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Constant Maturity Products Forward CMS Rate Adjustment

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Constant Maturity Products Forward CMS Rate Adjustment Risk Technical article Risk is the world's leading financial risk management magazine. Risk’s Cutting Edge articles are a showcase for the latest thinking and research into derivatives tools and techniques, with comprehensive coverage of the most recent advances in areas such as option pricing and hedging, market risk, credit risk, swaps and Monte Carlo methods. www.risk.net/technical 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 swap 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 swaptions. 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) yield curve, 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 derivatives market 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 swaption 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.
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