Swaps: Constant maturity swaps (CMS) and constant maturity. Treasury (CMT) swaps

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1 Swaps: Costat maturity swaps (CMS) ad costat maturity reasury (CM) swaps A Costat Maturity Swap (CMS) swap is a swap where oe of the legs pays (respectively receives) a swap rate of a fixed maturity, while the other leg receives (respectively pays) fixed (most commo) or floatig. A CM swap is very similar to a CMS swap, with the exceptio that oe pays the par yield of a reasury bod, ote or bill istead of the swap rate. More geerally, oe calls Costat Maturity Swap ad Costat Maturity reasury derivatives, derivatives that refer to a swap rate of a give maturity or a pay yield of a bod, ote or bill with a costat maturity. Sice most likely, treasury issued o the market will ot exactly match the maturity of the referece rate, oe eeds to iterpolate market yield. (rates published by the British Baker Associatio i Europe ad by the Federal Reserve Bak of New York) MARKEING OF HESE PRODUCS CM ad CMS swaps provide a flexible ad market efficiet access to log dated iterest rates. O the liability side, CMS ad CM swaps offer the ability to hedge log-dated positios. Great cliets have bee life isurers as they are heavily idebted i log dated paymet obligatios. Geerous isurace policies eed to be hedged agaist the sharp rise of the back ed of the iterest rate curve. ypical trade is a swap where they received the swap rate. O the asset side, corporate ad other fiacial istitutios have heavily

2 ivested i CMS market to ejoy yield ehacemet ad diversified fudig. I a very steep curve eviromet, swaps payig CMS look very attractive to cliets that thik that the swap rates would ot go as high as the market (ad the forward curve) is pricig. Alteratively, i a flat yield curve eviromet, swaps receivig CMS look very attractive to market participats thikig that swap rates would rise i the futures as a cosequece of the steepeig of the curve. I a swap where oe pays Libor plus a spread versus receivig CMS 0 year, the structure is maily sesitive to the slope of the iterest rate yield curve ad is almost immuized agaist ay parallel shift of the iterest rate yield curve. For all these reasos, it is ot surprisig that the CMS markets ad the CMS optios markets ow trade i large quatities, both iterbak ad betwee corporates ad fiacial istitutios. Pricig Because of the icreasig size of the CMS market, the market has see its margi erodig. Baks have developed more ad more advaced models to accout for the smile, resultig i first a more proouced smile ad also a icreasigly spread betwee CMS swap ad their swaptio hedge. here exist two differet methodologies for pricig CMS swaps: Parametric computatio of the CMS covexity correctio (See Hull(200), Behamou (999) ad (2000)). I this approach, oe assumes a model ad uses some (smart) approximatio methods to compute the expected

3 swap rate uder the forward measure. No parametric computatio of the swap rates. his approach assumes No parametric computatio of the CMS rates. his approach tries to miimize the amout of hypothesis betwee the computatio of the CMS rate (see the works of Amblard, Lebuchoux (2000), Pugachevsky (200)). Note also that practitioers focus heavily o the computatio of the forward CMS as they use these modified forwards ad the volatility read from swaptio market to compute simple optios o CMS (CMS cap ad floor, CMS swaptio). his practice is justified by the fact that the first order effect comes maily from the covexity corrected forwards as opposed to modified volatility assumptios. Usig the same vol is therefore right at first order approximatio, ad strictly right i a Black Scholes settig. Let use derive shortly the sketch lies of the two methods metioed above. First, oe ca rapidly see that pricig a CMS swap boils dow to price a simple swap rate received at time. his ca be doe uder the forward measure forward eutral measure, leadig to compute: [ Sw( )] E,,...,, (.) where E [] is the expectatio uder the forward eutral measure, ad Sw (,..., ) dates, the value at time of the swap rate with fixed paymet,...,.

4 We ca the use stadard chage of umeraire techique to chage the expressio above. he atural umeraire for the swap rate is the auity (also called level or dvo, defied as the pv of oe basis poits paid over the life of the forward swap rate) of the swap rate, deoted by LVL ( ). his leads to: E (, ) ( ) ( 0) ( 0, ) LVL B LVL [ ( )] Sw,,..., E * * Sw LVL B,..., (, ) = (.2) sice d d LVL (, ) ( ) B = LVL ( 0) ( 0, ) LVL * B. (.3) his shows that the CMS rate is equal to the swap rate plus a extra term fuctio of the covariace uder the auity measure betwee the forward swap rate ad the forward auity: E [ Sw(,,..., )] ( 0) B(, ) ( ) B( 0, ) LVL LVL = Sw( 0,,..., ) + Cov, Sw(,,..., ) LVL (.4) As a result, the CMS rate depeds o the followig three compoets: he yield curve via the swap rate ad the auity. he volatility of the forward auity ad the forward swap rate. he correlatio betwee the forward auity ad the forward swap rate. he first method relies o derivig a approximatio for the covariace terms. here are may ways of doig this, i particular, usig oe factor approximatio with logormal assumptios, Wieer chaos expasio or simply martigale theory. o be more specific, let us examie the logormal case. It assumes a logormal martigale diffusio for the swap rate uder the auity measure:

5 ds S (,,..., ) (,,..., ) = σ dw (.5) he oe factor approximatio relies o assumig that the level ca be represeted as a fuctio of the swap rate (which is rigorously true for cash settled swaptios). his leads to Oe ca show that the adjustmet is give by: ( ) f ( S( )) t t LVL =,..., (.6), Cov Sw LVL LVL( 0) B(, ) LVL( ) B( 0, ) LVL ( ) ( 0) 0,,..., B( 0, ), Sw exp (,,..., ) f f '( Sw( 0,,..., )) ( Sw( 0,,..., )) 2 σ Sw ( 0,,..., ) (.7) he secod approach relies o the fact that i the oe factor approximatio; the computatio boils dow to computig: E LVL f Sw(,,..., ) ( ( )) Sw 0,,..., (.8) But we kow that ay fuctio of oly the swap rate ca be evaluated as a portfolio of swaptios. his comes from the fact that a expectatio ca be traslated ito a itegral of the itegrad times the desity fuctio of the swap rate. We ca therefore evaluate the CMS swap rate as a portfolio of swaptios. As a matter of fact, replicatig CMS with cash settled swaptios is accurate, while oe eeds to make a oe factor approximatio to exted the replicatio argumet to physical settled swaptios. Usig regressio ideas, oe ca also exted the ideas of CMS replicatio to deferred paymet CMS structures. see Breede Litzeberger (979) result o the fact that the secod order derivatives of a call price with respect to the strike is simply the desity fuctio, hece the result

6 Eric Behamou 2 Swaps Strategy, Lodo, FICC, Goldma Sachs Iteratioal Etry category: swaps Scope: Ratioale for CMS swaps, Pricig, Covexity adjustmet Related articles: Swaps: complex structures; Swaps: taxoomy 2 he views ad opiios expressed herei are the oes of the author s ad do ot ecessarily reflect those of Goldma Sachs

7 Refereces Amblard G. Lebuchoux J (2000), Model For CMS Swaps, Risk, September. Behamou E. (999), A Martigale Result for the Covexity Adjustmet i the Black Pricig Model, Lodo School of Ecoomics, Workig Paper. Behamou E. (2000), Pricig Covexity Adjustmet with Wieer Chaos, Fiacial Markets Group, Lodo School of Ecoomics, FMG Discussio Paper DP 35. Hull, Joh C, Optios, Futures, ad Other Derivatives, Fourth Editio, Pretice-Hall, Pugachevsky, D. (200), Adjustmets for Forward CMS Rates, Risk Magazie, December.

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