bond back again, as the U.S. budget deficit climbs,
391
Sec. 9.2 Futures Contracts
for notes and bonds is given by the following equation:
where A1 is the accrued interest, N the number of cashflows, and the paramount is 100.Practitioners usually see yield movements as the driving force behindprice changes, so that, by convention, they quantify the first-order risk ofa bond by using the modified duration, which is defined as:
According to the price-yield relationship of Equation (9.2), bothmodified duration and convexity are positive definite, therefore the priceyield curve is convex, as shown in Figure 9.2 in a stochastic interest rateenvironment, if the yield change in the near future is symmetrically distributed, the price change will follow an asymmetric distribution with apositive expectation. This effect provides an advantage to the bondholder, and the expected gain is often called the convexity
Price
bYield
Figure 9.2. Price-Yield Relationship.
9.2
Futures Contracts
A futures contract is an agreement between two parties that obligates thebuyer to buy and the seller to sell an asset at a certain price on a future~~
~
The convexity value is only an illusion if in reality price changes are symmetricallydistributed.lZ2
392
Simple Interest Rate Products
Chapter 9
date as specified by the contract. In this book, we limit our study to theexchange traded Euro-dollar futures, Treasury note futures, and bondfutures.
9.2.1
Euro-dollars and LIBOR
U.S. dollars deposited in European banks (sometimes in other international banks) are called Euro-dollars. Similarly other currencies deposited in international banks are called Euro-currencies, such as EuroMarks and Euro Yens. LIBOR stands for London Inter-Bank OfSer Ratefor Euro-dollars. LIBOR interest rates are offered daily for a wide rangeof maturities from one day to a few years. The most popular one is the6-month LIBOR, which is often used as a benchmark for floating interestrates. Because of credit risks, LIBOR is usually higher than the corresponding term U.S. Treasury rate.9.2-2
Euro-dollar Futures
Spot LIBOR based contracts for short term investments are usually negotiated between banks and other financial institutions. The more liquid,exchange traded LIBOR instruments are the Euro currency futures contracts. In this section we use the E u r o - d o h futures (EDF) contracts,which are in fact the most popular financial products in the world, as anexample. The EDF contracts are the contracts on future 90 day LIBORrates, and they are specified as discount instruments for 90 calendar daysstarting on the Wednesdays closest to the fifteenth of the month forMarch, June, September and December. Therefore there are four contracts per year. The EDF contracts are settled on the next business dayfollowing the trade day, and are marked to market at the closing of everybusiness day. Usually the first 8 EDF contracts are very liquid and thereis still significant activity in the 40" contract that settles ten years in thefuture.The EDF contract is quoted by using the annualized discount pricebased on an actua1/360 day count convention. In actual practice, theEDF transaction is always cash settled. On the last trading day of thecontract, the spot 90 day LIBOR rate R can be different from the contractyield Y. For a par amount of $1 million, each basis point of difference is$25. On the settlement day, the bearer gets paidp = 2500(Y - R )(9.4)dollars from the seller. We can see that, if the buyer of the contract en-
Sec. 9.2
Futures Contracts
393
ters the market at a yield higher than the actually realized 90 day LIBOR,namely Y - R > 0 , he or she makes a profit. Otherwise he or she suffersa loss. For making longer term investment in the LIBOR market, onecan use consecutive EDF contracts to synthetically create a zero-couponbond.
9.2.3
Note and Bond Futures
While Euro-dollar futures are contracts on short term interest rate instruments, note and bond futures are futures contracts on longer terminterest rates. There are exchange traded futures contracts on the 2 year,5 year, 10 year Treasury notes, as well as contracts on Treasury bonds.In any year, there are four contracts with settlement in March, June, September, and December, and most of these nearest contracts are evenmore liquid than the underlying notes and bonds. The futures contractsare popular because they are convenient to work with. A transaction in afutures contract only requires minimal capital commitment in terms of amargin account deposit; whereas the purchase of a bond involves immediate cash payment and the subsequent handling of carry.The specification of note and bond futures contracts is designed tofacilitate the actual transaction. The bond contract specifies a basket ofunderlying bonds that can be delivered against the futures contract. Onthe futures settlement date, any bond with a remaining maturity or thenumber of years to the first call date if callable, longer than 14.5 years isqualified for delivery against the contract. Instead of quoting a futuresprice for every bond in the basket, the exchange lists a single price forthe contract, so that the specification of the futures price is greatly simplified. The trick is to define a standardized bond with a fixed 6% coupon and maturity, the futures price is quoted according to this syntheticbond. Then the invoice futures price for each deliverable bond is linkedto the quoted futures price by the bonds conversion factor C, which isdefined as the ratio of the bond price and the synthetic bond price. Onthe delivery day, the seller of the futures contract will deliver the bond iin exchange for a payment of
=CiF,(9.5)where F is the futures price specified when the futures contract is entered.
394
9.3
Simple Interest Rate Products
Chapter 9
Interest Rate Derivatives
Interest rate derivatives are derivatives written on interest rate sensitiveinstruments, such as bonds or on the interest rates themselves, such asinterest rate caps, floors, collars and swaps. Caps and floors are used toset the upper or lower limits on the interest rates for loans. Interest ratecollars set both limits. Caps and floors are essentially portfolios of callor put options on interest rate. (Add more information on caps, floorscollars swaps and their pay-off functions).
9.4
Interest Rate Swaps
An interest rate swap is another type of LIBOR instrument that providesthe bridge between shorter term floating interest rates and longer termfixed interest rates. Swaps are frequently used for assevliability management, hedging, and speculation. The LIBOR is used as the benchmark for floating interest rates.A swap involves cash flows between two counterparties, thus thereare credit risks: either party involved in the swap could default. Becauseof credit risks, LIBOR is usually higher than the corresponding term U.S.Treasury rate.9.4.1
Plain Vanilla Interest Rate Swap
Plain vanilla interest rate swap is the most common swap, thus sometimes it is called a generic interest rate swap. In this case, the buyer ofthe swap contract gets fixed rate coupon payments on a specified notional par amount at a series of specified dates in exchanging for a seriesof floating interest rate payments. The fixed interest rate and the payment schedule are all predetermined by the swap contract, whereas thefloating interest rate is normally the London Interbank Offer Rate (LIBOR) for Euro-dollars. LIBOR interest rates are offered daily for a widerange of maturities from one day to a few years. The most popular one isthe 6-month LIBOR, which is often used as a benchmark for floatinginterest rates. For example, on February 13th2003, the market quote forthe generic 5 year swap rate is 5.74%. The long position holder of theswap on $100 million notional amount will pay 6-month LIBOR rate andreceive $2.87 million every 6 months for 3 years.A variety of derivatives contracts can be written on swaps that serveas underlying instruments.
Sec. 9.4
9.4.2
Interest Rate Swaps
395
Forward Swap
A forward starting swap is a forward contract on a swap. The buyer andseller of the forward starting swap contract agree on a future date, thelength of the term, and the fixed coupon rate of the forward swap contract. Both parties use the contract to lock in a future swap rate. At initiation, the contract is priced at zero, and its value will change afterwardsas the prevailing market interest rates change. A forward swap contractis often traded in the over the counter (OTC) market.9.4.3
Basis Swap
A basis swap is a contract agreed upon by two parties that requires thetwo parties to exchange two floating rates on a notional amount on anumber of specified dates. It is often traded in the OTC market. For example, two parties A and B agree to enter into a basis swap contract on$100 million notional amount. The contract requires that Party A paysParty B the 3 month LIBOR on the first business days of March, June,September, and December for five years, and that in exchange Party Bpays Party A 6 month LIBOR minus X basis points on the same dates.The motivation behind this deal maybe that Party A would like to swapits quarterly income into 6 month LIBOR linked cashflow that matchbetter with its other liabilities, while Party Bya deal who can mange therisk.9.4.4
Constant Maturity Swap
A constant maturity swap (CMS) contract is a special case of basisswaps. In a typical CMS, the long position holder receives the specifiedterm swap rate and pays 3 month or 6 month LIBOR. For example, aCMS contract on the 10-year swap rate for five years is a contract thatrequires the long position holder to pay 6 month LIBOR plus 20 basispoints twice a year for 5 years in exchange for the 10 year swap rate forthe same periods.9.4.5
Swaption
Options on various types of swaps are traded in the OTC market. Theseoptions are called swaptions. A swaption gives the holder the right butnot the obligations to enter the underlying swaps at a fixed time in thefuture. The most liquid ones are the put and call options or swaptions onthe forward starting generic interest rate swaps. Swaptions are fre-
396
Simple Interest Rate Products
Chapter 9
quently used for hedging and speculation purposes.
9.5
Bond Options
Options on notes and bonds are traded actively in the OTC market. Theyare frequently used as hedges and for speculation. Portfolio managerssell bond and note options according to their market views in order toenhance performance. Another type of option on bonds is the optionallyembedded in the bond structure, such as the put and call provisions in theputable and callable bond issues.
9.5.7
O K Options
A put or call option on a Treasury note or bond is considered a rathersimple derivatives instrument. Because bond options, usually calledOTC options, are traded in the OTC market, their terms, strike prices,and prices are all negotiable. An example of a popular OTC option is a90 day at the money put on the on-the-run U.S. Treasury bond. For example, on February 13th 1998, the price of the current bond, T 6.12511/15/27, is 103-27+, with a yield of 5.849%. The 90 day put optionwith the strike price equal to 104-11 3/8, volatility of 8.5% is priced at19+/32. Usually the OTC options are priced by applying the BlackScholes formula to the forward price, which is assumed to be lognormally distributed at expiry.
Chapter I 0
10.1
Yield CurveModeling
Introduction
The fair value of any interest rate instrument depends on the market perception of the time value of money. An accurate and dependable yieldcurve model is crucial to the valuation of interest rate instruments, andcan serve as an indicator of market mispricing that may lead to profitabletrading strategies. A yield cuwe is a numerical representation of the interest rate term structure, and is also the foundation of derivatives pricingand other advanced analysis and applications. For example, we can derive volatility and correlation structures by studying the time series ofyield curves. The goal of yield curve modeling is to construct yieldcurves that accurately fit market observations and possess desirableproperties such as flexibility, stability, continuity and smoothness.The most basic yield curve is the zero-coupon bond yield curve, orspot rate curve. Let’s consider an investment in a zero-coupon bond inthe U.S. Treasury market. On the settlement date a zero-coupon bondwith T years to maturity is sold at a price of P < 100, and the buyer willget 100 back on the maturity date. The spot rate S(t) is yield of thebond that satisfies the following price-yield relationship for zero-couponbonds,-21
P=100(1+~)
.
(10.1)
If we calculate the spot rates by using zero-coupon bonds with different maturities, we can obtain a spot curve S ( t ) . Given the spot curve
Yield Curve Modeling
398
Chapter 10
S ( t ) , it is straightforward to calculate the price of any given stream ofcashflows, 4 , for i = 1,2, ...,n, by adding up each of the cashflows discounted at the spot rate corresponding to its maturity time:
P = C 4 D ( t i >,
(10.2)
i
where D ( T ) is known as the discount function or the discount factor anddefined as(1 0.3)
One can see that it is trivial to create the spot curve at a discrete setof maturity points if one already has a set of zero-coupon bonds withknown prices. One only needs to compute the yields of the zero-couponbonds and then connect them to obtain the yield curve. Sometimes oneneeds to fit a smooth curve through the points in order to produce estimates for spot rates between two successive known points.However, if one is given a set of coupon-bearing bonds, it becomes anontrivial problem to derive the spot curve that fits optimally to the market. The remaining section of this chapter explains how the task is accomplished by using the bootstrap and the orthogonal exponential splinemethods.
10.2
The Bootstrap Method
Let’s assume that we are given the prices of a set of n coupon-bearingbonds with strictly ascending maturities, tk , k = 1,2,-. n . We also assume that we are given a short term interest rate, So for a term fo < ti.The first section of the spot curve is thena ,
S ( t ) = So
t I to .
(10.4)
The first bond can have p and q cashflows before and after time torespectively. Obviously, we have p 2 oand q 2 1, We assume that thespot rate changes linearly from So to S, over time interval [to,t, ] :
Sec. 10.3 Orthogonal Exponential Spline Model
S ( t )= So +- t- to S,
399
t,Itlt,.
(10.5)
tl -to
Now the price of the first bond is equal to the sum of its cashflowsdiscounted at the spot rates given by Eq. (10.5):
By solving the above equations, we can get a unique solution for S, .The bootstrap method iterates this process to obtain all the spot ratesuntil maturity t, . To demonstrate how this is done, let us assume thatwe have already obtained the spot rates S ( t ) for t I t k , and want tocompute the spot rates on the interval [ t k , t k + l ] . We apply a linear interpolation formula for spot rate on this interval as follows,(10.7)Let us assume that bond k has p k and q k cashflows before and aftertime tk respectively, then its price is given by
can be derived by solving the above equations.We can see that the bootstrap method generates a spot curve that fitsexactly to the market prices. The spot rates are piece-wise linear, thecorresponding forward curve is discontinuous, and the longer maturityinputs do not affect shorter term spot rates.Sk+l
10.3
Orthogonal Exponential Spline Model
The main drawback of the bootstrap method is that it does not provide asmooth yield curve, and because it exactly fits the market, it cannot beapplied to identify market mispricings. Other alternatives are the popularpiece-wise cubic or exponential spline models. These models usually
400
Yield Curve Modeling
Chapter 10
partition the maturity range into a several sections, fit each section to themarket by using cubic polynomials or exponential functions, and thensmoothly concatenate them at the partition points. The details of thesemethods can be found in many books and research papers (Vasicek andFong, 1991). However, the piece-wise spline models also lead to thefollowing problems:The forward short rate curve is kinked;The spline curve is not stable over regions lacking input data.These models, by construction, are not stable over the passage oftime, because the set of partition points (node points) is arbitrary: it hasto change with time in order to ensure optimal performance.For a yield curve model to serve as a good market indicator, it mustmeet several criteria, such as:Smoothness of the model’s spot and forward curves;Stability of the model’s estimates;Identification of both individual assets and market sector asrichkheap;Consistency of the model’s estimates with both economic intuitionand market segmentation;Flexibility to perform well in a variety of interest rate markets.In this section we describe the Orthogonal Exponential Spline (OES)Model which satisfies all these criteria. The model incorporates the natural economic feature of exponential basis functions, fits the benchmarkinstrument prices with accuracy, and generates smooth spot and forwardcurves. It can serve as a solid foundation for relative value, volatility,and correlation analyses. The model has been successfully applied to theU.S. and all major non-dollar Treasury and swap markets, and has beenaccepted by the finance industry as a better alternative than the conventional bootstrap, cubic or piece-wise exponential spline models.
10.3.1 Exponential Basis FunctionsThe objective of the spline model is to derive the spot curve S (t) for agiven set of benchmark instruments. The discount function D (t) is thepresent value of one dollar at the future time t . There is a one-to-onecorrespondence between the spot curve and the discount function orfactor:
401
Sec. 10.3 Orthogonal Exponential Spline Model
.
s(t)=2 qt>-2,-1[
l
(10.9)
l
Or equivalently,(10.10)We choose to work with the discount function for convenience.In a flat interest rate environment the discount function decays exponentially with time to maturity. Naturally, we propose to use thesmoothly decaying set of exponential functions
as basis for a series expansion of the discount function. Optimal fittingefficiency and numerical stability can be achieved if we construct an orthogonal set of exponential basis functions EN{e,(t), n = 1,2;..N} suchthatem ( t )@ en (t>= 6m.n 7
(10.11)
where the inner product is defines as
rfm
1
if m=n=O
=[rfm(t)fn(t)dt
otherwise *
(10.12)
The orthogonal exponential basis functions can be generated throughthe standard Gram-Schmidt procedure. Figure 10.1 shows examples oforthogonal exponential basis functions.
Yield Curve Modeling
402
Chapter 10
1
0.8
0.6C
0
9 0.43
U.
.J 0.2
t
m0
-0.2
-0.4h0
5
10
15Time (Year)
20
25
30
Figure 10.1. Examples of orthogonal exponential basis functions.
We expand the discount function in terms of the basis functions(10.13)
where Ai is given by
Ai = D(t)0ei ( t )
(10.14)
Since lim en(t)= 0 , we have lim R,(t) = 0 for all a > 0 . The sen-tm
N-WC
ries expansion for the discount function is unbiased as N + W . Empiricalstudies indicate that, for all practical purposes, it will suffice to chooseN in the order of 10.In the construction of the basis functions, one must choose a valuefor the exogenous parameter a . After combining the same order exponential functions, the expansion given by Eq. (10.13) is equivalent to anexpansion in terms of the exponential h c t i o n s f,,directly:
Sec. 10.3 Orthogonal Exponential Spline Model
403
(10.16)
This expansion has the following propertylim ~ ( t+)C1e-" .
t+-
(10.17)
This identifies a with the instantaneous forward rate at distant maturities.123
70.3.2 Maximum Likelihood Estimates for Spline CoefficientsGiven a collection of benchmark bonds, B{B, ,n = 1,2, - - ,Ns} with market prices P{P,,n =1,2,--.,N,] we aim to construct the discount function D ( t ) as an expansion of the exponential family F up to some appropriate order N such that the residual R, is reasonably small (smallerthan allowed by the noise level in prices); our objective is to find the optimal estimate for the coefficient set A{ A,, n = 1,2,.. ,N B ] . The discount function thus constructed is a function of A :D(t) = D ( A , t ).For each benchmark bondB, , in addition to the market price, thenumber of cash flows q, and each cash flow { F' ,k = 1,2; .-,q,] is exactly known. The theoretical price P ' ~is defined as the present value ofthe cash flows discounted back by using the N-parameter discount function D ( A , t )( 10.18)i=l
k = l i=l
This equation can be written in matrix form as:
P ~ = H A
(10.19)
where His an N , x N matrix with
lZ3 Suppose one makes an investment at time t + 00 in the future for a period At, thediscount factor one would get for the investment period is D(t + At) / D ( t ) = e - M t ,which corresponds to a continuous compounding yield of a .
404
Yield Curve Modeling
Chapter 10
Tln
Hn,k = x F ( t i ) e k ( t i )i=l
(10.20)
and A is the column vector {Ai}.Assuming normal distribution for pricing errors, we can derive themaximum likelihood estimators for the spline coefficients. The likelihood function, ignoring certain constants, can be written as:L ( A ) =nexp(-?(Pih(A)-P,,)NB
(10.21)n=l
where w,, is the weight assigned to bond B,, . The goal is achieved byletting-lOgL(A)adA
=-
(
(A) W Pth(A) - p )dA= - H ~ W ( H A- P )=O
(10.22)
then we obtain the analytical solutionA =(H~wH)-~H~wP,
(10.23)
which, known as Generalized Least Squares (GLS), is the optimal solution (in the least-square sense) for an over-determined linear system.However, the economic nature of the discount function requires thatwe impose the time-zero constraint D(0) = 1 in the optimization. Let usdefine the N x 1 matrix M j = e j ( 0 ) for j = 1,2,...,N , so that the constraint can be written in matrix form
MTA'- 1= 0 ,
(10.24)
where A' is a new set of spline coefficients that incorporates the timezero constraint.We introduce a Lagrange multiplier p into the optimization procedure, and letd-(dA'logL(A') + p(MTA'- 1)) = 0
(10.25)
Sec. 10.3 Orthogonal Exponential Spline Model
405
Thus
- H ~ w ( H A ’ - P)+,UM =o
(10.26)
MTA‘-I = OTherefore
P=
I-M~A
M ( H T~~ )-’ M
(10.27)
A’ = A + ,u( H WH 1 - l ~70.3.3 Implementation of the Spline ModelThe implementation of the spline model depends on how it is to be used.For the construction of swap curves, we choose the number of basisfunctions equal to the number of inputs so as to exactly fit the market.For non-dollar markets with fewer liquid instruments and maturity gaps,we use fewer basis functions to achieve better stability.The parameter a controls the flexibility of the basis functions. Alarger a means more fitting power and generates a “softer” curve, whilea smaller a provides less flexibility and covers maturity gaps nicely.With good choices of the parameterization ( N , a ) , the model has beensuccessfully applied to the U.S., Canada, Japan, Australia, and all majorEuropean bond and swap markets. Financial applications often requirehistorical analyses. Once chosen, for consistency of results, the pair( N , a ) should be kept constant throughout the time period.For instance, a model for the U.S. Treasury market uses all the bulletnotes and bonds as input. Each instrument is assigned a weight inverselyproportional to its years to maturity. The weights for the currents arereduced by a factor of 10. The model uses 7 basis functions and choosesa = 7%. For trade date August 11, 1997, N , = 189. In other words, 7basis functions used to match the market prices of 189 instruments. Theaverage fitting error is 2.4 basis points in yield terms. The results are notsensitive to a when it takes any value between 3% and 12%. Progressingly smaller errors can be obtained by using progressingly more basisfunctions. Due to the existence of the analytical solution, the model alsoruns very fast and can produce the spline coefficients in less than 1/10 ofa second.
406
Yield Curve Modeling
Chapter 10
Due to the discount finction given by Eq. (10.16) is infinitely differentiable, the smoothness of the forward curve is always satisfied.70.3.4 Summary
Unlike other piece-wise spline models, the OES model does not introduce subjective inputs such as node points and particular partitioning ofthe maturity domain. In the piece-wise constant models, the inhomogeneous treatment of different yield curve sectors often distorts the volatility and correlation measurements, and as time passes, a cashflow of aninput instrument may move across a node poigt, causing a sudden jumpin the yield curve. The OES model provides smooth and reliable volatility and correlation structures. It can serve a$ a good foundation for treeand lattice based derivatives valuation and hedging models.Smooth and reliable correlation pattehs are crucial to principal components analyses of yield curves. A two-factor risk model has been developed based on correlation analyses of the exponential spline curves.The model reveals yield curve sector relative richness or cheapness, andis a powerful tool for hedging and index tracking.A remarkable feature of the OES model is its mathematical eleganceand simplicity. The natural exponential decay and the orthogonality ofthe basis functions enhance the model's numerical efficiency. A typicalcubic spline model for the U.S. Treasury market requires 7 partitions and14 spline coefficients, whereas the OES model only needs 7 basis functions and 7 coefficients. The model runs about 4 times faster. For realtime applications, speed is advantage.
10.4
Swap Curve
The swap curve is the very foundation for all swap related assets analysis. In order to price a stream of cashflows with LIBOR credit rating,one uses the zero-coupon swap curve to discount the cashflows. According to the arbitrage-free pricing theory, one can use the swap curve implied forward rate as a fair value projection for an unknown floating ratein the future. One can also compare the swap curve with the Treasurycurve for making cross-market relative value analyses.The market convention for constructing the swap curve is to use thespot LIBOR rate for the stub period from settlement to the start date ofthe nearest Euro-dollar futures contract, then use the first eight Eurodollar futures strips to create the zero-coupon curve to two years, and
Sec. 10.4 Swap Curve
407
then use the par-coupon swap rates for building the zero-coupon swapcurve to 30 years. The challenges are:The zero-coupon swap curve has to fit accurately to the market.Specifically, the zero curve has to fit exactly to the Euro-dollar strippoints for the first two years, and then the implied par-coupon curve hasto exactly pass through the input swap rate points.The swap curve and the implied forward short rate curve have to besmooth and stable.Convexity adjustment needs to be made to the Euro-dollar stripcurve.
70.4.7 Constructing Euro-dollar Strip CurveThe idea is to synthetically create a zero-coupon curve by using the stubrate and the consecutive Euro-dollar futures contracts. One usually constructs the discount function, and then converts it to a zero-coupon curve.The time period between the settlement date and the start date of thefirst (nearest) Euro-dollar futures contract is defined as the stub period z, ,in units of actual calendar days, which can be zero or up to 3 months.The interest rate for the stub period is called the stub rate, Rl . If the stubperiod is less than one week, one usually uses the one week LIBOR forthe stub rate; otherwise one interpolates the 1 month, 2 month and 3month LIBOR rates to obtain the stub rate. See Table 10.1.Table 10.1: Computation of the Stub RateStub PeriodStub Rate
c 1 Week
1 Week LIBOR
Between 1 Weekand 1 MonthInterpolate Between1 Week and 1 MonthLIBOR
Between 1 Monthand 2 Months
Between 2 Monthsand 3 Months
Interpolate BetweenInterpolate Between1 Month and 2 Month 2 Month and 3 MonthLIBORLIBOR
The discount factor for the stub period is then1d,=--.1+-W,360
(10.28)
Now let’s define z, as the time period, in units of actual days, fromthe settlement to the start date of the ith Euro-dollar futures, pi the priceof the ith EDF contract. Then the discount factor for the time periodfrom zi-,to zi is
408
Yield Curve Modeling
Chapter 10
(10.29)
The discount factor for a synthetic zero-coupon bond with zi days tomaturity is thereforei
D(Ti) = n d k .
(10.30)
k=l
The corresponding zero-coupon swap rate can be computed. ForUSD swap rates, the market uses the 30/360 day count, and the semiannual compounding convention. Let ti be the number of years corresponding to the period zi,the zero-coupon swap rate is then given by
1-7
S(ti)= 2 D ( T j ) 2ti -1
,
(10.3 1)
This provides the zero-coupon swap rates at discrete times with 3month increments. For any time (years to maturity) between thesepoints, we can use a linear interpolation or the spline method to obtainthe swap rate.70.4.2 Convexity Adjustment
Euro-dollar futures contracts usually trade actively to 5 years, and thereis a reasonable amount of activity even for contracts that start in 5 to 10years. Practitioners often use EDF contracts to construct hedges forswap or bond positions, or use the synthetic EDF zero-coupon bond as aninvestment alternative. If the hedges or investment strategies involveusing long term (say more than 2 years) EDF contracts, it is important tomake convexity adjust