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Copyright statement © Risk Waters Group Ltd, 2003. All rights This PDF is made available for personal use reserved. No parts of this article may be only. Under the following terms you are reproduced, stored in or introduced into any granted permission to view online, print out, retrieval system, or transmitted, in any form or store on your computer a single copy, or by any means, electronic, mechanical, provided this is intended for the benefit of photocopying, recording or otherwise the individual user. Should you wish to use without the prior written permission of the multiple copies of this article, please contact copyright owners. [email protected] Constant maturity products Forward CMS rate adjustment

The growing importance of constant maturity products as risk management tools has sparked a debate about pricing, and rumours about trading losses. Now, Dmitry Pugachevsky demonstrates that current models do not correctly adjust for convexity in forward constant maturity swap rates, and provides an improved adjustment formula

constant maturity swap (CMS) rate is a swap rate that E(P(T , T ) × D ) = E(D ) = P(0, T ) (2) will be observed at a swap settlement time in the fu- i j i j j ture. Seen from today, a CMS rate is a random variable Throughout the article, we consider a forward-starting plain vanilla swap whose ultimate value depends on the evolution of in- with settlement at T0 and n subsequent payments at dates T1, ..., Tn when terest rates between today and the date when it is re- we pay Libor and receive fixed rate C. For the sake of simplicity, assume alised. In contrast, forward swap rates (FSRs) and that the notional amount of the swap is one (unit) and that the payment A“forward” CMS rates are rates that are known today. FSRs are used in pric- frequencies and bases of the fixed- and floating-rate sides of the contract ing forward starting swaps and European-style . Forward CMS and of Libor are the same. rates are used in pricing CMS swaps and CMS caps and floors. While for- If we also assume that we pay and receive notional at maturity, then ward swap rates are easily calculated from the (swap) , forward the value today of the floating-rate side of the swap is equal to par at T0, CMS rates can be found by adjusting forward swap rates. and today’s value of the swap Sw(0, C) is: In the early to mid-1990s, when fixed markets first encountered CMS n =× ∆× + − swaps/caps, much literature was published both in journals and within Sw0,C() C∑ jj P0,T( ) P0,T() n P0,T () 0 banks (eg, Jamshidian, 1997, Pugachevsky, 1996, and Schmidt, 1996), show- j1= ∆ ing that the FSR is the expectation of the CMS rate under a (forward) swap where j is the day count for time interval [Tj – 1,Tj]. measure, while CMS swaps and caps require calculation of CMS under a We now want to answer the following question: what fixed rate C makes different, forward measure. But, until recently, most of the market partic- the value of the swap Sw(0, C) equal to zero? Because the swap is forward ipants settled with the “yield convexity adjustment” formula, which uses starting, this rate is called a forward (par) swap rate, let’s denote it as FSR. just the yield curve and CMS variance. However, increasing competition in It follows that: the CMS has made the inaccuracies of the convention- P0,T()− P0,T () FSR = n0 al yield formula more apparent. Thus, new attempts are made to find a n (3) better approximation for forward CMS rates, not only for forward-to-set- ∑ ∆×jjP0,T() tlement, but for the whole range (eg, Benhamou, 2000, and Hull, 2000). j1= This article derives the new formulas for adjustments for forward CMS When settlement time T0 = 0 (ie, today), the swap rate in (3) is a “spot rates, and compares them with the conventional formula and with exact val- swap rate”. As a practical matter, while spot rates for swaps of various ma- ues of forward CMS rates calculated numerically. The main result is given turities are quoted in the market and are used for constructing the swap below (“New formulas”), where we use the CMS identity and some extra as- yield curve, forward swap rates are not quoted, but they can be calculat- sumptions to derive adjustments for forward CMS rates. The resulting for- ed from yield curve using equation (3). mulas depend only on the yield curve and variance, which makes Let’s consider now the value of swap Sw(T0,C) at settlement time T0: them convenient for fixed-income traders to use. This new approach to CMS n =× ∆× + − rates is more accurate than the conventional yield adjustment formula. Sw(T,C)C0j0j0n∑ PT,T( ) PT,T() 1 (4) j1=

CMS and forward swap rates We can define the corresponding CMS rate as the time T0 swap rate. In Consider a sequence of dates 0 < T0 < T1 < T2 < ..., where time 0 is other words, the CMS rate is a spot swap rate for a forward-starting swap. today. Denote the risk-neutral stochastic discount from forward date Tj to Then, condition Sw(T0,CMs) = 0 and equation (4) imply: today as D . Denote today’s risk-neutral expectation as E, and expectation j − conditioned on information up to future time T as E . Then today’s price 1PT,T()0n i i CMS = of a zero-coupon bond maturing at time T is P(0, T ) = E(D ), and the price n j j j ∆× (5) at future time T (for i < j) of the same bond is: ∑ j0jPT,T() i j1= EDij() PT,T()ij= Then equation (4) can be rewritten as: Di n SwT,C()(=− C CMS ) ×∆×∑ P( T,T) which implies P(Ti, Ti) = 1. 0j0j(6) Throughout the article, we will also use the following probability iden- j1= tities. If X is Fi-measurable random variable (ie, its value is known at time which is the convenient representation of swap value. Ti), then: Bear in mind that CMS is a time T0 spot swap rate. Thus, it will be known E(X × P(Ti, Tj) × Di) = E(X × Dj) (1) only at T0, and as of today it is a random variable, and it cannot be calcu- which implies for X = 1: lated from today’s yield curve like spot and forward swap rates.

125 • RISK • MARCH 2001 Constant maturity products

Note, that the FSR and the CMS rate considered throughout this article CMSSw0i0,ii=−()() C FCMS ×∆× P 0,T always correspond to a same forward-starting swap, which settles at time T0 and matures at time Tn. This allows us to simplify notations. Note here that it can be shown that the forward CMS rate FCMSi is an Equation (5) and identities (1) and (2) imply: at-the-money strike for the CMS caplet that settles at time T0 and pays at n time Ti, i > 0. E CMS×∆×=∑ jj D P()() 0, T 0 − P 0, T n Let us introduce a set of forward measures, each of them related to dis- j1= counting_ from some time Tj, j = 0, ..., n. The expectation under this mea- Now, using equation (3), we get the following relationship between the sure E(j) can be defined as: FSR and the CMS rate: EX()× D (j) = j nn E(X): (12) ×∆×= ×∆× P0,T( j) ECMS∑∑jj D FSR j P0,T( j) (7)  j1== j1 where X is any random variable. Let’s introduce a new probabilistic measure that is related to the for- Comparing with equation (11), one can see that: ~ E ward-starting swap described above. Expectation under this measure FCMS= E(j) () CMS (13) can be expressed through the risk-neutral expectation E as: j

n ie, “time Tj forward CMS rate” is an expectation of CMS rate under Tj for- EX×∆×∑ jj D ward measure.  j1= It also follows from (7) that: E(X) = n (8) nn ∆× ∑ jjP0,T() ∑∑FCMSjj×∆ × P( 0,T j) = FSR × ∆ j × P( 0,T j) (14) j1= j1== j 1 where X is any random variable. thus: n Then equation (7) implies: ∑ ×∆ × ( )  FCMSjj P 0,T j FSR= E(CMS) (9) = j1= (15) FSR n so the forward swap rate is the expectation of a CMS rate under a corre- ∑ ∆×jjP0,T() sponding forward swap measure. j1= Let’s note here that it can be shown (eg, Jamshidian, 1996, Pugachevsky, ie, the FSR is the weighted average of forward CMS rates where weights 1996) that the value of receiver (payer) European-style swaption equals are discounts multiplied by daycount. the value of the put (call) option on the CMS rate under the swap mea- It can be shown (Schmidt, 1996) that the difference between two con- sure multiplied by the DV01 term (the term in the denominator in (8)). secutive forward CMS rates can be expressed through the covariance under This also implies that the at-the-money strike is an expectation of the CMS a corresponding forward measure as: rate under the swap measure, ie, the FSR. This fact is well-known to swap- P0,T()+ ×∆ −=−(j) ×j1 j tions traders. FCMSj1+ FCMS j Cov() CMS,R j (16) P0,T( j )

“Forward” CMS rates and measure where Rj denotes the rate for time interval [Tj, Tj+1] evaluated at time T0, ie: We now introduce CMS swaps and show that the values of these financial  instruments depend on the new types of expectation of the CMS rate, which 1 PT,T( 0j) R1=× − we dub “forward” CMS rates. We introduce the “forward” measure, and j ∆  j1+ PT,T()0j1+ show that forward CMS rates are expectations of the CMS under the cor- responding forward measure. Then we derive the relative inequality among Since the CMS and forward rate Rj are both interest rates settled at T0, forward swap rate and forward CMS rates, and consider the special case they are positively correlated, thus equation (16) implies that FCMSj > of one-period CMS. FCMSj+1. Combined with (15), we get that for n > 2 (the case where the Consider the case of a CMS swap. This is a swap where at every pay- number of payments n = 1 will be considered below), the value of FSR is ment date a payment calculated from a CMS rate (usually determined on located strictly between the values of last (forward-to-maturity) and first the preceding payment date) is exchanged for a fixed rate. Note that the (forward-to-settlement) CMS rates: term and payment frequency of the swap used to identify the CMS rate FCMS > FSR > FCMS (17) may be different from the term and payment frequency of the CMS swap 0 n itself. Thus, we can contemplate a five-year annual swap on a seven-year Consider the special case where n = 1, ie, where the underlying swap semi-annual CMS rate. has just one payment date T1 at which a Libor rate is exchanged with a fixed Consider one period of such a CMS swap that will start at time T0 and rate. Then the CMS rate is equivalent to Libor and to rate R0 in (16). It fol- terminate at time Ti, i > 0. Assume we receive at time Ti the fixed coupon lows that FSR is equivalent to FRA rate FRA0 corresponding to [T0, T1], ie: C accrued over the interval from T0 to Ti, and pay the CMS rate observed P0,T() at time T0 and accrued over the same interval. Thus today’s value of a one- 1 0 FSR==× FRA0  − 1 period CMS swap is: ∆11P0,T() CMSSw00,ii=− E(() C CMS ×∆× D ) (10) Moreover, the FRA is the forward-to-maturity CMS, and the expression ∆ where 0,i is the day count for the time interval [T0, Ti]. for the forward-to-settlement CMS (also called Libor in arrears) rate LA fol- Though forward contracts on CMS rates are not traded, the payout in lows from (16): (10) looks the same as the (FRA) that exchanges ∆×P0,T() LA=+ FRA Cov(1) () CMS,R ×11 at time Ti CMS rate (settled at T0) against fixed rate. Let’s call the par fixed 00 P0,T()0 rate in (10) “time Ti forward CMS rate” and denote it FCMSi. Then it fol- lows that: Replacing these CMS and R0 rates with Libor, L, we get an expression ECMSD()× i in terms of the FRA only: FCMSi = (11) (1) P0,T() Var() L ×∆1 i LA= FRA + (18) 0 ∆× and the value of the one-period CMS swap is: 1+10 FRA

126 • RISK • MARCH 2001 _ (1) Note that Var (L) is a variance of Libor under time T1 forward mea- proximation for forward-at-settlement CMS: sure, ie, it is a “caplet” variance. Con() FSR,FSR  FCMS≈− FSR × Var() CMS (25) Yield convexity adjustment formula 0 2× Dur() FSR,FSR We will show that a conventional yield convexity adjustment formula can As prices of European-style swaptions are equivalent to prices of be used as an approximation for a forward-to-settlement CMS. calls/puts under swap measure, the variance under swap measure ~ Let us write down the expression for time T0 price PB0 of the bond with Var(CMS) is the European-style swaption variance and can be obtained coupon FSR, settlement date T0 and payment dates T1, ..., Tn: from market swaption prices. The formula for the forward-to-settlement n CMS rate does not require anything apart from today’s yield curve and =×∆×+ PB00j0n FSR∑ P( T,T) P() T,T swaption prices. This makes it very convenient, and explains its popular- j1= ity among swaption traders. where ∆ =1/frequency. But there are some problems with conventional adjustment (25). First, Multiplying both parts by discount D0 from T0 to today, and taking being just a second-order approximation, this formula fails to capture the today’s expectation, we get from (3) (neglecting difference between day- “skew” of the CMS rate, which basically means that it gives the same ad- counts): justments independently of the distribution of the CMS (under forward n measure) as long as their swaption variances are the same. The second ()×= ×∆×+ EPB00 D E FSR∑ D jn D problem is that it basically replaces second moment under T0 forward mea- j1= sure in (24) by the second moment under swap measure. For swaps with n long maturities this difference can be significant. And the third, and maybe =×∆×+ = FSR∑ P() 0, Tjn0 P()() 0, T P 0, T the biggest, is that it gives adjustment only for forward-to-settlement CMS j1= rate, while CMS swaps and caps usually require forward-to-first-payment then the definition of the forward measure (12) implies: CMS rate. Note that there are some attempts (Hull, 2000) to build such ap- EPB1(0) ()= (19) proximations, but they require correlations between the CMS and Libor 0 that are not quoted. ie, the “forward” price of this bond is par. Next, we will demonstrate the new formulas that, as in equation (25), Let us now introduce yield-to-price function YTP(.), which gives a price use only the swaption variance and the yield curve, but produce very ac- for the bond for known coupon and yield (assume same maturity and fre- curate approximations not only for forward-at-settlement but for the whole quency as in bond above). For example, for coupon x and yield y, it has range of forward CMS rates. the form: n ∆ 1 New formulas YTP() x,y=× x ∑ + in(20) We can now use the new approach to derive adjustments for forward swap i1= ()1y+×∆() 1y +×∆ rates. The resulting formulas depend only on the yield curve and swap- This implies the well-known yield identity: tion variance, which makes them very convenient for fixed-income traders. YTP(x, x) = 1 (21) We show that in a linear case, the resulting adjustment for the forward-to- settlement CMS rate is close to the yield adjustment formula. Let’s denote as Y0 time T0 yield for a bond that pays coupon FSR. Thus Using the definition of CMS (5) and identity (2), we can write down: the bond’s price PB can be expressed as: PB = YTP(FSR, Y ). Then equa- 0 0 0 n tion (19) implies: k ECMS()×∆×∑ j0j0 PT,T( ) × D (0) = j1= EYTPFSR,Y1()0 (22) − =×−×ECMS(()k1 () 1PT,T () D) Expanding YTP(FSR,Y0) into Taylor series with respect to second variable 0n 0 around the FSR and keeping only terms up to the second order, we obtain: k1− =×−ECMS()() ( D0n D ) YTP()()() FSR, Y0 ≈+ YTP FSR,FSR Dur FSR,FSR Con() FSR,FSR Then, definitions of swap measure (8) and forward measure (12) togeth- ×− + ×− 2 ()YFSR00() YFSR er imply: 2 n ECMS ()k ×∆×∑ P0,T() where duration and convexity terms are the first and second derivatives of ( ) jj j1= YTP function with respect of the second argument, ie: =×−×k1−− k1 (26) ∂YTP() FSR,FSR E00nn(() CMS) P() 0, T E(() CMS) P() 0, T Dur() FSR,FSR = (23) ∂ y For k = 1, we get equation (14), and for k = 2 it gives: ∂2YTP() FSR,FSR  n Con() FSR,FSR = ECMS()2 ×∆×∑ P0,T( ) 2 jj ∂y j1=

Taking expectation of both parts of expansion and using (21) and (22), =×−×E00nn()() CMS P 0, T E()() CMS P 0, T gives: Con() FSR,FSR which can be rewritten using definition of variance as: (0)()≈− × (0) () − 2 (24) EYFSR E YFSR n 2× Dur() FSR,FSR  2 ()Var() CMS+×∆× FSR∑ jj P( 0,T ) which is the well-known convexity adjustment formula for “par forward” j1= (27) yield. =×FCMS00 P() 0,T −× FCMS nn P () 0,T In general, PB0 is not par (only of the bond paying FSR is par), thus Y0 is not exactly time T0 par yield (which equals the CMS rate). We will use this identity together with equation (14) for calculating for- But assuming that rates don’t change much between today and T0, we can ward CMS rates. Unfortunately, (unless n = 1) these two equations are not ≈ approximate Y0 CMS. Therefore, replacing the second moment term in enough, so we have to make some extra assumptions. (24) with the variance under swap measure, we get the following ap- Let’s note here that since identity (26) relates moments under forward

127 • RISK • MARCH 2001 Constant maturity products

thus formulas (25) and (31) are fairly close, at least for not very steep yield curves. 1. Approximation for first forward CMS In numerical tests we will also use the linear/exponential assumption: β P(0, T ) Φ= − × j jj0()TT  (32) 0.01 P(0, T0 ) 0.00 where exponent β is some chosen constant. –0.01 Numerical results –0.02 We now describe the two types of numerical tests that we ran: 1) to show (%)

0 that equation (29) gives a very good approximation to forward CMS rates, –0.03 for both real-life data and flat and yield curves; and 2) to com- –0.04 pare the forward-to-settlement equations from (29) and (25). dFCMS We compared results of the previous section with “exact” forward CMS –0.05 rates calculated from the term structure model, in this case a one-factor –0.06 Yield approximation “blend” short-rate tree model. For calculations in (25) and (29), we used FCMS approximation swaption volatilities implied by prices obtained from the same term struc- –0.07 ture model (when this model is calibrated to market data, these are mar- 1 4 7 10 13 16 19 22 25 28 ket swaption volatilities too). Maturity (years) Let us note here that adjustment formula (29) doesn’t imply any distri- bution of the CMS rate under the swap measure, so it can be used for any type of “blend” model, whichever is better for describing swaptions volatil- measure to higher moments under swap measure (which is known from ity skew. To capture the “skew” effect, we ran two sets of tests for two swaptions pricing) this identity can also be used for deriving adjustments “extreme” cases: when underlying rates in the tree model had normal and for variance of the CMS under forward measure, ie, CMS caplet variance. lognormal distributions. Numerical tests for the normal model showed that Let’s consider the general additive assumption: both the linear assumption formula (31) and yield adjustment formula (25) worked very well, and gave adjustments almost indistinguishable from the FCMSj0j=+α⋅Φ FCMS (28) “exact” model. ie, the difference between jth and first forward CMS rates is proportional For the lognormal model, we used linear/exponential form (32) with to some known function (in general, of times, discounts and swaption vari- exponent β = 0.27, though in general it can be shown that this parame- ance): ter is roughly proportional to the value of swaption variance. For the first  test, we considered a 30-year annual CMS rate with a settlement time of ΦΦ=j0()T ,...,T n ,P() 0,T 0 ,...,P ()() 0,T n ,Var CMS , 0 0 five years, ie, n = 30, T0 = 5, T1 = 6, …, Tn = 35. We ran two tests: for Then we have only two unknowns: first forward CMS FCMS0 and co- real-life curves (from closing of September 21, 2000) and for flat curves efficient α. Substituting (28) into equations (14) and (27), we obtain the (yield curve – flat 6%; short rate volatility curve – flat 20%), which show main formula: that approximation (29) works very well (the difference is within 3 basis n points) for both cases.  ×Φ−Φ×∆× Var() CMS∑( jk) j P( 0,T j) For the second test, we fixed swap settlement time T = 5 years, and = 0 =+ j1 were changing swap maturity from one to 30 years. We compared differ- FCMSk FSRn , ×Φ×∆× +Φ× ence dFCMS between the “exact” first forward CMS rate and its approxi- FSR∑ jj P() 0, T j n P() 0, T n (29) 0 j1= mation calculated using yield formula (25) and the new forward CMS ()k= 0,1,...,n formula (29). Figure 1 shows that the new formula works better for all ma- turities (for this test we used real curves). ■ which allows us to calculate all FCMS rates easily. Let’s consider the special case of (28) where: Dmitry Pugachevsky is a director, OTC research at Deutsche Bank in New York. He would like to thank Puru Voruganti Φ=jj0TT − and Michelle Faissola for helpful comments and suggestions Comments on this article can be posted on the technical discussion forum which implies that adjustments for forward CMS are linear in time, ie: on the Risk website at http://www.risk.net

FCMS=+α×− FCMS() T T (30) j0j0 REFERENCES where α is a linearity coefficient. Then it follows from equation (29): Benhamou E, 2000  n Pricing convexity adjustment with Wiener chaos Var() CMS×−×∆×∑( Tjk T) j P( 0,T j) Working paper, London School of Economics j1= FCMS=+ FSR Hull J, 2000 k n (31) FSR×−×∆×+−×∑() T T P () 0, T()() T T P 0, T Pricing securities j0 j j n0 n Prentice Hall, fourth edition j1= Jamshidian F, 1997 It is interesting to compare this formula for k = 0 (ie, for forward-to- Libor and swap markets models and measures settlement rate) with yield adjustment formula (25). By replacing discounts Finance and Stochastics 1, pages 293–330 with yield terms, it can be shown that: Pugachevsky D, 1996 n CMS rates and adjustments for stochastic discounts FSR Bankers Trust Topics in Derivatives Analytics 1 −≈×−×∆×+−×Dur() FSR,FSR∑( Tj0 T) j P( 0,T j) ( T n T 0) P( 0,T n) P0,T()= 0 j1 Schmidt W, 1996 Pricing irregular interest cashflows ∂Dur() FSR,FSR 1 n Working paper, Deutsche Bank OTC Derivatives Research Con() FSR,FSR=≈−×∆× 2∑( T T) P( 0,T ) ∂ j0 j j x P0,T()0 j1=

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