ollars are overwhelmingly traded in the United States ( 97 percent in volumeover 1994 through 1999), whereas the biggest share of EURIBOR is traded in London 88 percent over 1999). SIMEX/TIFFE is therefore the only example of a sizablerepartition of liquidity.The charts show respectively the open interest on SIMEX as a percentage of thetotal open interest (SIMEX + TIFFE), and the difference closing price of SIMEX minusthe closing price of TIFFE.
208
THEORY and TOOLS
Figure 4-26 TIBOR Price Difference(Source: Compiled by author from Datastream data.)
ideal in terms of informational content, and the overall calculation ismuch simpler than for swaps. There is a small caveat, however: Farmaturities tend to be much less liquid; therefore, medium- to longterm swaps do remain the best option to access long-term zerocoupons.Transforming euro rates into zero-coupon rates is based on abootstrap method. We consider all euro contracts to be chainedtogether, and because we know each individual rate, we can compound these rates and extract zero coupons for each future maturity.From there, a simple linear interpolation leads to the standard yieldcurve.As an illustration, consider the spreadsheet in Table 4-12 withmarket prices of eurodollar contracts. Each contract has a ®nal settlement on the second Wednesday of March, June, September, andDecember. Settlement convention is T  2; therefore, the March 2000contract expiring on March 15, 2000, trades until March 13, 2000. Fromthere we apply the following calculations:. The bid and ask shaded prices are market prices. Three-month rates
are deducted from the de®nition: 100 price. Because of the subtraction, a ``bid'' price on the future is an ``ask'' price on the rate. We
TABLE 4-12(Today: February 15, 2000. Settlement: February 17, 2000.)Settlement
Price Bid
Price Ask
Rate Bid
Rate Ask
Days
From Today
Factor Bid
Factor Ask
ZC Bid
ZC Ask
Mar. 13, 2000
Mar. 15, 2000
93.8400
93.8450
6.155%
6.160%
Jun. 19, 2000
Jun. 21, 2000
93.4350
93.4400
6.560%
27
1.00418
1.00428
5.575%
5.700%
6.565%
98
125
1.02101
1.02112
6.050%
Sep. 18, 2000
Sep. 20, 2000
93.1700
93.1750
6.081%
6.825%
6.830%
91
216
1.03794
1.03806
6.323%
6.343%
Dec. 18, 2000
Dec. 20, 2000
92.9550
Mar. 19, 2001
Mar. 21, 2001
92.8650
92.9600
7.040%
7.045%
91
307
1.05584
1.05598
6.548%
6.565%
92.8700
7.130%
7.135%
91
398
1.07463
1.07479
6.751%
Jun. 18, 2001
Jun. 20, 2001
6.765%
92.7650
92.7750
7.225%
7.235%
91
489
1.09400
1.09417
6.920%
Sep. 17, 2001
6.933%
Sep. 19, 2001
92.7100
92.7200
7.280%
7.290%
91
580
1.11398
1.11418
7.075%
7.087%
Dec. 17, 2001
Dec. 19, 2001
92.6350
92.6450
7.355%
7.365%
Mar. 18, 2002
Mar. 20, 2002
91
671
1.13448
1.13471
7.215%
7.228%
91
762
1.15557
1.15584
7.350%
7.363%
CHAPTER 4 MONEY MARKETS
Expiry
209
210
THEORY and TOOLS
consider these rates to be actual/360 and make the approximationthat they are valid for the full period between each settlement date.61. The ®rst line of ZC rates are deducted from the prevalent shortterm money-market rates, typically the LIBOR. We need thembecause we have to have a rate up to the ®rst contract's settlement.. ``Days'' represents the number of days from contract settlementdate to contract settlement date. ``From today'' refers to the numberof days from today's settlement date, February 17, 2000, to eachcontract's settlement date.. ``Factor bid'' (``factor ask'') represents the compounding factor. Forexample, for the June expiry, we have to compound 5.575 percent(5.7 percent) on 27 days, 6.155 percent (6.16 percent) on 98 days,according to the diagram in Figure 4-27.27  5:575%98  6:155%1:1 1:02101Fbid1360360Fask1 Similarly:Fbid2 
27  5:7%98  6:16%: 1 1:021121360360
27  5:575%98  6:155%91  6:56%: 1: 11360360360
 1:0379427  5:7%98  6:16%91  6:565%Fask:1:112360360360 1:03806First Eurodollar6.155%, 98 days
Today
Second Eurodollar6.56%, 91 days
Market ZC5.575%, 27 days
Figure 4-27 ZC from Euro61
This is an approximation because rigorously speaking, three-month eurodollars areexactly 90 days, no more, no less. Here we consider that the rate embedded, forexample, in the March contract is valid from March 15, 2000, to June 21, 2000Ðthat is, 98 days.
CHAPTER 4 MONEY MARKETS
211
. The ZC is then extracted from the compound factor, taking into
account the total number of days:360bid 6:05%rZC-1st euro  1:02101 1 27  98360 6:081%rask1 ZC-1st euro  1:0211227  98360bid 6:323%rZC-2nd euro  1:03794 1 27  98  91360 6:343%rask1:038061Z -2nd euro27  98  91
From there we obtain a ®rst zero-coupon yield curve based on eurofutures expiries. The next step is to translate this curve into a standardset of expiriesÐ1M, 2M, 6M, and so onÐthrough a linear interpolation,as shown in Figure 4-28. For example, we know the rate for 27 days(5.574 percent/5.70 percent) and 125 days (6.05 percent/6.081 percent);therefore, it is easy to average them to obtain the rate for 29 days,which is exactly one month from today (February 17, 2000, to March17, 2000):rbidZC-1M raskZC-1M 
125
29  5:575%  29 27  6:05% 5:585%125 2729  5:70%  29 27  6:081% 5:708%125 27
125
and for two months:rbidZC-2M raskZC-2M 
125125
60  5:575%  60 27  6:05% 5:735%125 2760  5:70%  60 27  6:081% 5:828%125 27
Applying the interpolation to all maturities leads to the result shown inTable 4-13. The only dif®culty in that approach is that we need fourFirst Eurodollar ZC6.05%/6.081%, 125 days
1M ZC29 days
Second Eurodollar ZC6.323%/6.343%, 216 days
2M ZC60 days
6M ZC182 day
Figure 4-28 ZC from Euro
212
THEORY and TOOLS
TABLE 4-13Expiry
Settlement
Days
ZC Bid
ZC Ask
1M2M3M6M9M1Y2Y
Mar. 17, 2000Apr. 17, 2000May 17, 2000Aug. 17, 2000Nov. 17, 2000Feb. 19, 2001Feb. 18, 2002
296090182274368732
5.585%5.735%5.880%6.221%6.467%6.684%7.305%
5.708%5.828%5.945%6.246%6.485%6.699%7.318%
eurofutures for each year we look ahead because eurofutures ``price''rates three months at a time. For eurodollars this is a minor issuebecause more than 40 expiries trade at any given time; therefore, wecan accurately extract zero coupons for the next 10 years. For othercurrencies, however, euro contracts are not listed so far in the future orthey are too illiquid to really be indicative, and in that case, swapsremain the only reliable source of information. Generally speakingthe yield curve deducted from eurofutures should be very close tothat extracted from swap rates.
Eurofutures Basis
As regular futures, euro contracts trade at a price different from theprice of their underlying. It is particularly interesting to examineclosely the magnitude and sign of this difference, an exercise we willpractice with index futures also when considering index arbitrage. InFigures 4-29 through 4-32 are, for example, the differences betweenthree-month contracts and the corresponding short-term rates for eurodollar, euroyen, eurosterling, and euribor futures62:Despite many particularities relative to each individual market, ingeneral a negative basis tends to be associated with decreasing interestrates and vice versa. This situation indicates that the futures contract iswidely used by market participants as a proxy for how the currentrates will evolve in the future, de®nitely something to keep in mindwhen trading interest rate products. Note also that the existence of aliquid cash and futures market potentially creates arbitrage opportunities for whoever has cheap access to money markets.62
The term basis in the charts refers to the difference between the rate implied by thefuture price minus the prevalent three-month rate. It is positive if the FRA implied bythe future is higher than the current money market rate, and negative otherwise.
CHAPTER 4 MONEY MARKETS
Figure 4-29 Euro Basis, USD(Source: Compiled by author from Datastream data.)
Figure 4-30 Euro Basis, YEN(Source: Compiled by author from Datastream data.)
213
214
THEORY and TOOLS
Figure 4-31 Euro Basis, GBP(Source: Compiled by author from Datastream data.)
Figure 4-32 Euro Basis, EUR(Source: Compiled by author from Datastream data.)
CHAPTER 4 MONEY MARKETS
215
Eurofutures and Interest Rate Exposure
We mentioned earlier that eurofutures were often used as complements to swaps to manage interest rate exposure. In practice, this isdone by trading a euro ``strip''Ðthat is, a series of chained contracts.63Consider, for example, an investor who needs to borrow cash regularly over a given period and wants to be protected against anincrease in short-term rates. The investor needs to enter into a transaction that will turn pro®table if rates increase, offsetting the additionalinterest expense incurred on the loans. We saw earlier that a shortposition in eurodollars would behave exactly this way; therefore, theinvestor has the option to sell eurodollar futuresÐif the original exposure is in dollars. For example, if the rolling nominal at risk is$20,000,000 every three months for a total period of one year, 20 contracts need to be sold of the four nearest maturities to create an offsetting futures position64 as shown in Figure 4-33.Generally speaking, buying or selling a series of consecutive eurofutures protects against a decline or rise in interest rates for a periodequal to the total period covered from the ®rst to the last contract. Tosee this more clearly, consider, for example, the ®rst contract. If it weretraded at 95.00, the implied FRA would be 5 percent. Now supposethat the actual LIBOR is 5.5 percent upon expiration of the future. The®nal settlement price is 100 5:5  94:50. Assuming we were short at95.00, the 0.5 percent difference in rate would generate a gain of 0.50points per contract. If we borrow at the spot rate of 5.5 percent, thisgain means that the effective borrowing cost is actually 5.5 percent±0.5
1Y Eurodollar Strip, 4 * 3M
To day
st
1
Eurodollar, 3M
Figure 4-3363
Euro Strip
The terminology may be different in some cases. For example, the CME lists eurodollar ``bundles'' and ``packs,'' which are essentially strips. Bundles are strips of aninteger number of yearÐ1, 2, 3, 5, 7, 10Ðwhereas packs are strips of only fourconsecutive contracts. See the CME Web site for more details.64In reality, the calculation would be more complex because there must be severaladjustments to account for the mismatch in dates between the rolling loans and thetwo nearest future expiries.
216
THEORY and TOOLS
percent  5 percent, which was the original implied rate when thefutures contract was sold. Applying the same reasoning to each individual contract shows that the rate for the entire period was indeed setupon selling the strip.This strip protection covers a period of one year and thereforehedges a one-year rate. Furthermore, because it starts with the ®rstmaturity, which is typically less than three months away, the protection is effective against the current one-year rate because any movetoday in that rate would be re¯ected in the prices of the futures contracts. In contrast, if we started the strip with a contract in one year, wewould still protect a one-year rate but one year away. In essence, therate hedged is not the spot rate anymore but the FRA, as shown inFigure 4-34. Naturally there are many other variations possible usingeurofutures, which is why these contracts are so popular and so widelyused.
Eurofutures and Swaps
Consider again a regular one-year euro strip, as shown in Figure 4-35.Assume we sell the strip and that at each and every expiry, we enter aborrowing transaction equal to the notional of the strip. The result is aone-year money market operation at a rate ®xed by the compoundedimplied rates of each contract.This operating modeÐregular borrowing at market ratesÐisexactly equivalent to borrowing liquidity through a swap with a variable leg matching the eurofutures schedule. In terms of risk and exposure, swaps and eurofutures strips are therefore exactly equivalent, butthey differ in the implementation of this exposure. Because swaps havetwo legs, they are priced such that the net present value of all ¯ows iszero at the time of the transaction.65 A eurofutures strip, on the other1Y
1Y Eurodollar Forward Strip
5th Eurodollar
To day
Figure 4-34 Euro Strip
65
Note that this is generally true of all derivative contracts because their pricing is byde®nition based on an absolute convergenceÐthat is, an arbitrage-free situation.
CHAPTER 4 MONEY MARKETS
217
1st Eurodollar1Y Eurodollar Strip
Toda y
1Y Implied Fixed Rate
Figure 4-35
Euro Strip
hand, is not a two-legged product, so its net present value at initiationis equal to the borrowing cost over the periodÐthat is, the implied®xed rate from the compounding of each individual contract.Because of the differences between the swap and eurofutures markets, notably the fact that one is an OTC market while the other is listed,one may be more appropriate to trade than the other depending on theparticular circumstances and on the nature of the transacting institution. However, from a functional perspective, they are entirely equivalent even though the actual ¯ows resulting from each are different.
FOREIGN-EXCHANGE SWAPS AND FORWARDSDe®nitions
Foreign exchange swaps are exactly identical in principle to interestrate swaps, but each leg is pegged to a different currency. For example,a dollar/yen swap is a transaction in which one party agrees to pay agiven interest rate on a dollar notional, while the other agrees to pay ayen interest rate on an equivalent notional in yen. A forward is simplya deferred foreign exchange transaction at an exchange rate set today.Foreign exchange derivatives are slightly more complex thanstraight interest rate products because of the introduction of an additional currency, which implies that the spot exchange rate and a secondset of interest rates must be included in the pricing of the product. Asan illustration, consider, for example, a simple one-year dollar/yenswap, a three-month LIBOR on each side, for a notional of N$ , asshown in Figure 4-36.66 Nothing is exchanged today, and only theinterest is paid every three months. If we denote f¥=$ as the currentyen/dollar exchange rate, we have the additional relationship66
A variant of this swap would include the actual exchange of N¥ for N$ at initiationand conversely at expiry. The inclusion of these ¯ows depends on the particular needof the transacting parties.
218
THEORY and TOOLSN$ * r1$
N$ * r2$
N$ * r3$
N$ * r4$
N¥ * r2¥
N¥ * r3¥
N¥ * r4¥
Figure 4-36
Forex
3MToda y
N¥ * r1¥
N¥  N$  f¥=$ . There are multiple variations of this simple structure,notably in terms of frequency and rate reference. For example, one ofthe legs may be a ®xed rate or a six-month LIBOR.
Markets
Even more than interest rate derivatives, foreign exchange transactionsare mostly traded over the counter (OTC). However, outstanding positions are signi®cantly lower with $14,899 billion as of June 1999 versus$54,072 billion for interest rate products as a whole. The U.S. dollar isnot surprisingly, even more dominant with a market share of 44 versus30 percent for interest rates. Table 4-14 presents an overview of themarket.Interestingly the vast majority of foreign exchange contracts areshort-term products: 83.5 percent of outstanding positions are belowone year versus 37.5 percent for interest rates. It also appears thatmarket participation is broader, with reporting ®nancial institutionsholding only 36.7 percent of outstanding positions. This is in contrastto the ratio of 50 percent for interest rate transactions. Non®nancialinstitutions are also more actively involved: 20.2 percent versus 10.8percent.These statistics show clearly that the market for foreign exchangederivatives is signi®cantly different from that of interest rate products.The nature of the needs and of the population they serve are different,if only because numerous non®nancial corporations are exposed toforeign currencies as part of their day-to-day business. In contrastthe interest rate market is a more delineated arena in which tradersand speculators have a larger share.
PricingSwapsForeign exchange swaps are fairly straightforward to price by discounting all ¯ows at the prevailing zero-coupon rates, as shown inFigure 4-37. If F$i and F¥i stand for the ¯ows on each leg, N for their
CHAPTER 4 MONEY MARKETSTABLE 4-14
219
The Global OTC Foreign Exchange Derivatives Marketsa,b(Amounts outstanding in billions of U.S. dollars)Notional Amounts
Gross Market Values
EndJune1998
EndDec.1998
EndJune1999
EndJune1998
EndDec.1998
EndJune1999
Total contracts18,719With other reporting dealers7,406With other ®nancial institutions 7,048With non®nancial customers4,264Up to one yearc16,292Between one and ®ve yearsc1,832595Over ®ve yearscU.S. dollar16,1678,168EurodJapanese yen5,579Pound sterling2,391Swiss franc1,104Canadian dollar660Swedish krona351Other3,019
18,0117,2847,4403,28815,7911,62459215,8107,6585,3192,6129375944192,674
14,8995,4646,4293,00712,4441,77268313,1814,9984,6412,2818236364352,805
799314299186ÐÐÐ74719335155352911179
786336297153ÐÐÐ6982233706230349146
582200246136ÐÐÐ51920617163332714130
57
80
Ð
Ð
Ð
Memorandum item:Exchange-traded contractse
103
a
See footnote a to Table 4-3.Counting both currency sides of every foreign exchange transaction means that the currencybreakdown scans to 200% of the aggregate.cResidual maturity.dBefore end-June 1999; legacy currencies of the euro.eSee footnote f to Table 4-3.b
(Source: Bank of International Settlements, ``The Global OTC Derivatives Market at the End ofJune 1999.'')
number, ri $ and ri ¥ for the corresponding zero-coupon rates, and f¥=$for the current exchange rate, the value V of the swap is simply67:!!NNXXF$iF¥iV$  f$=¥  V f$=¥ 1  r¥i1  r$ii1i167
$ and ¥ have been chosen for simplicity as a generic notation for any two currencies.We employ indifferently f¥=$ and f$=¥ with f¥=$  1=f$=¥ .As we did for a generic interest rate swap pricing formula, we implicitly considerthat all rates are midmarket; therefore, V$ is also a midmarket valuation.
220
THEORY and TOOLSr4r2
r3
r1
Fi$
Fi¥
Toda y
Figure 4-37
Forex Swap
Incidentally, this formula can be generalized to handle any type ofswap, regardless of its characteristics. If N denotes the number of ¯owson one leg, M the number on the other leg, V becomes:!!NMXXF$iF¥iV$  f$=¥  V¥ f$=¥ 1  r¥i1  r$ii1i1where F$i and F¥i may have totally different schedules or even calculation methodsÐ®xed versus variableÐprovided that r$i and r¥i arechosen appropriately for each ¯ow.In the particular case of each leg being pegged to a variable rate, forexample, the LIBOR, we can operate the same simpli®cation we usedfor the variable leg of an interest rate swap and substitute two ¯ows ofnotional to each unknown variable ¯ow.68 The whole structurebecomes as shown in Figure 4-38.In the ®gure, rZC stands now for the zero-coupon rate until maturity. V can be expressed here as:!!N$N¥f$=¥  N¥V$  f$=¥  V¥  N$1  r¥ZC1  r$ZCConsidering that N$  f$=¥  N¥ , this expression can be further simpli®ed:!r$ZCr¥ZCr¥ZC r$ZCV$  N$ ::N$1  r$ZC 1  r¥ZC1  r¥ZC   1  r$ZC The valuation of the swap depends on the interest rate differentialbetween the two currencies.6968
Note that the transformation requires only that each leg be reset on a market variablerate. Therefore, both sides need not have the same frequency.69Note that in a variant of the swap where N¥ and N$ would be exchanged at initiationand back at expiry, the value of the swap would be exactly 0 because all ¯owsdisappear from the structure.
CHAPTER 4 MONEY MARKETS
221
rZCN$
N¥
N¥
N$
Figure 4-38
Forex Swap
As a numerical application, consider a one-year dollar/yen swapfor $1,000,000, for which r¥ZC  0:29 percent and r$ZC  6:87 percent. Ifwe apply the formula for V$ , we get: 0:29%  3656:87%  365360360   $62,175:66V$  $1,000,000  0:29%  3656:87%  365 11360360V$ < 0 indicates that the buyer of this swapÐthat is, the receiver ofdollar rates/the payer of yen ratesÐwould have to pay close to $62,176to enter the transaction. This is a very intuitive result: Considering thedifference between 0.29 percent and 6.87 percent, it only seems naturalthat the differential appears very clearly in the value of the deal.
FUTURES REVISITEDThe payoff to the holder of a future position can be expressed as:P  Q  FSQ
FH 
Q is the number of units in contracts, FSQ the ®nal settlement value ofthe future, and FH the price at which it was traded originally.70 Thispayoff is usually denominated in the currency of the underlying, butthere exist other options. We examine below three types of futures:standard, ADR, and quanto.
Standard Futures
To price a standard futures contract on a security S, consider the diagram in Figure 4-39 in which:70
We neglect again here the margin calls and deposits on the futures contract. As wasmentioned on several occasions these ¯ows only generate ®nancing costs which weleave aside.
222
THEORY and TOOLSt0, S
t1 FSQ
P
rZC
d
Figure 4-39
rd
Futures Contract
. t0  time of pricing and transaction, typically today. t1  expiry
. P  payoff to the holder of the future position. d  dividend paid on the underlying security. rZC zero-coupon rate from t0 to t1 71
. rd  forward zero-coupon rate from the dividend payment date to
t1 72. S  price of the underlying security on t0. FSQ  price of the underlying security on expiryÐthat is, future SQprice. Q  number of units of futures traded on t0 at FHConsider now the following elementary transactions from the perspective of the seller of the future who needs to hedge the short:On t0 :. Borrow an amount N  Q  S from t0 to t1 @ rZC .. Buy Q units of S @ S0 , for a total nominal N.. Sell one future @ FH .
The resulting cash ¯ow is:C0  N
QS0
On t1 :. Pay back loan N 
1  rZC .