the forward market at the one-year forward rate of 1 = $1.35 for total proceeds of From this sum, the investor would repay the euro that she borrowed for 1.06, leaving a total profit of This result is clearly an arbitrage profit, which indicates a violation of IRPT. Thus, the system of exchange rates and interest rates constitutes an inconsistency that can be resolved by the adjustment of any of the four terms. Barring market frictions, the activity of arbitrageurs would generate market forces that would restore the system of exchange rates and interest rates to conformity with the IRPT.

3 FOREIGN EXCHANGE DERIVATIVES 117 As we have seen, the PPPT does not necessarily hold very well in actual practice. In contrast, IRPT holds extremely well, and FX prices correspond extremely closely to the conditions specified by the IRPT. FOREIGN EXCHANGE FORWARD AND FUTURES CONTRACTS The foreign exchange forward market is an over-the-counter (OTC) market that trades on a worldwide basis and is dominated by large financial institutions. Typical contract amounts are quite large, and the market sees very few individual investors. By contrast, there are foreign exchange futures markets in many countries. In comparison with the forward market, these futures markets are quite small. The discussion of these markets will focus on the forward market, which will be contrasted with a single foreign exchange futures market, the Chicago Mercantile Exchange in the United States. 1 The Bank for International Settlements (BIS 2007, p. 4) reports estimates of market size in its surveys of central banks. For 2007, the BIS reported that the average daily turnover for foreign exchange forwards was $362 billion. Exhibit 8.1 shows the main currency pairings in the foreign exchange forward market. In spite of the emergence of the euro as an important international currency, the U.S. dollar continues to dominate the market, with fully 71 percent of all foreign exchange transactions involving the dollar as one of the currencies in each pair. As Exhibit 8.1 shows, the dollar/euro currency pair is the most heavily traded type of transaction in the FX forward market. Contract maturities in the OTC FX forward market are determined by agreement between the contracting parties and can be of any duration. Typical terms are 90, 180, or 270 days and one year, and the normal contract size is reckoned in the millions to many millions of dollars. Most FX forward contracts are settled by actual delivery. The Futures Industry Association (2008) reported that worldwide volume in foreign exchange futures in 2007 was 335 million contracts. With total futures contract volume across the world and all kinds of instruments of billion, foreign exchange futures account for only 2.2 percent of the total. As mentioned at the outset of our discussion, we take the foreign exchange futures market of the Chicago Mercantile Exchange (CME) as an example. The CME trades a wide dollar/euro dollar/yen dollar/sterling dollar/other all other Exhibit 8.1 Foreign Exchange Forwards by Currency Pairs (%)

4 118 Types of Financial Derivatives variety of futures contracts. Compared to the OTC market for forwards, all of the futures volumes are quite small. The main volume is concentrated in the currencies of major industrialized countries: the United States, European Union, Switzerland, United Kingdom, and Japan. However, the CME lists futures for many relatively obscure currency pairs, such as the dollar/ruble and the dollar/shekel. In general, these markets see little volume and are relatively illiquid. In contrast with the OTC FX market, the CME FX futures trade with specified contract maturities, usually on the March/June/September/December cycle, with each contract having a fixed quantity. Contract sizes are relatively small, with $100,000 being a representative figure for the underlying currency value of each contract. As with all futures, the CME imposes margin requirements, and frequent cash settlements to avoid accumulating credit exposure. In contrast with OTC FX forwards, most FX futures are settled by offset rather than delivery. For both FX forwards and futures, pricing generally follows the strictures of the IRPT very closely. FX forward and futures prices for the same currency pairs and maturities are virtually identical, with almost all of the small difference being attributable to the different institutional features of the markets, notably the requirement for margins and daily resettlement in the futures market but not in the OTC market. As is typical with almost all futures and forwards, FX futures and forwards serve as vehicles for both speculation and hedging. Speculative motives turn on anticipation of changes in interest rates, inflation rates, central bank interventions, and speculative attacks on currencies with fixed or pegged exchange rates. Hedging motivations are usually classified as being oriented toward either transaction exposure or translation exposure. Transaction exposure refers to an anticipated future exchange of currencies. For example, an importer might expect to pay for a shipment of goods in a foreign currency at a future date. The importer might hedge the currency risk associated with this future transaction by using the forward or futures market. Translation exposure arises when transactions denominated in a foreign currency must be translated, or restated, in the local currency. Thus, translation exposure is an accounting vulnerability. Yet some firms enter the FX futures or forward markets, thereby engaging in a transaction of economic significance, to avoid the risk of having reported earnings adversely affected by a disadvantageous currency fluctuation. FOREIGN EXCHANGE OPTIONS Much of the preceding discussion of FX forwards and futures also apply to FX options. The OTC FX options market dwarfs exchange-traded options. The OTC FX options market is a worldwide market dominated by institutional investors, with contract maturities and quantities being flexible and determined by negotiation. BIS (2007, p. 14) gauges daily average turnover of FX options on the OTC as $212 billion in As with forwards and futures, the main markets for FX options involve the currencies of the large industrial countries. However, virtually any kind of option with any pair of currencies is available in the OTC market upon request. Exchanges generally trade FX options on all of the currency pairs for which they list FX futures. The common pricing relationships between options and futures stimulate traders to trade FX options and futures in concert. While most

5 FOREIGN EXCHANGE DERIVATIVES 119 of the trading volume of FX options is concentrated in plain vanilla options, there is a robust market in exotic FX options as well. As with forwards and futures, the OTC market for foreign exchange dwarfs trading on organized exchanges, with average daily turnover in OTC-traded FX options exceeding $31 billion in 2007 (BIS 2007, p. 15). FX OPTION PRICING For plain vanilla FX options, pricing generally conforms quite closely to appropriately adjusted Black-Scholes model values, and models of this type are used throughout the market to price FX options. While the Black-Scholes options pricing model strictly pertains only to options on stocks that pay no dividends, FX option models using the Black-Scholes technology developed quickly following the publication of the original Black-Scholes paper. 2 The essential insight was to see that the existence of a continuous payoff from an asset, whether it is a continuous dividend rate or a rate of interest, was to treat that outflow as a diminution of the interest rate in the original Black-Scholes model. C t = e r F (T t) FC N(d 1 ) Xe r d (T t) N(d 2 ) P t = Xe r F (T t) N( d 2 ) FCe r F (T t) N( d 1 ) d 1 = ln(fc/ X)(r D r F + 0.5σ 2 )(T t) σ T t d 2 = d 1 σ T t where C t = price of a call option (priced in the domestic currency) on foreign currency FC P t = price of a put option (priced in the domestic currency) on foreign currency FC FC = a quantity of the foreign currency r D,r F = domestic and foreign interest rates, respectively X = exercise price T t= time until expiration σ 2 = variance of the foreign currency value N( ) = cumulative normal function The essential difference between this equation and the pricing of a stock paying a continuous dividend rate is that the interest rate on the foreign currency, r F, takes the place of the dividend rate. This extension of the original Black-Scholes model functions extremely well for FX options, and this model is an industry standard. In addition to the model of the equation, binary methods also function extremely well and are widely used for pricing more complex options. PLAIN VANILLA FOREIGN EXCHANGE SWAPS There are two essential types of plain vanilla swap agreements: interest rate swaps and foreign exchange swaps. In a plain vanilla interest rate swap, one party pays

6 120 Types of Financial Derivatives a floating rate of interest, while the other pays a fixed rate of interest on the same quantity of a currency (the nominal amount). Further, at each payment date, the obligations of the two parties typically are netted out, and only the netted amount actually changes hands. In a plain vanilla interest rate swap, these periodic payments are the only cash flows; the nominal amount does not change hands. In a plain vanilla foreign exchange swap, there are always two currencies involved, and the parties generally exchange the foreign currencies that constitute the nominal amounts as well. In contrast to interest rate swaps, plain vanilla foreign exchange swaps often are motivated by the desire of the parties actually to acquire the other currency, and the full amounts of the periodic payments are made by both parties in the respective currencies. This market is extremely robust, with average daily turnover of nominal amounts exceeding $2.2 trillion in 2007 (BIS 2007, p. 85). Today, there are no officially imposed restrictions on the movement of most major currencies. In the not too recent past, however, central banks in many industrialized countries imposed active restrictions on the flow of currency. The parallel loan market developed to circumvent restrictions imposed by the Bank of England on the free flow of British pounds. British firms wishing to invest abroad generally needed to convert pounds into U.S. dollars. The Bank of England required these firms to buy dollars at an exchange rate above the market price. The purpose of this policy was to defend the value of the pound in terms of other currencies. Firms, naturally, were not interested in subsidizing the Bank of England by paying the above-market rate for dollars required by the Bank of England s policies. Attempts to evade these currency controls led directly to the development of the market for currency swaps. In this environment of foreign exchange controls, consider two similar firms, one British and one American, each with operating subsidiaries in both countries. By cooperating with a U.S. firm that has operations in England, the British firm can evade the currency controls. The British firm lends pounds to the U.S. subsidiary operating in England, while the U.S. firm lends a similar amount to the British subsidiary operating in the United States. This is a parallel loan two multinational firms lend each other equivalent amounts of two different currencies on equivalent terms in two countries. A parallel loan is also known as a back-to-back loan. The development of swaps stemmed directly from these incentives to create parallel loans. Although the projected cash flows from the parallel loan and the plain vanilla currency swap are identical, there are still some subtle but important differences. In the parallel loan, both parties need to pretend that the transactions are completely distinct. If so, default on one of the loans would not justify default by the other party. In an interest rate swap agreement, there are cross-default clauses. (The parallel loan cannot contain those cross-default clauses, because the two parties are trying to pretend that the parallel loans are distinct.) Also, a swap agreement would have lower transaction costs than arranging two separate loans. In a plain vanilla FX swap, the parties exchange currencies at the outset of the swap s tenor and pay interest on the currency received (which can be at either a fixed or floating rate). At the termination of the swap, the parties again exchange the identical nominal currency amounts. Considering fixed and floating rates, there

7 FOREIGN EXCHANGE DERIVATIVES 121 are four possible basic payment patterns which can be illustrated by reference to two parties, A and B, and an FX swap between dollars and euros: 1. Party A pays a fixed rate on dollars received, and Party B pays a fixed rate on euros received. 2. Party A pays a floating rate on dollars received, and Party B pays a fixed rate on euros received. 3. Party A pays a fixed rate on dollars received, and Party B pays a floating rate on euros received. 4. Party A pays a floating rate on dollars received, and Party B pays a floating rate on euros received. Although all four patterns of interest payments are observed in the market, the predominant quotation is of the second type: pay floating on dollars/pay fixed on the foreign currency. This is known as the plain vanilla currency swap. Of these, the simplest kind of currency swap arises when each party pays a fixed rate of interest on the currency it receives, such as type 1. The fixed-for-fixed currency swap involves three different sets of cash flows. First, at the initiation of the swap, the two parties actually exchange cash. Typically, the motivation for the currency swap is the actual need for funds denominated in a different currency. This differs from the interest rate swap in which both parties deal in a single currency and can pay the net amount. Second, the parties make periodic interest payments to each other during the life of the swap agreement, and these payments are made in full without netting. Third, at the termination of the swap, the parties again exchange the principal. Pricing of FX swaps must reflect the term structure of interest rates for both currencies as well as the term structure of FX spot and forward rates. These relationships are mediated by the interest rate parity theorem discussed earlier. Fair value FX swaps leave the two parties with no anticipated change in wealth given these pricing relationships. Of course, deviations in interest and exchange rates from their respective forward rates will determine the actual wealth outcomes for the swap. FLAVORED CURRENCY SWAPS As we have noted, a plain vanilla currency swap calls for the two counterparties to exchange currencies at the outset of the swap and to make a series of interest payments for the currency that is received. However, currency swaps are subject to many elaborations. More complicated swap structures can be created by allowing the notional principal to vary over the tenor of the swap. For example, currency swaps can be amortizing (the notional amount decreases over the swap s tenor), accreting (the notional amount increase over the swap s tenor), seasonal (the notional amount fluctuates over the calendar year in a specified pattern), and rollercoaster swaps (the notional amount varies based on prenegotiated changes in the notional principal over the tenor of the swap). For example, a U.S. firm might import clothes from Hong Kong, with much higher imports in the winter, and pay for these in Hong Kong dollars. The firm

8 122 Types of Financial Derivatives might structure a currency swap with a seasonal notional principal to match its greater anticipated need for funds in winter months. Two fixed-for-floating swaps can be combined to create a fixed-for-fixed currency swap. As a second example, a CIRCUS swap is a fixed-for-fixed currency swap created by combining a plain vanilla interest rate swap with a plain vanilla currency swap. (CIRCUS stands for combined interest rate and currency swap.) A dual-currency bond has principal payments denominated in one currency, with coupon payments denominated in a second currency. For example, an issuing firm might borrow dollars and pay coupon payments on the instrument in euros. When the bond expires, the firm would repay its principal obligation in dollars. This dual-currency bond can be synthesized from a regular single-currency bond with all payments in dollars (a dollar-pay bond) combined with a fixed-for-fixed currency swap. A currency annuity swap is similar to a plain vanilla currency swap without the exchange of principal at the initiation or the termination of the swap. It is also known as a currency basis swap. For example, one party might make a sequence of payments based on British London Interbank Offered Rate (LIBOR) while the other makes a sequence of payments based on U.S. LIBOR. As we will see, the currency annuity swap generally requires one party to pay an additional spread to the other or to make an up-front payment at the time of the swap. Variations of this structure can be created by allowing one, or both, parties to pay at a fixed rate. In pricing these swaps, the key is to specify a spread or up-front payment that makes the present value of the cash flows incurred by each party equal. In a cross-index basis note, or quanto note, the investor receives a rate of interest that is based on a floating rate index for a foreign short-term rate but is paid in the investor s domestic currency. For example, a U.S. investor might buy a note with an interest rate based on European interest rates, but with all payments on the note being made in U.S. dollars. From the investor s point of view, the quanto note allows exposure to foreign interest rates without currency exposure. Also, the quanto note allows the investor to speculate on relative changes in the foreign and domestic yield curves. From the point of view of the issuer, the quanto note can offer investors attractive investment opportunities that might not be available elsewhere. Also, the issuer can issue a quanto note but use swaps to transform its risk exposure to a perhaps more congenial form. CONCLUSION This chapter has provided a brief overview of the range of foreign exchange derivatives current in the market today. The basic no-arbitrage pricing principles common to all financial derivatives apply with full force to FX derivatives. As we noted, the Black-Scholes model, adapted for the two interest rates involved in an FX option, performs extremely well as a practical method for pricing FX options. As we have seen, the full range of derivative types is available for foreign exchange as one asset class among many. These range from forwards to futures, to options and swaps. For example, there is a robust market for exotic options based on foreign exchange as well, with active volume for these instruments in the OTC market.

9 FOREIGN EXCHANGE DERIVATIVES 123 ENDNOTES 1. Even though the strong dominance of OTC trading continues in the FX market, there is a movement toward more exchange-based trading, especially on electronic platforms. See Acworth (2007). 2. The essential advancement over the original Black-Scholes article was made by Robert C. Merton (1973), and this consisted of extending the model to account for continuous dividends. Specific application of this framework to currency options was made by Garman and Kohlhagen (1983). REFERENCES Acworth, W Foreign Exchange Trading: New Trading Platforms Reshape FX Markets, Futures Industry (September October): Bank for International Settlements Triennial Central Bank Survey: Foreign Exchange and Derivatives Market Activity in 2007 (December). DeRosa, D. F Options on Foreign Exchange.NewYork:JohnWiley&Sons. Futures Industry Association Futures Industry (March/April): 16. Garman, M. B., and S. V. Kohlhagen Foreign Currency Option Values, Journal of International Money and Finance (December): Kolb, R. W., and J. A. Overdahl Futures, Options and Swaps, 5th ed. Oxford: Blackwell Publishers. Lipton, A Mathematical Methods for Foreign Exchange: A Financial Engineer s Approach. Hoboken, NJ: John Wiley & Sons. Merton, R. C Theory of Rational Option Pricing, Bell Journal of Economics and Management Science (Spring): Weithers, T Foreign Exchange: A Practical Guide to the FX Markets, Hoboken, NJ: John Wiley & Sons. ABOUT THE AUTHOR Robert W. Kolb is professor of finance at Loyola University Chicago, where he also holds the Frank W. Considine Chair of Applied Ethics. Dr. Kolb received two PhDs from the University of North Carolina at Chapel Hill in philosophy and finance, (philosophy 1974, finance 1978). Previously he held full-time academic positions at the University of Florida, Emory University, the University of Miami, and the University of Colorado. Over his career, he has published more than 50 academic research articles and more than 20 books, most focusing on financial derivatives and their applications to risk management. In 1990, he founded Kolb Publishing Company to publish finance and economics university texts. He sold the firm to Blackwell Publishers of Oxford, England, in 1995.