jod-wu:jod-wu 11/9/10 3:12 pm page 20 modifying...

20 MODIFYING THE LMM TO PRICE CONSTANT MATURITY SWAPS WINTER 2010

Within the framework of the LIBOR market model,this article presents a new approach for finding theapproximate distribution of constant maturity swap(CMS) rates under forward martingale measures. Withthis approach, many popular CMS-type interest ratederivatives, such as CMS, CMS caps, CMS floors,CMS steepeners, and CMS range accruals, can bepriced under the LIBOR market model and their riskcan be managed consistently with LIBOR-type interestrate derivatives. We use this approximation to pricethree types of CMSs, namely, CMS-for-CMS, CMS-for-LIBOR, and CMS-for-fixed CMSs. The result-ing pricing formulas are shown to be robustly accurateand time saving by comparison with Monte Carlosimulations based on the market data over a recentthree-year period.

Aconstant maturity swap (CMS) is anagreement that designates two coun-terparties to periodically exchangetwo payment streams (commonly

called “legs”) over a specified period. Based onan agreed notional principal, one leg pays(receives) a CMS rate, while the other receives(pays) a fixed or floating rate. The floating ratemay be a London Interbank Offer Rate (LIBOR)or another swap rate with a different tenor.CMSs started trading in the mid-1990s, andtheir trading volume has continued to growrapidly.

There are two main types of CMS end-users. The first type includes investors who

wish to take profits from a change in the shapeof the yield curve. In a flat yield curve envi-ronment, market participants may take a viewthat long-term CMS rates will rise in the futureas the yield curve steepens and thus take profitsby entering CMS positions to receive long-term CMS rates and to pay short-term CMSrates. For example, as shown in Exhibit 1, thespreads between the 10-year and 2-year U.S.dollar CMS rates in January 1995, June 2000,and March 2006 were lower than 30 basispoints (bps), which is significantly narrowerthan they used to be. Due to the flatness of theyield curve, investors were able to generateattractive returns as the yield curve steepened.In July 2003, the spread reached its peak andwas higher than 260 bps. Thus, CMSs couldalso be used to take profits in a very steep yield-curve environment.

The second type of end-users includescorporations and investors who seek to flex-ibly and efficiently maintain a constant assetor liability duration. For example, life insurersmay be heavily indebted to long-term pay-ment obligations, encountering risk if the backend of the yield curve rises sharply. To hedgethis risk, they may use CMSs to swap theirassets from those receiving short-term interestrates (e.g., 6-month LIBOR) to those receivinglong-term swap rates (e.g., 10-year swap rate).With all these uses, it is not surprising thatCMSs are actively traded in financial markets.

Modifying the LMM toPrice Constant Maturity SwapsTING-PIN WU AND SON-NAN CHEN

TING-PIN WU

is an associate professor inthe Department of Financeat the National CentralUniversity in Taoyuan,[email protected]

SON-NAN CHEN

is a professor of finance atthe Shanghai AdvancedInstitute of Finance (SAIF)at Shanghai Jiao Tong Uni-versity in Shanghai, [email protected]

JOD-WU:JOD-WU 11/9/10 3:12 PM Page 20

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As indicated by Jamshidian [1997], under a forwardswap measure, a forward swap rate is a martingale and,thus, the expectation of a CMS rate. But CMSs are usu-ally priced under forward measures rather than under for-ward swap measures, thereby making their pricing morecomplicated. Actually, within most interest rate models,the distribution of CMS rates under forward measures isunknown. Therefore, the expectation of cash flows linkedto CMSs cannot be computed analytically.

In practice, two main approaches are used to com-pute the expectation of CMS rates. The first approachis based on Monte Carlo simulation within an interestrate model. While this approach is flexible enough tocompute prices of almost every kind of interest ratederivative, its disadvantage is that it is too time con-suming, especially in a competitive financial market.The second approach is to compute the expectation ofCMS rates via an approximate analytic formula, namely,the forward swap rate (obtainable from market data)multiplied by a convexity adjustment. Many CMS con-vexity adjustment formulas are available in the litera-ture, such as Benhamou [2000], Pugachevsky [2001],Brigo and Mercurio [2006], and Hull [2006]. Some ofthem are based on less theoretically sound assump-tions and may lead to some pricing errors. Increasing

competition in the CMS market, however, hasmade the inaccuracies of the conventional CMSconvexity adjustments more apparent. Accord-ingly, this article attempts to develop an alterna-tive method to compute the expectation of CMSrates under forward measures.

The LIBOR market model (LMM) and theswap market model (SMM) are well known to beincompatible, and thus the distribution of swap rateswithin the LMM framework is unknown. We pre-sent an alternative approach for finding a lognormaldistribution to approximate the distribution of afuture CMS rate under a forward measure. Withthis approach, many popular CMS-type interest ratederivatives, such as CMSs, CMS caps, CMS floors,CMS steepeners, and CMS range accruals, can bepriced in the LMM and their risk management canbe conducted consistently with LIBOR-typeinterest rate derivatives. This article intends to pricethree types of CMS contracts within the LMM,namely, CMS-for-CMS, CMS-for-LIBOR, andCMS-for-fixed-CMS. The resulting pricing for-mulas are shown to be robustly accurate and time

saving by comparison with Monte Carlo simulations basedon the market data over the recent three years.

This article is organized as follows. The next sectionreviews the LMM and introduces some useful techniques,such as the change of measure and the lognormalizationof LIBOR rates and swap rates. The third section pre-sents three types of CMSs and their pricing formulas. Thefourth section examines the accuracy of the approximateformulas via Monte Carlo simulations, and the last sectionsummarizes the results.

REVIEW OF THE LMM AND AN APPROXIMATE DISTRIBUTION OF A SWAP RATE IN THE LMM

In the first subsection, we briefly review the LIBORmarket model and some useful techniques, such as thechange of measure and the lognormalization of LIBORrates. In the second subsection, we introduce a new log-normalization approach for swap rates under the LMM.

Review of the LMM

This subsection briefly reviews the LMM developedby Brace, Gatarek, and Musiela (BGM) [1997], Miltersen,

WINTER 2010 THE JOURNAL OF DERIVATIVES 21

Note: The time series data of the spread in basis points between the 10-year and 2-yearCMS rates from July 3, 1989, to October 31, 2008.

E X H I B I T 1The Spread between the 10-Year and 2-Year CMS Rates

JOD-WU:JOD-WU 11/9/10 3:12 PM 1

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Sandmann, and Sondermann [1997], and Musiela andRutkowski [1997]. The LMM has been widely used in themarketplace for several reasons. It directly specifies thebehavior of the market-observable rate (LIBOR) ratherthan the abstract rates in the traditional interest rate models,thus making the model richer in financial intuition. TheLIBOR modeled in the LMM follows a lognormal dis-tribution, which avoids negative rates and pricing errors.1

Moreover, the pricing formulas of caps and floors withinthe LMM framework are Black’s formulas, thus makingthe calibration procedure easier. In addition, most activelytraded interest rate products can be priced within theLMM framework so that their hedges can be managedconsistently and efficiently.2

The LMM is briefly introduced as follows, and wewill employ it to price CMSs in the next section. Assumethat trading takes place continuously over an interval [0, T ],0 < T < ∞. The uncertainty is described by the filtered spotmartingale probability space (Ω, F, Q, {Ft}t∈[0,T ]), and anm-dimensional independent standard Brownian motionW(t) = (W1(t), W2(t), …, Wm(t)) is defined on this filteredprobability space. The information flow, accruing to all theagents in the economy, is represented by the filtration{Ft}t ∈[0,T ], which satisfies the usual hypotheses.3 We intro-duce some notations as follows. Q denotes the spot mar-tingale probability measure. P(t,T ) denotes the time-t priceof a zero-coupon bond paying $1 at time T. L(t,T ) is forwardLIBOR contracted at time t for the period [T, T + δ ]. QT

denotes the forward martingale measure with respect to thenuméraire P(., T ). The relationship between L(t,T ) andP(t,T ) can be expressed by

L(t,T ) = (P(t,T ) – P(t,T + δ ))/δP(t,T + δ ) (1)

where δ stands for a compounding period. ForwardLIBOR can be derived via Equation (1) from the pricesof zero-coupon bonds.

BGM [1997] modeled interest rate behavior in termsof forward LIBOR based on the arbitrage-free conditionspresented in Heath, Jarrow, and Morton [1992]. We brieflyspecify their results in the following proposition.

Proposition 2.1. LIBOR Dynamics under theMeasure Q

The dynamics of LIBOR L(t, T) under the spot martin-gale measure Q are as follows:

(2)dL t T

L t Tt T t T dt t T dWP

( )

( )( ) ( ) ( ) (

,,

= , ⋅ , + + , ⋅γ σ δ γ tt)

where 0 ≤ t ≤ T ≤ T and σP(t, .) are defined as follows:4

(3)

where [δ –1 (T – t)] denotes the greatest integer that is less thanδ –1 (T – t), and the deterministic function isbounded and piecewise continuous.

According to the derivation process of BGM [1997],{σP (t, T )}t∈[0, T ] stands for the volatility process of thebond price P(t, T ), which is stochastic rather than deter-ministic. Thus, the stochastic differential Equation (2)is not solvable, and the distribution of L(T, T ) is unknown.Given a fixed initial time assumed to be zero, however, wecan approximate σP (t, T ) by , which is defined by

(4)

where 0 ≤ t ≤ T ≤ T. The calendar time of the process{L(t, T – jδ )}t∈[0,T–jδ ] in Equation (4) is frozen at its ini-tial time 0, and thus the process becomesdeterministic. By substituting for σ P(t, T + δ )in the drift terms of Equation (2), we can solve it and findthe approximate distribution of L(T, T ) to be lognor-mally distributed.

This technique was first used by BGM [1997] forpricing interest rate swaptions. It was developed furtherby Brace, Dun, and Barton [1998] and formalized by Braceand Womersley [2000]. It was also used by Schlögl [2002].This approximation has been widely used in the market-place and has been shown to be very accurate.

Proposition 2.2. The Lognormalized LIBORMarket Model

The dynamics of lognormalized forward LIBOR L(t,T )under the spot martingale measure Q are given by

σ

δ δδ δ

δ

P

j

T t

t T

L t T j

L t T j

( )

( )

(

[ ( )]

, =

, −+ , −=

−−

∑1

1

1 ))( ) [ ]γ δ δ

δ

t T j t T

T

, − ∈ , −

− >

⎧

⎨

⎪ 0

0

0

and

otherwise

⎪⎪⎪

⎩

⎪⎪⎪

P t T0σ δ( ), +{ ( )} [ ]P t Tt T0

0σ , ∈ ,

P

j

T t

t T

L T j

L T j0

1

1

0

1 0

σ

δ δ

δ

δ

( )

( )

(

[ ( )]

, =

, −

+ , −=

−−

∑δδ

γ δ δ

δ

)( ) [ ]t T j t T

T

, − , ∈ , −

− >

⎧

⎨

⎪ 0

0

0

and

otherwise

⎪⎪⎪

⎩

⎪⎪⎪

P t T0σ ( ),

γ : →+R Rm2



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(5)

where is defined in Equation (4).In pricing interest rate derivatives, it is sometimes

more convenient to compute under a specific probabilitymeasure rather than the spot martingale measure Q. Thus,we need to know the processes of LIBOR rates underother martingale measures. The following propositionspecifies the general rule under which LIBOR dynamicsare changed following a change of the underlying prob-ability measure.

Proposition 2.3. The Drift Adjustment Techniquein Different Measures

The dynamics of forward LIBOR L(t, T) under an arbi-trary forward martingale measure QS are given as follows:

(6)

where 0 ≤ t ≤ min(S,T ).5

An Approximate Distribution of aSwap Rate in the LMM Framework

Jamshidian [1997] presented the SMM underwhich the forward swap rate is a martingale and swap-tions can be priced via Black’s formula. It is well knownthat the LMM and the SMM are incompatible, becauseswap rates and LIBOR rates cannot be lognormally dis-tributed under the same probability measure. Therefore,choosing either of the two models as a pricing founda-tion is problematic. Because of mathematical tractability,Brace, Dun, and Barton [1998] suggested that the LMMshould be adopted as the central model, and we followtheir suggestion.

We chose the LMM to price CMSs. Accordingly, thefirst problem we encounter is how to find the approxi-mate distribution of swap rates under a forward measure.As we will show later, a swap rate is roughly a weightedaverage of LIBOR rates, and the LIBOR rates under theLMM are approximately lognormally distributed. As aresult, the distribution of a swap rate is roughly a weightedaverage of lognormal distributions.

P t T0σ δ( ), +

dL t T

L t Tt T t T dt t T dWP

( )

( )( ) ( ) ( )

,,

= , ⋅ , + + , ⋅γ σ δ γ0 (( )t

dL t T

L t Tt T t T t S dP P

( )

( )( ) ( ( ) ( ))

,,

= , ⋅ , + − ,γ σ δ σ0 0 tt t T dW t+ , ⋅γ ( ) ( )

This subsection presents a new approach to findingthe approximate distribution of a swap rate under theLMM framework.

Define an n-year forward swap rate, observed at t tomature at time Ti with 0 ≤ t ≤ Ti, as follows:

(7)

where

(8)

and and denote,respectively, reset and payment dates with a constant-yearfraction (i.e., δ = Ti+j – Ti+j–1, j = 1, 2, …, qn). Brace andWomersley [2000] showed that the variability of thewas small compared to the variability of forward LIBORL(t, Ti+j), and their conclusion was further empiricallyconfirmed by Brigo and Mercurio [2006]. Thus, we canfreeze the value of the process to its initial value,

, and obtain

(9)

By observing Equation (9), Sn(Ti,Ti) is a weightedaverage of lognormally distributed variables, and there-fore its distribution is unknown. Although Sn(Ti,Ti) isnot a lognormal distribution, it can be well approximatedby a lognormal distribution with the correct first twomoments.7 Based on this approximation, we assume thatlnSn(Ti,Ti) has a normal distribution with mean M andvariance V 2. The moment-generating function forlnSn(Ti,Ti) is given by

(10)M E explnS n i ihh S T T Mh V h( ) [ ( ) ]= , = +⎛

⎝⎜⎞⎠⎟

1

22 2

w tni j, ( )

wni j, ( )0

S t T w L t Tn ij

q

ni j

i j

n

( ) ( ) ( ), ≅ ,=

−,

+∑0

160

w tni j, ( )

{ , }T T Ti i i qn, , ...+ + −1 1 { , }T T Ti i i qn+ + +, , ...1 2

w tP t T

P t T

qn

ni j i j

kq

i k

n

n

, + +

=−

+ +

=,

,

=

( )( )

( )1

01

1Σ

δδ

S t T w t L t Tn ij

q

ni j

i j

n

( ) ( ) ( ), = ,=

−,

+∑0

1



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Where h = 1 and h = 2 in Equation (10), respec-tively, we obtain the two conditions to be solved for Mand V 2 as follows:

(11)

(12)

where E[Sn(t,Ti)] and E[Sn(t,Ti)2] are computed in the

Appendix.The accuracy of this technique has been examined

by Mitchell [1968]. Furthermore, many areas of sciencehave verified the lognormal approximation to be highlyaccurate by the sum of lognormal random variables (e.g.,Aitchison and Brown [1957], Crow and Shimizu [1988],Levy [1992], Limpert, Stahel, and Abbt [2001], andBorovkova, Permana, and Weide [2007]). In the next tolast section, we will provide detailed empirical results toshow that this technique is robustly accurate in pricingCMS derivatives.

PRICING CONSTANT MATURITY SWAPS

In this section, we use the approximation methodfrom the previous section to price three types of CMSs,namely, CMS-for-CMS, CMS-for-LIBOR, and CMS-for-fixed.

Pricing CMS-for-CMS

A CMS-for-CMS constant maturity swap is calleda two-way CMS. It is mainly sensitive to the slope of theyield curve and thus can be used to take a profit from achange in the difference between long-term and short-term interest rates. As the yield curve steepens, investorsmay think that the long-term swap rates will not remainas high in the future as the market reveals, and thus theymay take a position in CMSs by paying long-term CMSrates and receiving short-term CMS rates. Alternatively,in a flat-yield-curve market, investors may believe thatlong-term interest rates may rise in the near future andwish to take a position in CMSs by paying short-termCMS rates and receiving long-term CMS rates.

To provide a general pricing formula for two-wayCMSs, we define a general CMS contract as follows.

M S T T S T Tn i i n i i= , − ,21

22lnE lnE[ ( )] [ ( ) ]

V S T T S T Tn i i n i i2 2 2= , − ,lnE lnE[ ( ) ] [ ( )]

Consider a τ-year two-way CMS starting at time T0 withpayment dates T1 ≤ T2 ≤ … ≤ Tqτ. For simplicity, weassume that δ = Ti – Ti–1 for i = 1, 2, …, qτ, qτ = τ/δ,and the notional principal is $1. At each payment date Ti(in some variants at Ti + 1) for i = 1, …, qτ , one partypays (receives) δ(Sn(Ti,Ti) – K1) to the counterparty andreceives (pays) δSm(Ti,Ti) from the counterparty, whereSn(Ti,Ti) and Sm(Ti,Ti) denote, respectively, n-year and m-year swap rates observed at time Ti, and K1 is a premium(or a discount). For the party who pays δSm(Ti,Ti) andreceives δ(Sn(Ti,Ti) – K1), the cash flow stream is givenas follows:

The pricing formula of the two-way CMS is pre-sented in the following theorem, and its proof is providedin the Appendix.

Theorem 3.1. The Approximate Pricing Formulaof the Two-Way CMSs

For the party who pays m-year CMS rates and receivesn-year CMS rates, the price of the τ-year two-way CMS attime t, T0 ≤ t ≤ T1, is given by

(13)

where

(14)

(15)

(16)w tP t T

P t Ti j i j

kq

i k∗

+ +

=−

+ +

( ) =∗

,( , )

( , )1

01

1Σ

ζ( ) ( )t T T u T T dui j i t

T

t i j i

i, ; = Δ , ;( )+ +∫exp

At time

At time

1T S T T S T T K

T

n m: , ,δ 1 1 1 1 1( ) − ( ) −⎡⎣ ⎤⎦

22

At time

: , ,

:

δ

δτ

S T T S T T K

T

n m

q

2 2 2 2 1( ) − ( ) −⎡⎣ ⎤⎦� �

SS T T S T T Kn q q m q qτ τ τ τ, ,( ) − ( ) −⎡⎣

⎤⎦1

11

CMS E

E

( ) ( )( [ ( ) ]t P t T S T Ti

q

iQ

n i i t

Ti= , , |

−=∑δ

τ

F

QQm i i t

Ti

S T T K[ ( ) ] ), | −F 1

EQi i t

j

qi j

i j

Ti

S T T w t L t T[ ( ) ] ( ) (∗=

−

∗,

+, | = ,∗

∑F0

1

)) ( )ζ t T Ti j i, ;+



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(17)

*∈ {m, n} (18)

(19)

It is worth noting that all parameters in the pricingformula can be easily extracted from market data, and thusthe formula is tractable and feasible for market practi-tioners. In addition, market practitioners usually set anexpected CMS rate equal to its corresponding forwardCMS rate modified by a convexity adjustment, which canbe seen as an overall correction for the forward CMS rate.Our pricing formula, however, separately modifies for-ward LIBOR involved in the CMS rate by different con-vexity adjustments, ζ. By observing the pricing formula inEquation (13), two-way CMSs are almost immunized fromany parallel shift in the interest rate yield curve, but theyare sensitive to change in the slope of the interest rate yieldcurve. These features will be examined in the next section.

As a CMS is priced at contract initiation, T0, its priceis reflected in the premium K1. By adjusting the premiumK1, the initial price of the two-way CMS can be set tozero and trading becomes a fair game. This fair premiumK1 is provided by

(20)

where

By observing Equation (20), the fair premium isrepresented by the difference between two weighted aver-ages of CMS rates over the life of the transaction.

Pricing CMS-for-LIBOR

A CMS-for-LIBOR constant maturity swap is sim-ilar to the two-way CMS described in the previous

q∗ =∗δ

wP T T

P T Ti i

kq

i

τ τ

0 1 0

0 01

, ( , )

( , )−

=

=−Σ

K w S T T

w

i Q

i

q

n i i T

i

Ti

10 1

1

0 1

0=

−

−

=

−

∑ τ

τ

τ,

,

[ ( , )| ]E F

EEQ

i

q

m i i T

Ti

S T T=∑

10

τ

[ ( , )| ]F

Δ , ; = , ⋅ , −+ + + +t i j i i j Pt

i j Ptu T T u T u T( ) ( ) ( ( ) (γ σ σ1 uu Ti, ))

subsection, except that one of the CMS rates is replacedby LIBOR in the cash flow stream. At each payment dateTi for i = 1, …, qτ , one party pays (receives) δ(Sn(Ti,Ti) – K2)to the counterparty, where K2 is a premium (or a discount),and receives (pays) from the counterparty δL(Ti,Ti) (orδL(Ti–1, Ti–1) in some variants) where L(Ti,Ti) is LIBORfor the period [Ti, Ti+1]. For the party who pays δL(Ti,Ti)and receives δ(Sn (Ti,Ti) – K2), the cash flow stream is givenas follows:

The pricing formula of the CMS-for-LIBOR CMSis presented in the following theorem.

Theorem 3.2. The Approximate Pricing Formulaof CMS-for-LIBOR CMSs

The price of the τ-year CMS-for-LIBOR CMS at time t,T0 ≤ t ≤ T1, is given by

(21)

where and ζ(t, Ti, Ti) are defined, respec-tively, in Equations (14) and (15), and L(t, Ti) denotes the for-ward LIBOR rate that is observable from market data.8

It is well known that a one-period swap rate is actu-ally a LIBOR rate, and thus a CMS-for-LIBOR CMS isa special case of a two-way CMS. As the tenor, m, of thepaid swap rate Sm(Ti, Ti) is set to δ, Equation (13) reducesto Equation (21). Similar to Equation (13), CMS-for-LIBOR CMSs are sensitive to a change in the slope ofthe interest rate yield curve, but almost immunized fromany parallel shift in the interest rate yield curve.

End-users usually use CMS-for-LIBOR CMSs toadjust their asset or liability duration. For example, lifeinsurers may be indebted to long-period payment oblig-ations, but own short-duration assets (may be receivingLIBOR payments). To remove interest rate risk, they mayuse CMS-for-LIBOR CMSs to transform their LIBOR

EQn i i t

Ti

S T T[ ( ) ], | F

CMS t P t T S T T

Li

q

iQ

n i i t

Ti

21

( ) ( )( [ ( ) ]= , , |

−=∑δ

τ

E F

(( ) ( ) )t T t T T Ki i i, , ; −ζ 2

At time

At time1

2

T S T T L T T K

Tn: [ ( , ) ( , ) ]

:

δδ

1 1 1 1 2− −[[ ( , ) ( , ) ]

: [ (

S T T L T T K

T S T

n

q n q

2 2 2 2 2− −� �

At time τ δ ττ τ τ τ, ) ( , ) ]T L T T Kq q q− − 2



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cash inflows into long-period CMS cash inflows. Con-versely, debts to commercial banks, usually from customers’deposits, may have short-term duration while their assets,usually from commercial loans, may have long-term dura-tion. Via CMS-for-LIBOR CMSs, commercial banks canhedge their interest rate risk.

Similar to two-way CMSs, pricing CMS-for-LIBOR CMSs at contract initiation is determined by itspremium K2. By adjusting K2, the initial price of the CMScan be set to zero and the fair premium K2 is solved asfollows:

(22)

The fair premium K2 can be regarded as the differ-ence between the weighted averages of CMS rates andLIBOR rates over the life of the transaction.

Pricing CMS-for-Fixed

A CMS-for-fixed constant maturity swap is similarto the two-way CMS described in the subsection onpricing CMS-for-CMS CMSs, except that one of theCMS rates is replaced by a fixed-rate R in the cash flowstream. At each payment date Ti for i = 1, …, qτ , one partypays (receives) δSn(Ti, Ti) to the counterparty and receives(pays) from the counterparty δR. For the party who paysδR and receives δSn(Ti, Ti), the cash flow stream is givenas follows:

The pricing formula of the CMS-for-fixed CMSsis presented in the following theorem.

Theorem 3.3. The Approximate Pricing Formulaof CMS-for-Fixed CMSs

At time

At time1

2

T S T T R

T S T Tn

n

: ( , )

: ( ,

δδ

1 1

2 2

−⎡⎣ ⎤⎦))

: ( , )

−⎡⎣ ⎤⎦

−⎡⎣ ⎤⎦

R

T S T T Rq n q q

� �

At time τ τ τδ

K w E S T T

w

i

qi Q

n i i T

i

q

Ti

21

0 1

1

0= , |

−

=

, −

=

∑

∑

τ

τ

τ [ ( ) ]F

ττ ζ0 10 0

, − , , ;ii i iL T T T T T( ) ( )

The price of the τ-year CMS-for-fixed CMS at time t,T0 ≤ t ≤ T1, is given by

(23)

where is defined in Equation (14).As the paid swap rate Sm(Ti, Ti) degenerates to a

constant rate R, Equation (13) reduces to Equation (23),and thus a CMS-for-fixed CMS is also a special case of atwo-way CMS. Likewise, CMS-for-fixed CMSs are alsosensitive to change in the slope of the interest rate yieldcurve. Unlike the aforementioned CMSs, CMS-for-fixedCMSs are sensitive to any parallel shift in the interest rateyield curve. These features will also be examined in thenext section.

The prices of CMS-for-fixed CMSs are reflected inthe fixed rates. By adjusting R, the initial price of theCMS can be set to zero and the fair rate R is solved as

(24)

The fair rate R is thus the weighted average ofexpected CMS rates over the life of the transaction.

Corporations and financial institutions usuallyemploy CMS-for-fixed CMSs to enhance yields. Investorswho receive fixed-rate cash inflows, and expect the backend of the yield curve to rise steeply, may enter into thisCMS to take profits by receiving long-term CMS rateswhile paying fixed rates.

We have provided the pricing formulas of three typesof CMSs by employing the approximation methods intro-duced in the second section. The accuracy of these pricingformulas will be examined in the next section by com-parison with Monte Carlo simulations.

NUMERICAL STUDIES

This section first presents a method to calibrate theparameters in the LMM and then provides some numer-ical examples to examine the accuracy of our pricing for-mulas. In the last subsection, we examine the sensitivityof CMSs to changes in yield curves, volatilities, and timesto maturity.

R w S T Tj

qi Q

n i i T

Ti= , |=

, −∑1

0 1

0

τ

τ E [ ( ) ]F

E S T TQn i i t

Ti

[ ( ) ], | F

CMS t P t T S T T Ri

q

iQ

n i i t

Ti

31

( ) ( )( [ ( ) ]= , , | −=∑δ

τ

E F ))



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Calibration Procedure

The most important advantages of the LMM are itstractability and feasibility. The cap and floor pricing for-mulas within the LMM framework are Black’s formulas,and thus the model volatilities can be extracted directlyfrom quoted implied volatilities of caps. Since the stan-dard pricing formula of caplets involves only a single for-ward LIBOR rate, the correlation matrix of the forwardLIBOR rates cannot be extracted from quoted cap prices.There are two main methods widely used in the market-place to calibrate the correlation matrix between LIBORrates. The first method is based on the price quotationsof swaptions, and the second is presented by Rebonato[1999], who applied a historical correlation matrix toengage in calibration.9 We adopt the Rebonato methodto engage in a simultaneous calibration of the LMM tothe volatilities and correlation matrix of the forwardLIBOR rates. This method is also adopted by Wu andChen [2007a, 2007b] to price equity swaps.

We briefly introduce the method as follows. Assumethere are n forward LIBOR rates in the m-factor frame-work. Assume that each forward LIBOR rate, L(·, Ti ), hasa constant instantaneous volatility, namely, for i = 1, …, n,γ (·, Ti ) = νi. The setting is presented in Exhibit 2.10 Thus,if the market-quoted volatility for the T1-year cap is ξ1,then ν1 = ξ1. Next, for i = 2, …, n, if the Ti-year cap isξi, then .

In addition, we use the historical data of the forwardLIBOR rates to derive a market correlation matrix Φ.Φ is an n-rank, positive-definite, and symmetric matrix thatcan be written as

Φ = HΓH

where H is a real orthogonal matrix and Γ is a diagonalmatrix. Let A ≡ HΓ1/2, and thus Φ = AA′. In this way, we

v T Ti i i i i= − − −ξ ξ2 21

21

2

can find an m-rank (m ≤ n) matrix B so that ΦB = BB′ isan approximate correlation matrix for Φ.

The advantage of finding B is that we may replacethe n-dimensional original Brownian motion dZ(t) withBdW(t) where dW(t) is an m-dimensional Brownian motionsuch that an approximate correlation structure is given by

BdW(t) (BdW(t))′ = BdW(t)dW(t)′B′ = BB′dt = ΦB dt

We must still find a suitable matrix B. Rebonato[1999] proposed a method to find B, which is describedas follows. Assume that the ikth element of B, fori = 1, 2, …, n, is specified as

Thus, ΦB = BB′ is a function of Θ = {θi, k}i=1,…,n;

k=1, …, m–1. An optimal solution is obtained by solvingthe following optimization problem:

(25)

where Φi,j is the ijth element of Φj, and is the ijth element of ΦB, specifically defined as follows:

By substituting into B, we obtain an optimalmatrix such that is an approximate corre-lation matrix for Φ.

We use to distribute the instantaneous totalvolatility, νi, to each Brownian motion without changingthe amount of the instantaneous total volatility.11 That is,

B̂

B̂B BBˆ ( ˆ ˆ )Φ = ′Θ̂

Φi jB

i k i kk

m

b b, , ,==∑

1

Φi j,B

minΘ

Φ Φi j

n

i jB

i j, =

, ,∑ | − |1

2

Θ̂

bk m

i k

i k jk

i j

j,

, =−

,== , , ..., −cos sin ifθ θΠ

Π11 1 2 1

==−

, =

⎧⎨⎪

⎩⎪ 11k

i j k msin ifθ


E X H I B I T 2Instantaneous Volatilities of L(t, ·)


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where Tk–1 ≤ t < Tk, k = 1, …, i and i = 1, 2, …, n.The calibration procedure leaves the choice of the

number of random shocks, m, for end-users. The numberof random shocks, may depend on the range of the termstructure of interest rates involved in financial products.12

For example, we may use a three-factor model (i.e., m = 3)to capture the shift and twist of the entire yield curve.The first two random shocks can be interpreted as theshort-term and long-term factors respectively, causing ashift in different maturity ranges on the yield curve. Thecorrelation between the short-term and long-term interestrates is specified by the third random shock. The numer-ical examples in the following section are based on thisthree-factor model.

Numerical Studies

In this subsection, we provide some practical exam-ples that use Monte Carlo simulation as a benchmark toexamine the accuracy of the CMS pricing formulasderived in the preceding section. Since CMS-for-LIBORand CMS-for-fixed CMSs are special cases of two-wayCMSs, we provide only the examination of Theorem 1for the sake of parsimony.

In the numerical examples, we consider two-wayCMSs receiving 10-year CMS rates and paying 2-yearCMS rates with times to maturity of 1, 3, 5, and 7 years.The CMS rates are reset semiannually, and the premium(discount) rate K1 is set to zero. To examine the accuracyand robustness of the derived pricing formulas for dif-ferent market scenarios, the two-way CMSs are priced onthe semiannual dates chosen for the three years 2005–2008,namely, 2005 March 1; 2005 September 1; 2006 March 1;2006 September 1; 2007 March 1; and 2007 September 3.The market data on these dates are omitted for the sakeof brevity but can be obtained upon request from theauthors. Based on the market data observed on these dates,the two-way CMSs are priced, and the pricing results arethen compared with a Monte Carlo simulation. Thenotional principal is assumed to be $1, and the simulationis based on 10,000 paths.

In addition, we also provide a comparison of ourapproximation formula with the convexity adjustment for-mulas introduced by Brigo and Mercurio [2006]. Brigo and

v b b … bt T

i i k i k i k m( ˆ ˆ ˆ )

( (− + , − + , − + ,, , ,

= ,1 1 1 2 1

1γ ii i m it T … t T) ( ) ( )), , , , ,γ γ2

Mercurio introduced two convexity adjustment formulas intheir Equations (13.15) and (13.16) that are widely used inthe marketplace. We find that one of the two formulas alwaysslightly overestimates, and the other slightly underestimates.Therefore, we use an average of the two, which is morestable, as a proxy of the convexity adjustment approach.

The pricing results are listed in Exhibit 3, which showsthat our pricing formulas are sufficiently accurate and robustin all market scenarios examined as compared with theMonte Carlo simulation. Without closed-form pricing for-mulas for CMSs, market practitioners would resort to time-consuming Monte Carlo simulation. In contrast, ourapproximate pricing formulas provide sufficiently accurateprices with significant time savings, which is an importantadvantage, especially in a competitive financial market.Moreover, our pricing formulas yield values very close tothose computed from the convexity adjustment approach,but in comparison with Monte Carlo simulations ourpricing errors are less than in the convexity adjustmentapproach, particularly for long-term CMSs.

Our pricing approach has another significant meritin that it can derive the approximate distribution of swaprates, the convexity adjustment approach only providesthe first moment of swap rates. Therefore, our approachcan also be used to price options written on swap rates.With our approximation approach, both LIBOR-typeand swap-type interest rate derivatives can be priced con-sistently within the LMM framework. Because of the ver-satility of the LMM framework, the risk of interest ratederivatives can be managed efficiently and consistently.

Property of CMSs

We have previously mentioned that two-way andCMS-for-LIBOR CMSs are sensitive to change in theslope of the interest rate yield curve but are almost immu-nized to any parallel shift in the curve. CMS-for-fixedCMSs, however, are sensitive to both a change in the slopeand parallel shift in the interest rate yield curve. In this sub-section, we provide some market scenarios to examinethese conclusions.

Consider the three scenarios of initial forwardLIBOR curves in Exhibit 4: curves 1 and 2 are flat curves,namely, 2% and 8%, and curve 3 is spread uniformlybetween 2% and 8%. The difference between curves 2 and1 stands for a parallel shift in the curve; the differencebetween curves 3 and 1 stands for a change in the slopeof the yield curve. In addition, we consider three flat



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volatility levels, namely, 10%, 20%, and 30%, and CMStimes to maturity from 1 year to 10 years. The CMSs con-sidered in this subsection are the two-way CMS receiving10-year CMS rates and paying 2-year CMS rates, the CMS-for-LIBOR CMS receiving 10-year CMS rates, and paying6-month LIBOR rates, and the CMS-for-fixed CMSreceiving 10-year CMS rates and paying a fixed rate.13

For each volatility level and time to maturity, wefirst compute the prices of two-way CMSs via the pricingformulas in Theorem 1 by using the initial forwardLIBOR curves 1, 2, and 3, respectively, and then computethe price differences of the two-way CMSs calculated byusing curves 2 and 1 and the price differences by using

curves 3 and 1. Similarly, we compute the price differ-ences of CMS-for-LIBOR and CMS-for-fixed CMSs.

Exhibit 5 shows the results. The first row in Exhibit 5presents price differences using curves 2 and 1, and thesecond row presents price differences using curves 3 and 1.The first column in Exhibit 5 presents price differencesusing flat volatility 10 and the other columns are 20 and30, respectively. Each figure contains three types of CMSprice differences.

As anticipated, Exhibit 5 indicates that two-way andCMS-for-LIBOR CMSs are sensitive to change in theslope of the interest rate yield curve but are almost immu-nized from any parallel shift in the curve. CMS-for-fixed


Note: The 1-, 3-, 5-, and 7-year two-way CMSs receiving 10-year CMS rates and paying 2-year CMS rates are priced based on six market scenarios that aresemiannually chosen over the three-year period 2005–2008. The market data are available upon request from the authors. The CMS rates are reset semiannu-ally. The notional principal and the premium rate K1 are assumed, respectively, to be $1 and 0. The simulation is based on 10,000 paths. MC stands for theresults of the Monte Carlo simulation presented in basis points, and the standard error of each simulation is about one basis point. THM1 and CA stand for theprices in basis points computed from Theorem 1 and the convexity adjustment, respectively. RE1 and RE2 stand for the percentage difference between THM1and MC, and CA and MC, respectively. For example, RE1 = |(THM1 – MC)/MC|, and RE2 is defined accordingly. The computation run times for theMonte Carlo simulations of the 1-, 3-, 5-, and 7-year two-way CMSs are about 18, 131, 375, and 771 seconds, respectively.

E X H I B I T 3The Numerical Examples of Two-Way CMSs in (bps)


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CMSs, however, are sensitive to both change in the slopeand any parallel shift in the interest rate yield curve. Theseeffects become more profound as the time to maturityand the volatility level increase.

CONCLUSION

We provide a technique to approximate the distrib-ution of forward swap rates under the LMM and use it toprice three types of CMS contracts, namely, CMS-for-CMS, CMS-for-LIBOR, and CMS-for-fixed. The resultingpricing formulas are shown to be sufficiently accurate bycomparison with Monte Carlo simulation. We also examinesome properties of CMSs with respect to initial LIBORzero curves, volatility levels, and times to maturity. PricingCMSs via our pricing formulas is time saving and accurateand provides a good alternative to time- consuming MonteCarlo simulation. Therefore, the pricing models are worthrecommending to market practitioners.

A P P E N D I X

The Proof of Theorem 1

Based on the risk-neutral valuation, the time-t price ofthe two-way CMS can be computed under forward martingalemeasure as follows:QTi

(A-1)

Since the derivation processes of the two expectations inEquation (A-1) are the same, we compute only the first expec-tation; the second can be computed similarly.

According to Equation (9), the first expectation can bederived as follows:

(A-2)

where is frozen to its initial value,According to Proposition 2.3, under forward martingale

measure , the dynamics of L(·, Ti+j) are given as follows:

and via Itô’s lemma, we have

(A-3)

where

Inserting Equation (A-3) into Equation (A-2), we have

where

ζ( ) ( )t T T exp u T T dui j i t

T

t i j i

i, ; = Δ , ;( )+ +∫

EQn i i t

j

q

ni j

i j

Tin

S T T F w t L t T[ ( ) ] ( ) (, | = ,=

−,

+∑0

1

)) ( )ζ t T Ti j i, ;+

11

CMS T E

E

( ) ( )( [ ( ) ]t P t S T Ti

q

iQ

n i i t

Ti= , , |

−=∑δ

τ

F

QQm i i t

Ti

S T T K[ ( ) ] ), | −F 1

Δ , ; = , ⋅ , −+ + + +t i j i i j Pt

i j Ptu T T u T u T( ) ( ) ( ( ) (γ σ σ1 uu Ti, ))

L T T L t T exp u T T dui i j i j t

T

t i j i

i

( ) ( ) ( ), = , Δ , ;(+ + +∫−− ,

+ , ⋅

∫

∫

+

+

12

2

t

T

i j

t

T

i j

i

i

u T du

u T dW u

� �γ

γ

( )

( ) ( )))

E EQn i i t

j

q

ni j QTi

n Ti

S T T w t L T[ ( ) ] ( ) [ (, | ==

−,∑F

0

1

ii i j tT, |+ ) ]F

dL u T

L u Tu T u Ti j

i ji j P

ti j

( )

( )( ) ( (

,,

= , ⋅ ,+

++ + +γ σ 1 )) ( ))

( ) ( )

− ,

+ , ⋅+

Pt

i

i j

u T dt

u T dW u

σ

γ

QTi

wni j, ⋅( ) w tn

i j, ( ).


E X H I B I T 4Three Scenarios of LIBOR Zero Curves

Note: Curves 1 and 2 are flat curves, namely, 2% and 8%; curve 3 isuniformly spread between 2% and 8%.


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ENDNOTES

1As examined by Rogers [1996], the Gaussian term struc-ture model has an important theoretical limitation: the rate canattain negative values with positive probability, which may causepricing errors in many cases.

2The LMM has been adopted to price many exotic interestrate options, (e.g., Wu and Chen [2008, 2009a, 2009b], and Benner,Zyapkov, and Jortzik [2009]).

3The filtration {Ft}t∈[0,T] is right continuous, and F0 containsall the Q-null sets of F.

4For ease of computation in Equation (3), δ may be fixed (forexample, δ = 0.5).

5We employ W(t) to denote an independent m-dimensionalstandard Brownian motion under an arbitrary measure withoutcausing any confusion.

6Brace and Womersley [2000] also showed that is amartingale under some probability measure. Therefore, it is notunreasonable to approximate with its initial value

7As indicated by Brigo and Mercurio [2006], forward swaprates obtained from lognormal forward LIBOR rates are not farfrom being lognormal under the relevant measure.

8If the LIBOR payment is paid in arrears, the term L(t, Ti)ζ(t, Ti, Ti) in Equation (21) becomes L(t, Ti–1).

9For details about the first method, please refer to Brigo andMercurio [2006].

10For other assumptions of volatility structures, please referto Brigo and Mercurio [2006].

11Note that the Euclidean norm of each row vector of Bis one.

w tni j, ( ) wn

i j, ( ).0

w tni j, ( )


E X H I B I T 5CMS Price Change Relative to Change of Yield Curve. Time to Maturity, and Volatility

Note: The first row presents price differences using curves 2 and 1 and the second row presents price differences using curves 3 and 1. The first column presents price differ-ences using a flat volatility of 10, and the second and third columns use volatilities of 20 and 30, respectively. Each figure contains three types of CMS price differences.


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12For more details regarding the performance of single- andmultifactor models, please refer to Driessen, Klaassen, and Melen-berg [2003] and Rebonato [1999].

13The value of the fixed rate has no impact on the result, andtherefore we do not specify it.

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