Convexity adjustment for constant maturity swaps in a multicurve framework
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Abstract
In this paper we propose a double curving setup with distinct forward and discount curves to price constant maturity swaps (CMS). Using separate curves for discounting and forwarding, we develop a new convexity adjustment, by departing from the restrictive assumption of a flat term structure, and expand our setting to incorporate the more realistic and even challenging case of term structure tilts. We calibrate CMS spreads to market data and numerically compare our adjustments against the Black and SABR (stochastic alpha beta rho) CMS adjustments widely used in the market. Our analysis suggests that the proposed convexity adjustment is significantly larger compared to the Black and SABR adjustments and offers a consistent and robust valuation of CMS spreads across different market conditions.
Keywords
Convexity adjustment Constant maturity swaps Multicurve framework Yield curve modelling Money market instruments1 Introduction
The recent financial crisis has led, among others, to unprecedented behavior in the money markets, which has created important discrepancies on the valuation of interest rate financial instruments. Important reference rates that used to be highly correlated and moving together for a long period of time, started to diverge from one another. A characteristic example, that has been widely studied recently, is the widening of the spread between deposit rates (Libor/Euribor) and overnight index swap (OIS) rates of the same maturity. At the same time, the market started observing nonzero spreads between swap rates of the same maturity, but based on different frequencies of the underlying Libor rate, or between forward rate agreement (FRA) rates and forward rates implied by consecutive deposits. These examples indicate that financial players consider each tenor as a separate market, incorporating different credit and liquidity premia, and as such, each one of them is driven by its own dynamics.
Such discrepancies have, above all, questioned the methodology used to bootstrap the yield curve, which has created a layer of uncertainty on the methods used to price and hedge interest rate financial instruments. There are three main issues associated with the precrisis approach, which make it inconsistent. First, the information incorporated into the basis spreads is not taken into account. Second, using a single yield curve does not allow us to consider the different dynamics introduced by each underlying rate tenor, making hedging and pricing of interest rate derivatives less stable. Finally, the noarbitrage assumption indicates that a unique discounting curve needs to be used, regardless of the number of the underlying tenors.
In order for market participants to comply with the mentioned market features, they started building a separate forward curve for each given tenor, so that future cash flows are generated using the appropriate curve associated with the underlying rate tenor. At the same time, a single and unique discounting curve had to be used, in order to calculate the present value of contract’s future payments. This led financial players to start using the OIS swap curve, rather than the Libor curve, for the construction of a riskless term structure. The reason behind their choice was mainly twofold. First, OIS is believed to contain very little credit and liquidity risk premia compared to Libor rates. Second, the fact that most trades in the interest rate market are (mainly cash) collateralized makes the funding cost for a financial institution no longer equal to the Libor rate, but to the collateral rate instead. For that reason, the Libor rate that was widely used as a proxy for the riskfree (discounting) rate, is now replaced by the collateral rate, which is assumed to coincide with the overnight rate (i.e. fed fund rate for USD, Eonia for EUR, etc.).
The literature on the valuation of interest rate derivatives based on separate curves, for generating future rates and for discounting, is growing rapidly. Previous contributions focus on the valuation of cross currency (basis) swaps (see, Boenkost and Schmidt 2005; Kijima et al. 2009; Fujii et al. 2010; Henrard 2010). Henrard (2007b), is the first to apply this methodology to the single currency case, whereas Bianchetti (2010) is the first to deal with the postcrisis situation. Furthermore, Ametrano and Bianchetti (2009), Chibane and Sheldon (2009) and Morini (2009), develop new methodologies for bootstrapping multiple interest rate yield curves. On the other hand, many contributions focus on extending pricing models under the multicurve framework. Kijima et al. (2009), apply the methodology to study two short rate models, the Vasicek model and the quadratic Gaussian model, and use them for the valuation of bond options and swaptions. Mercurio (2009, 2010) and Grbac et al. (2015) extend the libor market model (LMM) to be compatible with the multicurve practice and price caplets and swaptions, while more recently, Pallavicini and Tarenghi (2010), Crépey et al. (2012), Moreni and Pallavicini (2014) and Cuchiero et al. (2016) extend the classical Heath–Jarrow–Morton (HJM) framework to incorporate multiple curves in order to price interest rate products such as forward starting interest rate swaps (IRS), plain vanilla European swaptions and CMS spread options. Finally, important contributions include Crépey et al. (2015) who develop a Levybased HJM model for credit value adjustment (CVA) and Fanelli (2016) who develop a defaultable HJM model for pricing basis swaps in a multicurve setup.
In this paper, we follow the approach described in Mercurio (2010) and Pallavicini and Tarenghi (2010) to price a CMS. A CMS exchanges a swap rate with a fixed time to maturity against fixed or floating. In a common CMS, one would swap a quarterly (e.g. 3month Libor) or semiannual rate against a 5 or 10year swap rate. Whereas a regular floating rate (e.g. 6month Libor) contains information about shortterm interest rates, a CMS rate (e.g. 10year swap rate) contains information about the overall level of the yield curve. This makes CMS a popular instrument among investors and portfolio managers. It gives investors the ability to place bets on the shape of the yield curve over time. Generally, a constant maturity payer will benefit from a flattening or inversion of the yield curve and is exposed to the risk of the yield curve steepening. It also helps portfolio managers to hedge a floating rate debt without introducing duration risk from the hedging instrument.
The mix of short and long term rates in the structure of the CMS makes its value depend on the shape of the yield curve. Standard approaches for its valuation involve the calculation of a convexity adjustment. Such convexity adjustment cannot be computed exactly, so previous literature uses either adhoc approximations or utilising unrealistic assumptions. A common assumption used in the relevant literature is that the term structure of interest rates is flat and only parallel shifts are allowed.
There are two main avenues towards pricing a CMS. In the first, one sets up a term structure model and uses some approximation method to compute the expected swap rate, under the forward measure. More specifically, Lu and Neftci (2003) follow this direction and work with two or more forward rates jointly. Using the forward libor model, they price a CMS swap and compare its empirical performance with the standard convexity adjustment proposed by Hagan (2005). They find that the convexity adjustment overestimates CMS swap rates. Similarly, Henrard (2007a) uses onefactor LMM and HJM models to approximate CMS swaps, while Brigo and Mercurio (2006) use a twofactor Gaussian short rate model (G2++ model) to model bond prices associated with CMS products. Finally, in a recent work, Wu and Chen (2010) price different CMStype interest rate derivatives within the LMM framework. They present a new approach for finding the approximate distribution of a CMS under the forward martingale measure.
In the second direction, one uses replication arguments and the problem is formulated under the swap measure. The price is based on the implied swaption volatilities which play the role of the distribution of swap rates. For the replication procedure, the change from the forward to the swap measure is needed and the Radon–Nikodym derivatives need to be approximated. Pelsser (2003) is the first to show that the convexity adjustment can be interpreted as the side effect of a change of numeraire. He approximates the measure change by proposing a linearization of the swap rate and obtains analytical solutions to the CMS price. Hagan (2005) obtains closedform formulae for the pricing of CMS swaps and options by relating them to the swaption market via a static replication approach. Finally, Mercurio and Pallavicini (2006) use a strike extrapolation to statically replicate CMS swap/options by modelling implied volatilities of European swaptions using the SABR model of Hagan et al. (2002). Finally, in a recent work, Zheng and Kuen Kwok (2011) propose a generalised static replication approach to hedge exotic swap contracts and annuity options using different swaptions.
The main problem with previous contributions is that the yield curve is assumed to be flat and only parallel shifts are allowed. However, in a swap where one pays Libor plus spread and receives a 10year CMS rate, the structure is mainly sensitive to the slope of the interest rate yield curve and is almost immunised against any parallel shift. In this paper, following Hagan (2005), we apply the commonly used convexity adjustment in a new framework of double curving. We then develop a new convexity adjustment, by departing from the restricting assumption that the term structure is flat, and we allow for a tilt. Using market data for Euro money market instruments (Eonia, Euribor), CMS spreads and swaption volatilities, we find out that the new convexity adjustment is significantly larger than the one commonly made in the literature. We finally compare our approach with the SABR CMS adjustment, introduced by Mercurio and Pallavicini (2005, 2006), which is widely used in the market, and we find that our approach provides a better fit to the market’s CMS spread prices.
The remaining of this paper is organised as follows. Section 2 presents the valuation framework for the main instruments (FRA, IRS, CMS) considered. Section 3 shows the main result of our work, that is a new convexity adjustment that takes into account the tilt in the term structure, under a double curving framework. Section 4 briefly depicts the smileconsistent convexity adjustment using SABR model. Sections 5 and 6 present the market data that have been used and describe numerical calculations. Finally, Sect. 7 concludes.
2 The valuation framework
This section introduces the definitions of basic instruments under the muticurve environment. It mostly follows the works of Brigo and Mercurio (2006) and Mercurio (2009).
Definition 1
Consider times t, \(T_{1}\), \(T_{2}\), with \(t \le T_{1} < T_{2}\). The timet FRA rate \(\textit{FRA}(t;T_{1},T_{2})\) is defined as the fixed rate to be exchanged at time \(T_{2}\) for the Libor rate \(L(T_{1},T_{2})\), so that the swap has zero value at time t.
Proposition 1
Any simple compounded forward rate spanning a time interval ending in \(T_{i}\), is a martingale under the \(T_{i}\)forward measure, \(F(u;T_{i1},T_{i})= {\mathbb {E}}^{Q_{d}^{T_{i}}} \left[ F(t;T_{i1},T_{i})  {\mathscr {F}}_{u} \right] \), for \(0 \le u \le t \le T_{i1} < T_{i}\).
In the multicurve framework, however, Eq. (4) does not hold. The forward rate \(F_{ f }(t;T_{1},T_{2})\) is not a martingale under the forward measure \({\mathbb {Q}}_{d}^{T_{2}}\), and the FRA rate is different from the forward rate, \(\textit{FRA}(t;T_{1},T_{2}) \ne F_{ f }(t;T_{1},T_{2})\). Therefore, the present value of a future Libor rate is no longer obtained by discounting the corresponding forward rate, but by discounting the corresponding FRA rate. According to Mercurio (2010), the FRA rate is the natural generalization of a forward rate to the multicurve case. This has a straightforward implication, when it comes to the valuation of Interest Rate Swaps.
2.1 Interest rate swap
2.2 Constant maturity swap
A constant maturity swap contract, is a swap where one of the legs pays (receives) periodically a swap rate with a fixed time to maturity, c, while the other leg receives (pays) either fixed or floating. More commonly, one term is set to a short term floating index such as the 3month Libor rate, while the other leg is set to a long term fixed rate such as the 10year swap rate.
At this point it is important to emphasize the fact that, naturally, the expectation used to calculate the above payoff, is associated with the payment dates \(T_{i}\). However, under the forward measure, \({\mathbb {Q}}_{d}^{T_{i}}\), the swap rate, \(S_{T_{i1}}^{t_{i1,j}}\), is not a martingale. The convexity adjustment arises since the expected payoff is calculated in a world which is forward risk neutral with respect to a zero coupon bond. In that world, the expected underlying swap rate (upon which the payoff is based), does not equal the forward swap rate. The convexity is just the difference between the expected swap rate and the forward swap rate.
When we consider pricing CMStype derivatives, it is convenient to compute the expectation of the future CMS rates under the forward measure, that is associated with the payment dates. However, the natural martingale measure of the CMS rate is the underlying forward swap measure. Convexity correction arises when one computes the expected value of the CMS rate under the forward measure that differs from the natural swap measure with the underlying forward swap measure as numeraire.
3 Convexity adjustment
3.1 Flat term structure with parallel shifts
3.2 A term structure with tilts
4 Smileconsistent convexity adjustment
In order to test the proposed CMS convexity adjustments, we compare them with the smileconsistent convexity adjustment, which is widely used in the market. In the presence of a market smile, when the term structure is not flat, but may tilt, the adjustment is necessarily more involved, if we aim to incorporate consistently the information coming from the quoted implied volatilities. The procedure to derive a smile consistent convexity adjustment is described in Mercurio and Pallavicini (2006) and Pallavicini and Tarenghi (2010), and is the one we will use here.
5 Market data

For the discounting curve, we use Eonia Fixing and OIS rates from 3months to 30years.

For the 3month curve, we use Euribor 6months fixing, FRA rates up to 15 months, and swaps from 2 to 30 years, paying an annual fix rate in exchange for the Euribor 3month rate.

For the 6month curve, we use Euribor 6months fixing, FRA rates up to 2 years, and swaps from 2 to 30 years, paying an annual fix rate in exchange for the Euribor 6month rate.
6 Empirical results
In this section, we compare numerically the accuracy of the approximations for the CMS convexity adjustments against the Black and SABR models convexity adjustments presented in Sect. 4.
6.1 An empirical illustration
This table reports market quotes of swaption volatilities (in %) for different strikes K
Expiry  Tenor  \(\)200  \(\)100  \(\)50  \(\)25  25  50  100  200 

5 years  10 years  6.54%  2.30%  0.93%  0.41%  \(\)0.30%  \(\)0.51%  \(\)0.68%  \(\)0.39% 
This table presents the market price of the CMS spread against the calibrated price of the spread using the (case 1) Blacklike convexity adjustment (parallel shift), the (case 2) second convexity adjustment (allows for tilt) and the SABR model
Market  Case 1  Case 2  SABR  

Price  0.00649  0.006237  0.006402  0.006406 
Difference (in bps)  2.53  0.89  0.83 
This table presents the market price of the CMS spread against the calibrated price of the spread using the (case 1) Blacklike convexity adjustment (parallel shift), the (case 2) second convexity adjustment (tilt) and the SABR model
Market  Case 1  Case 2  SABR  

Price  0.00649  0.006352  0.0065001  0.006406 
Difference (in bps)  1.38  0.1  0.83 
This table presents the convexity adjustment for all four different cases, i.e. the Blacklike ‘flat’ term structure with and without spread, and the ‘tilt’ term structure with and without spread
i  CA (case 1)  CA (case 2)  CA (case 1—with spread)  CA (case 2—with spread) 

1  0.000036  0.000032  0.000037  0.000033 
2  0.000074  0.000083  0.000076  0.000092 
3  0.000114  0.000198  0.000116  0.000218 
4  0.000154  0.000313  0.000158  0.000343 
5  0.000196  0.000428  0.000201  0.000469 
6  0.000238  0.000542  0.000245  0.000593 
7  0.000282  0.000656  0.000289  0.000718 
8  0.000326  0.000769  0.000335  0.000842 
9  0.000373  0.000883  0.000383  0.000967 
10  0.000420  0.000995  0.000431  0.001090 
11  0.000468  0.001107  0.000480  0.001213 
12  0.000514  0.001216  0.000528  0.001334 
13  0.000564  0.001326  0.000579  0.001455 
14  0.000615  0.001436  0.000631  0.001576 
15  0.000667  0.001544  0.000685  0.001697 
16  0.000721  0.001652  0.000739  0.001817 
17  0.000775  0.001759  0.000795  0.001936 
18  0.000831  0.001865  0.000852  0.002054 
19  0.000888  0.001970  0.000911  0.002172 
20  0.000946  0.002074  0.000970  0.002289 
Our numerical results suggest that in all cases market data are well reproduced. As Table 2 reports, the SABR model performs slightly better than our new convexity adjustment (case 2), with 0.89 bps compared to 0.83 bps, when the spread is not taken into account, and much better compared to the Blacklike formula (case 1), 0.83 bps against 2.53 bps. However, this is not the case when we take into account the swap spread. The absolute difference (in bps) between our new convexity adjustment model and the market is significantly smaller compared to the SABR case, i.e. 0.1 bps compared to 0.83 bps. Furthermore, as expected, Black’s model calibration results, although better than the nonspread case (1.38 bps against 2.53 bps), still fail to fit the data compared to the other two cases. In addition, in Table 4, we report the convexity adjustments for all four cases. We observe that the convexity adjustment in the ‘tilt’ case is significantly larger than the ’flat’ case, especially, when the swap spread is incorporated. Furthermore, in the nonflat case, convexity adjustment presents a curvy shape compared to the earlier Blacklike case, where the shape behaves in a more static way.
6.2 Numerical examples
This table reports market CMS swap spreads (in bps) and market ATM swaption volatilities data for specific dates
Date  Market CMS  Market volatility (%) 

10/08/2007  41.6  12.30 
31/10/2008  104.5  12.98 
28/05/2010  178.1  19.30 
03/06/2011  136.4  19.80 
09/03/2012  158.65  25.15 
This table reports market swaption volatility smiles for different dates
Date  \(\)200 (%)  \(\)100 (%)  \(\)50 (%)  \(\)25 (%)  25 (%)  50 (%)  100 (%)  200 (%) 

10/08/2007  2.75  1.01  0.44  0.20  \(\)0.18  \(\)0.29  \(\)0.47  \(\)0.70 
31/10/2008  3.81  1.08  0.40  0.20  \(\)0.20  \(\)0.40  \(\)0.72  \(\)1.22 
28/05/2010  8.53  3.12  1.29  0.64  \(\)0.45  \(\)0.86  \(\)1.38  \(\)2.09 
03/06/2011  6.96  2.51  1.09  0.46  \(\)0.42  \(\)0.76  \(\)1.22  \(\)1.78 
09/03/2012  9.61  3.02  1.23  0.56  \(\)0.44  \(\)0.77  \(\)1.30  \(\)1.81 
Date  10/08/2007  31/10/2008  28/05/2010  03/06/2011  09/03/2012 

SABR parameters  
alpha  0.0585  0.0579  0.0885  0.0912  0.1139 
beta  0.7539  0.7161  0.768  0.7643  0.7794 
rho  \(\)0.2341  \(\)0.277  \(\)0.3957  \(\)0.3433  \(\)0.2364 
epsilon  0.1926  0.2177  0.2848  0.2867  0.231 
f (Nelson–Siegel)  
a  0.001  0.005  \(\)0.03  \(\)0.01  \(\)0.002 
b  0.002  0.0012  0.0047  0.0043  0.0038 
k  0.003  0.015  0.001  0.05  0.004 
This table compares market CMS spread prices against each different case
Date  10/08/2007  31/10/2008  28/05/2010  03/06/2011  09/03/2012 

Market  41.6  104.5  178.1  136.4  158.65 
Case 1 (Flat)  37.92  112.88  165.78  125.73  151.16 
Case 2 (Tilt)  40.94  107.73  173.15  131.29  154.99 
SABR  39.95  111.77  168.81  128.21  152.5 
(Mkt–SABR)  1.65  7.27  9.29  8.19  6.15 
(Mkt–Tilt)  0.66  3.23  4.95  5.11  3.66 
Finally, convexity adjustments for all different cases, the (Blacklike) flat term structure in red colour, the ‘tilt’ term structure in green colour and the SABR model in blue colour, are presented in Figs. 1, 2, 3, 4, 5 and 6. All cases take into account the spread on the swap rate. Furthermore, the outcome of the calibration procedure under the SABR model (i.e. the whole volatility smile against different strikes) is presented in the lower panel of the figures, where we observe that the SABR model is perfectly calibrated across different dates. The only exception is October of 2008, i.e. the peak of the financial crisis, where markets were under severe pressures, that the SABR model struggles to fit the volatility smile. Regarding the convexity adjustments, in all cases and across different periods, we depict similar characteristics. We observe that convexity adjustment with ‘tilt’ term structure is significantly larger than in the other two cases. Furthermore, the shape of the nonflat case presents a slope compared to the Blacklike case where the convexity adjustments are flat and static. This helps the model perform well, especially in periods of market turmoil.
7 Conclusion
In this paper we have developed a new CMS convexity adjustment in a doublecurve framework, that separates the discounting and forwarding term structures. The motivation of our study comes from the unprecedented increase in the LiborOIS spread that was experienced during the financial crisis, which has questioned the legitimacy of considering both (Libor and OIS) quotes as riskfree, and has raised valid issues in the construction of zerocoupon curves, which clearly, can no longer be based on traditional bootstrapping procedures. In that vein, our work fills the gap of the shortcomings of single yield curve model adjustments, widely used in the literature, when one deals with the issue of convexity in money market instruments.
In the doublecurving environment that we describe, we have derived the convexity factor requirement in the conventional case that the term structure of interest rates is flat, and its dynamic evolution allows only for parallel shifts, and we have expanded our setting to incorporate the more realistic and challenging case of term structure tilts. The new term appears to be approximately linear in this parameter. In all computations, our results conclude that the convexity adjustment of the ‘tilt’ term structure case is significantly larger than the convexity adjustments implied by the Black and SABR models.
As an empirical illustration, we have calibrated both convexity adjustments to real market data, by using swaption volatilities, and calculated the differences between market quotes and our model implied CMS spreads. We further compared our results with the widely used by market practitioners smileconsistent CMS adjustment, using the SABR model. We considered a different SABR model for each swap rate contained in the CMS payoff, and we performed a calibration of all the SABR parameters to swaption volatility smile and CMS spreads quoted by the market. In all cases the swaption volatility smiles are very well recovered by the calibrated SABR models. Furthermore, our results demonstrate that the proposed convexity adjustments offer a market consistent and robust valuation of CMS spreads, and suggest that CMStype of products should be priced under a multicurve framework.
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