On Valuing Constant Maturity Swap Spread Derivatives*

Journal of Mathematical Finance, 2012, 2, 189-194 http://dx.doi.org/10.4236/jmf.2012.22020 Published Online May 2012 (http://www.SciRP.org/journal/jmf)

Leonard Tchuindjo US Department of the Treasury, Washington DC, USA Email: [email protected]

Received December 4, 2011; revised February 4, 2012; accepted February 12, 2012

ABSTRACT Motivated by statistical tests on historical data that confirm the normal distribution assumption on the spreads between major constant maturity swap (CMS) indexes, we propose an easy-to-implement two-factor model for valuing CMS spread link instruments, in which each forward CMS spread rate is modeled as a Gaussian process under its relevant measure, and is related to the lognormal martingale process of a corresponding maturity forward LIBOR rate through a Brownian motion. An illustrating example is provided. Closed-form solutions for CMS spread options are derived.

Keywords: CMS Spread; Market Model; Brownian Motion; Forward Measure

1. Introduction expected an inversion of the long end of the yield curve. After the bond issuance Fannie Mae did not want to bear A Constant Maturity Swap (henceforth CMS) spread the yield curve slope non-inversion risk, and then got derivative is a financial instrument whose payoff is a into a cancellable CMS spread swap in which it paid 3- function of the spread between two swap rates of differ- month LIBOR minus a fixed spread every quarter.2 The ent maturities (e.g., the 10-year swap rate minus the 2- proceeds from the swap receiving leg were entirely year swap rate). This type of derivative, which is be- transferred to the bond holders. This Fannie Mae bond coming increasingly popular among insurance companies contains an embedded Bermudan call option on a CMS and pension funds, is traded by parties who wish to take spread bond and a multitude of embedded daily CMS advantage of, or to hedge against, future changes in the spread digital options. The hedging swap with Lehman slopes of specific parts of the yield curve. The most com- Brothers contains an embedded CMS spread swaption. mon CMS spread instruments are CMS spread notes/ The valuation of these CMS spread instruments is an bonds (steepener or flattener), CMS spread range accrual important subject of research for both practitioners and notes/bonds, and CMS spread caps and floors. There are academics. The difficulty arises from the fact that unlike other CMS spread derivatives that are not commonly a single interest rate, a CMS spread rate can allow both traded—such as CMS spread call and put options on positive and negative values, as the yield curve moves in bonds, CMS spread digital options, and CMS spread a way that any part can be either flat, upward or down- swaptions—but are embedded in other financial instru- ward sloping. This feature adds an extra complication in ments. the pricing of derivative instruments for which a CMS A concrete example is the 15-year CMS spread range spread rate is the underlying. Various attempts have been accrual bond—issued by Fannie Mae [1] on February 27, made to value financial derivatives on spread rates. Car- 2008 under the reference CUSIP 31398ANE8—semi- mona and Durrleman [2] provide an extensive literature annually callable after the first year, having a notional of review on the pricing of spread options on fixed income 100 million US dollar and a coupon of 8.45% that ac- instruments, as well as on equity, foreign exchange, com- crues every day the CMS 30-year minus the CMS 10- modities, and energy. 1 year is positive. At origination, buyers of this bond ex- In the existing literature of valuing CMS spread de- pected the long end of the yield curve to be upward slop- rivatives based on the LIBOR market model, it is coming most of the time, while the issuer—Fannie Mae— monly assumed that each rate used to calculate the spread is lognormally distributed, and there may be a nonzero * The views expressed herein are the author’s and should not be inter- correlation between them. Recent studies in this direction preted as reflecting those of the US Department of the Treasury. 1Fannie Mae previously issued a 10-year CMS spread 10 yrs - 2 yrs are those of Belomestny et al. [3] and Lutz and Kiesel [4] effective on 25th July 2007, and a 15-year CMS spread 30 yrs - 2 yrs rd effective on 23 January 2008 under the bond CUSIP 31398AEQ1 and 2The counterparty was Lehman Brothers which went bankrupt a couple 31398ALA8 respectively. of months after the deal was effective.

Copyright © 2012 SciRes. JMF 190 L. TCHUINDJO who approximate the value of CMS spread options in the joint hypothesis of both the skewness and excess kurtosis standard lognormal LIBOR market model with determi- being zero. Any deviation from the normal distribution nistic and stochastic volatilities respectively. This current increases the JB which the statistic is given by the fol- approach has the advantage to help understand the influ- lowing formula JB 2 32 4 N 6 , where  ence of various model parameters—in particular the cor- is the sample skewness,  is the sample kurtosis, and relation between the two rates used to calculate the N is the sample size. Historical CMS data have been spread. But it has a limited analytical tractability, as the obtained from BloombergTM database.3 Table 1 presents linear combination of lognormal variables has an un- the values of the statistics as results of the JB test. known distribution. Closed-form solutions for CMS spread The first row shows CMS spread rates for which the options can be obtained only in rare cases, such as the JB test has been done. The second row presents the test case of caplets and floorlets with zero strike in which statistics results with their associated p-values when we Margrabe [5] exchange option formula can be used. use 1-year weekly average of CMS spread data—from 1 Our approach is to model the CMS spread rate directly January 2007 to 31 December 2007. The third shows the with a distribution that allows for both positive and nega- test statistics results for 10-year quarterly average of CMS tive values in its range. We center the CMS spread rate spread data—from 1 January 1998 to 31 December 2007. on its forward value, and each forward CMS spread rate Overall, at 5 percent level of significance the normality is assumed to be driven by a Gaussian stochastic process assumption is accepted for each CMS spread and for both under its relevant measure, and is related to the log- cases of the 1-year and the 10-year historical data. Based normal martingale process of a forward LIBOR rate of on this level of significance it is realistic to model the the same maturity through a Brownian motion. Models CMS spread rate as normally distributed in the valuation based on our approach have two relative advantages. of both short and long maturity derivatives. The follow- Firstly they are more flexible for analytical tractability ing section then proposes a Gaussian market models to and can lead to close-form solutions for CMS spread op- value interest derivatives on CMS spread rates. tions. Secondly they can be calibrated directly with CMS spread instruments, and hence they reduce arbitrage risks 3. The Model in the valuation of more complex CMS spread deriva0 Let us consider a finite time horizon [0,T ] in which tives. trading is done. We assume the uncertainty in our econ- The rest of this paper is structured as follows. The next omy is modeled by a complete filtered probability space, section test and confirm the normality assumption of ,, , : , in which  is the set of some CMS spread rates. Section 3 presents the proposed  FTt F0tT  M model and shows how forward CMS spread rates for all possible states of nature, Ft is a filtration that different maturities can be simultaneously modeled under 0tT a single measure. Section 4 provides a numerical illus- satisfies the usual conditions and it is generated by two trates of the model. Closed-form solutions for CMS spread independent source of risk (two standard Brownian mo-  caplets, floorlets, and digital options are derived in Sec- tions), and is a probability measure that belongs to , the class of equivalent probability measures on tion 5. A final section concludes the study. M ,FT  . Let T N 1 be an increasing sequence of dates from 2. A Test of Normality for CMS Spreads i i1 which reset dates of financial derivatives will be taken. In this section we use Jarque-Bera (JB) test to access the ii: TT1  i will denote the day-count fraction between normality assumption on the spreads between the CMS at times Ti and Ti1 . Let Ptii11:0 tT be the key maturity points of the yield curve—2, 5, 10, and 30 price process of the risk-free discount bond paying one years. These points are the maturities of the most traded monetary unit at time Ti1 . According to the asset pric- fixed income instruments. Furthermore, the CMS spreads ing theory one can find a probability measure, i1 , 5-year minus 2-year, 10-year minus 5-year, 30-year mi- equivalent to  , for which Ptii11:0 tT is the nus 10-year, and 30-year minus 2-year can be viewed as numeraire as in Geman et al. [6]. Following Jamshidian representing the short-end, the middle, the long-end, and i1 [7],  will be called Ti1 -forward measure. the entire yield curve respectively. At any given time t Ti , let us define The JB test can be used to access the normality as- Sti  :0 t Ti to be the stochastic process of the sumption of the spread between these indexes. The JB forward CMS spread rate for the maturity date Ti . Here test statistic is distributed as a Chi-square random vari- St denotes the view of investors at time t of what i   able with two degrees of freedom, and measures the de- 3 parture of the skewness and kurtosis of a series from The CMS tickers are represented as USSWAPyy, where yy is the year indicator. For Example the tickers for CMS 30 yrs and CMS 2 yrs are those of the Normal distribution. The null hypothesis is a USSWAP30 and USSWAP02 respectively.

Table 1. JB statistics and associated p-values.

CMS Spread 5 yrs - 2 yrs 10 yrs - 2 yrs 30 yrs - 2 yrs 10 yrs - 5 yrs 30 yrs - 5 yrs 30 yrs - 10 yrs

1 year weekly 3.85859 3.67926 3.72086 4.67236 4.75436 4.32966 average (14.53%) (15.89%) (15.56%) (9.67%) (9.28%) (11.48%)

10 years quarterly 5.51943 5.45094 5.21446 5.19099 4.77346 3.83152 average (6.33%) (6.55%) (7.37%) (7.46%) (9.19%) (14.72%)

will be the level of the CMS spread rate at time Ti . Be- tional source of risk. This assumption has the implication cause the value of the forward CMS spread rate can be that traders can have different views on the entire yield either positive or negative, we assume its dynamics is curve on the one hand and on the slope of a part of the driven by a Gaussian process. A martingale spread meas- yield curve on the other hand.4 The forward LIBOR rate ure for this forward CMS spread exists. But as noticed by is then assumed to be driven by the two Brownian mo-

Antonov and Arneguy [8], the corresponding numeraire tions and its dynamics under the Ti1 -forward measure is process for this measure is difficult to calculate. To over- given by the following stochastic differential equation come this difficulty, we define the dynamics of this for- ii11  ddLtiiii   tLt    t W t  i td W t , ward CMS spread directly under the Ti1 -forward measure as a Brownian motion with a drift. This drift arises (2) from the convexity adjustment and the change of meas- subject to the initial forward LIBOR rate Li 0 , and ure. Hence the stochastic differential equation of the for- where  i :[0,T ]  is a deterministic bounded func- ward CMS spread rate is represented as tions that is square integrable on [0,T ], and  i1 i1 Wt  :0tTi  is a standard Brownian motion dddStii  t t i t W t, (1) under the Ti1 -forward measure. This additional Brow- i1 nian motion is independent of Wt:0 tTi . The subject to the initial forward CMS spread rate Si 0 , function i :[0,T ] [ 1, 1] is the correlation coeffi- and where  i :[0,T ]  and  i :[0,T ]  are deterministic bounded functions that are integrable and cient between Sti   and ln Lti   , and 12 square integrable on [0,T ] respectively, and 2 iitt1  is the orthogonal complement of i1   Wt:0 tTi  is a standard Brownian motion i t . under the Ti1 -forward measure. At any given time tT let us define Equation (1) defines the stochastic differential equa- i tion of the forward CMS spread rate under its relevant Lti:0 t Ti to be the stochastic process of the measure, the T -forward measure. This equation can  -tenor forward LIBOR rate maturing at time T . Here i1 i i lead to closed-form solutions to value financial deriva- Lt denotes the interest rate available at time t for a i tives that depend on a CMS spread rate at a single matur- risk-free loan which is effective at time T and matures i ity date such as CMS spread caplets, floorlets, and digital at time T . As in the standard LIBOR market model, i1 options. However, for valuing financial derivatives that let us assume the dynamics of Lt under the T - i i1 involve forward CMS spread rates at more than one ma- forward measure to be a lognormal martingale. turity date, all rates needed to be modeled simultaneously, It is important to note that under the T -forward i1 i.e., under a single measure as in the following proposi- measure if the forward LIBOR rate is driven only by the tion. same source of risk that drives the forward CMS rate, Proposition 1. Under the T - forward measure as- both stochastic processes will be perfectly correlated, and M 1 sociated to the numeraire Pt:0 tT : hence it will be economically redundant to trade CMS  MM11  spread derivatives, as investors can obtain the same result 1) for iM , the expression of dSti  is given by by trading the forward LIBOR rate. Therefore, we as- Equation (1). sume the forward LIBOR rate to be driven by an addi- 2) for iM ,

M ttLt    dSt t tkk k k d t   td WM 1 t, (3) iii   i  ki1 1 kkLt

4This will be illustrated in the numerical example.

 tL t M jj  j   ddLt  tLt  t  t    t  t t kkk  jkjk jk1 1 jjLt (4)  MM11  kktL  t k  t d W  t  k  t d W  t .

3) and for iM ,

i ttLt    ddSt t tkk k k t   td WM 1 t, (5) iii    i  kM1 1 kkLt where

k jjtLj  t d(Ltkkk  tLt  jkjk t  t t)  td t  1  Lt jM1 jj (6) tL t t d WMM11 t  t d W t . kk  k   k  

Proof: market data are not easily available. Therefore, for sim- Let us consider two consecutive forward LIBOR rate plicity of the illustration we assume all model parameters processes, Ltii11:0 tT and Ltii:0 t T, to be constant, i.e.,  i t   , i t   ,  i t  , which their stochastic dynamics are described by Equa-  i t   , and i   . In the first column we consider tion (2) under the Ti -forward measure and the Ti1 -for- the initial term structures of forward LIBOR and forward ward measure respectively. It is straightforward to prove CMS spread rates to be flat. Even though we started the that by applying Ito’s lemma on the Radon-Nikodym simulation with flat curves for both the forward LIBOR derivative that allows the change of measure from the rates and the forward CMS spread rates, the end results

Ti1 -forward measure to the Ti -forward measure, and shows that the expected forward LIBOR rates are in- using Cameron-Martin-Girsanov theorem (as in e.g. Pels- creasing while the expected forward CMS spread rates ser [9]) one obtain the following relationships are fluctuating around zero.

ii1 iittLt  i  i   ddWt W  td t 5. Closed-Form Solutions for CMS Spread 1()iiLt  Options  ttLt    ddWtii W1 tii i i d t This section presents the derivation of closed-form solu-    1iiLt  tions for valuing simple instruments which payoffs are functions of a CMS spread rate at a single maturity date, The result of the theorem is then obtained apply the such as CMS spread caplets and floorlets. A CMS spread above relationships repeatedly backward and forward on caplet (floorlet) is a call (put) option on a CMS spread Equations (1) and (2). rate. At maturity the buyer receives a payment from the 4. A Numerical Example seller if the CMS spread was above (below) the agreed strike rate. CMS spread caplets (floorlets) are not gener- For illustration purpose Table 2 shows the results of a ally traded. However they are useful as they are building one thousand paths of Monte Carlo simulation for gener- blocks of over-the-counter traded CMS spread caps ating forward LIBOR rates and forward CMS spread (floors). rates with a quarterly frequency over a two-year period. Let Gtii  :0 t T be the price process of a cap- The results of this simulation can be used to valuing let or a floorlet that resets at time Ti and pays off at CMS spread derivatives that mature within two years and time Ti1 with a strike rate K . We have that involve forward CMS spread rates at more than one  Gt P ti1 ST  K , (7) maturity date. This simulation is done under the terminal iiitii1    measure, i.e., the measure associated to the discount bond maturing in two years. where   1 for caplet and floorlet respectively, and i1 Although the CMS spread derivatives are increasing in t  represents the expectation with respect to the popularity, they are still OTC instruments, and their Ti1 -forward measure and the sigma-algebra Ft . The

Table 2. Average of one thousand sample paths of forward LIBOR and forward CMS spread rates in the terminal measure with σ = 20%, ρ = 10%, ψ = 0.5%, γ = 0.1%, and δ = 0.25.

Ti (in yrs) T0  0 T1  0.25 T2  0.5 T3  0.75 T4 1 T5 1.25 T6 1.5 T7 1.75

L0 Ti 2.50%

L1 Ti 2.50% 2.6483%

L2 Ti 2.50% 2.6484% 2.9422%

L3 Ti 2.50% 2.6486% 2.9426% 2.9961%

L4 Ti 2.50% 2.6487% 2.9429% 2.9967% 3.0208%

L5 Ti 2.50% 2.6489% 2.9432% 2.9973% 3.0216% 3.6086%

L6 Ti 2.50% 2.6490% 2.9436% 2.9978% 3.0224% 3.6098% 4.2913%

L7 Ti 2.50% 2.6492% 2.9439% 2.9984% 3.0232% 3.6110% 4.2931% 4.7647%

ST0 i 0.10%

ST1 i 0.10% −0.0699%

ST2 i 0.10% −0.0699% 0.1600%

ST3 i 0.10% −0.0699% 0.1600% 0.2344%

ST4 i 0.10% −0.0699% 0.1601% 0.2345% −0.0388%

ST5 i 0.10% −0.0699% 0.1601% 0.2345% −0.0387% −0.0421%

ST6 i 0.10% −0.0699% 0.1601% 0.2346% −0.0387% −0.0420% −0.0212%

ST7 i 0.10% −0.0699% 0.1602% 0.2346% −0.0386% −0.0420% −0.0211% 0.1335% stochastic differential e quation of the forwar d CMS generally traded. However t hey are usef ul as they are spread in Equation (1) implies building blocks of range accrual CMS spread notes that

Ti are over-the-counter traded. If Dtii:0 t T repre- ST St()  ud Wi1 u, (8) ii i t i  sents the price process of a digital option that pays one Ti monetary unit at time T in case the CMS spread rate where St St ud u is the convexity ad- i1 ii t i   is greater or less than a strike rate K at time Ti , then justed forward CMS spread for the maturity Ti , as seen at time t . Note that the second term of the right hand i1 Dtii   P1  t  ST ii K side of Equation (8) is normally distributed, as it is a sto- , (10) chastic in tegral of a deterministic function times a  PtNdtii1   Brownian motion. Hence ST is normally distributed ii where   1 if the CMS spread is greater than and with mean Sti  as the Ito integral is a martingale, and T less than the strike respectively, and dti  is given in variance 22tuu i d by Ito’s isometry. Thus, iit   Equation (9). Equation (7) can be rewritten as 6. Concluding Remarks  2  Gt P teu 2 St K u td u iii1   2π  i i  We proposed a two-factor model for valuing C MS spread 2 derivatives assuming the CMS spread rate is driven by Ptiiii1  t dt  dtNdt i  i  , one of the risk sources that drive the LIBOR rate. Our (9) proposed model is simultaneously calibrated to LIBOR where dtii St K i t , and  and N  instruments and CMS spread instruments and is flexible are the standard normal density and cumulativ e distribu- enough to take into account various deterministic volatil- tion functions respectively. ity and convexity adjustment functions. Furthermore, this Another important CMS spread option is the digital model is easy to implement, and can be used to value option. It is an instrument in which, at maturity, the Constant Maturity Treasury (CMT) spread derivatives. holder receives a payment if the CMS spread crosses a An area of improvement would be to show the consis- certain barrier level. CMS spread digital options are not tency between the forward LIBOR rates and the forward

CMS spread rates. [5] W. Margrabe, “The Value of an Option to Exchange One Asset for the Another,” Journal of Finance, Vol. 33, No. 1, 1978, pp. 177-186. doi:10.2307/2326358 REFERENCES [6] H. Geman, N. Karoui and J. C. Rochet, “Change of Nu- [1] Fannie Mae, “Fannie Mae Universal Debt Facility,” Fed- meraire, Change of Probability Measure and Option Pric- eral Home Loan Association, Washington DC, 2008. ing,” Journal of Applied Probability, Vol. 32, No. 2, 1995, http://www.fanniemae.com/markets/debt/pdf/CUSIP_PS3 pp. 443-458. doi:10.2307/3215299 1398ANE8.pdf [7] F. Jamshidian, “Bond and Option Evaluation in the Gaus- [2] R. Carmona and V. Durrleman, “Pricing and Hedging of sian Interest Rate Model,” Research in Finance, Vol. 9, Spread Options,” SIAM Review, Vol. 45, No. 4, 2003, pp. 1991, pp. 131-170. 627-685. doi:10.1137/S0036144503424798 [8] A. V. Antonov and M. Arneguy, “Analytical Formulas for [3] D. Belomestny, A. Kolodko and J. Schoenmakers, “Pric- Pricing CMS Products in the Libor Market Model with ing Spreads in the Libor Market Model,” Weierstrass In- Stochastic Volatility,” Numerix Software Ltd., London, stitute for Applied Analysis and Stochastics, Berlin, 2008. 2009. [4] M. Lutz and R. Kiesel, “Efficient Pricing of CMS Spread [9] A. Pelsser, “Efficient Methods for Valuing Interest Rate Options in a Stochastic Volatility LMM,” Institute of Mathe- Derivative,” Springer-Verlag, New York, 2000. matical Finance, Ulm University, Ulm, 2010.