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Constant Maturity Asset Swap Convexity

Correction

Mario Pucci

Banca IMI, Milan

2012-02-27 11:08:31Z

Contents

1 Introduction 2

2 Constant Maturity Asset Swaps 2

3 Definitions 3

4 Pricing Framework 4

5 One-Factor Linear Model 4

6 Hedging Considerations 7

7 Numerical Example 7

8 Model Limitations 8

9 Conclusions 8

Abstract

As a consequence of the high volatility regime recently establishedin the government bond market, investors may seek hedging their expo-sure to floating asset swap spreads. We obtain an analytical convexitycorrection for the asset swap spread which is instrumental to the pric-ing of constant maturity asset swaps.

I am grateful to Giuseppe Fortunati, Sebastiano Chirigoni, Emiliano Carchen, Giorgio

Facchinetti and Salvatore Crescenzi for their inspirational momentum. A special mention

for Marco Bianchetti, Damiano Brigo and Mara [email protected]

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AssetSwapConvexity.tex 4412 2012-02-27 12:57:10Z puccim1a February 27, 2012

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1 Introduction

Most market players trade in the government bond market: Europeanfinancial institutions are heavily exposed to long positions in eligibledebt to raise funding at ECB rates, pension funds traditionally includetreasury bonds in their asset portfolio to obtain flows offsetting theirliabilities. More generally, sovereign debt has always been seen as asafe haven for investors seeking an acceptable yield at a once perceivedrelatively low risk, while the interbank market was providing tools tohedge the bond-driven fixed income interest rate exposure at will: plainvanilla swaps, constant maturity swaps and swaptions. Likewise, whenthe inherent credit risk associated to treasury could not be overlooked,wary investors could seek refuge in the CDS market but, whereas aCDS provides, up to legal intricacies, perfect protection upon default,

basis risk (see [6]) hinders P&L immunization to credit migrations andcredit spread movements. The asset swap spread is instead an indicatorthat is closely related to bond P&L. As such, we believe that asset swapcontingent derivatives may be a match to market participants seekingto hedge P&L volatility due to swings in the government bond levelsas dramatically disclosed in recent market regimes.

In this paper we present a methodology to price a simple asset swapcontingent instrument, the constant maturity asset swap spread. Anapplication of this may be as follows: An investor holding a long po-sition in Italian government bonds, expecting to maintain a rollingapproximate duration of 5Y for the next 3 years willing to protect hisportfolio from may enter into a 3Y long swap paying a fixed rate leg

while receiving a floating constant maturity asset swap spread. Adeteriorating credit worthiness of Italy, affecting her/his portfolio, willfind compensation in the floating leg of the swap.

2 Constant Maturity Asset Swaps

An asset swap (ASW) is a trade where the investor enters a packageconsisting of a discount bond paid at par and a payer swap where s/hereceives Libor plus spread Rasw in exchange for amounts equivalent tothe bond coupons. The equivalence restricts to the non-credit compo-nent of the bond, i.e., in case of a credit event for the bond, the swapdoes not cancel (although it is usually unwound at market value). The

spread (the asset swap spread) is meant to make the package fair inthat it compensates the investor for the purchase at par of a bondthat is below par due to default risk of the issuer or market conditionsrelative to the promised coupon structure.

Because, by definition, the asset swap is determined relative to areference security (the underlying bond) one may devise a note or aswap leg, a Constant Maturity Asset Swap (CMASW), indexed to itsprevailing ASW spread.

This is similar to a CMS swap, except that, in a CMASW theindexing is relative to the asset swap spread of a specific referencebond in lieu of a rolling rate.

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In the following, we will focus on the pricing of a CMASW-let, or

ASW-let, paying, up to a coverage factor, at Tp the asset swap spreadfixed at T0 on a bond with price process Bt.

3 Definitions

Let T0 < T1 < < Tn a payment schedule of a bond, issued (orwith a coupon period starting) at T0, whose price process is (Bt)t0.

If

Brisk-freett0

is the price process of the risk-free version of the same

bond, the forward asset swap spread is defined t T0 as

Raswt T0 Brisk-freet Bt

At(1)

The process At is the risk-free annuity which we recall is defined as

At At(T1, . . . , T n) =ni=1

yf(Ti1, Ti)Dt Ti

Dt Ti is the price of the risk-free zero coupon bond expiring at Ti andyf(Ti1, Ti) the year fractions. The year fractions of the bond may dif-fer from those of the asset swap but we ignore this immaterial flexibility.Other technical introductions to ASW can be found in [7] and [11].

Expression (1) tacitly embeds the assumption that there is a risk-free (or less risky) discount curve that acts as a benchmark for the bondin hand (which also translates into a convenient Raswt T

0

> 0 t T0).In other words, the asset swap spread is the compensation, paid tothe investor over the residual life of the bond, for purchasing a (socalled discount) bond at par. We note that definition (1) generalizesthe concept of ASW in a variety of ways:

The ASW package may be forward starting, namely T0 > 0. Thereference instrument may be a bond issued at T0 or a residualbond with a coupon period starting at T0.

Brisk-freet may be replaced by any other bond acting as a bench-mark to our reference bond. Admittedly, we have made no refer-ence to the degree of similarity between the two bonds. Hence,in the ASW package the exchanged flows may not match the

structure of the reference bond as market practice prescribes. A credit event on the reference bond does not invalidate the for-

ward ASW package and the ASW spread definition. In otherwords, after the credit event the process Bt may well, literally,default to (a potentially even null) recovery value accruing atcash account rate.

Regardless of the above considerations, we will happily take ad-vantage of (1), as it makes forward ASW process a martingale underthe swaption measure so that modelling an ASW spread may be asstraightforward as for a forward swap rate. Yet, in choosing the right

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assumptions one must bear in mind that, while the positivity is a rea-

sonable assumption for a process describing the forward swap rate, forthe asset swap spread it may not be a sure bet. Indeed we recall thatasset swap spread may be negative (for the so called premium bonds).This used to be the case for treasury bonds, reflecting the, presentlynot so fashionable, situation where the interbank deposit may be per-ceived less risk-free than treasury.

Being T0 fixed throughout the rest of this note, we will drop thepostfix and write Raswt for R

aswt T0

.

4 Pricing Framework

Let Tp T0 a payment date (usually Tp = T1). Because the ASW-

let pays at time Tp an amount proportional to RaswT0 , we restrict tocomputing the normalized arbitrage-free price

N0EN0

RaswT0 DT0 Tp/NT0

Nt is the price of the numeraire and the expected value is under itsassociated martingale measure. Equivalently, and consistently withthe conventions used for CMS products, we will describe the pricingmethodology focusing on the ASW convexity correction defined as

CC N0

D0TpEN0

RaswT0 DT0 Tp/NT0

Rasw0

Thanks to (1), in what follows we pick Nt = At so that Raswt T0 is a

martingale under the corresponding martingale measure, the swaptionmeasure.

In the following we will make further model assumptions and ap-proximations to obtain pricing formulas. To this aim, we pick a simplemodel that allows for a closed-form solution.

5 One-Factor Linear Model

We assume that the ratio DT0 U/AT0 depends on the swap rate onlyfor all the relevant maturities, i.e. U {Tp, T1, . . . , T n}

DT0 U/AT0 = GU

Rswp

T0

for some functions GU of the swap rate Rswpt starting at T0 and with

tenor structure T1, . . . , T n. An assumption of this type is used in [12]for CMS products.

In order to specify GU, we use an approximation described in [8,The linear swap-rate model], used and justified in [4], [13] and [5] (andfurther in [3] for an application to constant maturity CDS), consistingin the choice of a linear function

GU(x) = + Ux (2)

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Under the linear swap-rate model the convexity adjustment reads

CC =A0

D0TpEN0

RaswT0 GU

RswpT0

Rasw0 (3)

=A0

D0TpEN0

RaswT0

+ TpR

swpT0

Rasw0

=

A0

D0Tp 1

Rasw0 +

A0TpD0Tp

EN0

RaswT0 R

swpT0

(4)

The unknowns and U can be calibrated ([8, 13.3.3 The linearswap-rate model]) as

=1

ni=1 yf(Ti1, Ti)

(5)

U =D0U/A0

Rswp0(6)

We recall the proof:Proof. First, the equality

1 =ni=1

yf(Ti1, Ti)D0Ti

A0

=

ni=1

yf(Ti1, Ti)

+ Rswp0

ni=1

yf(Ti1, Ti)Ti

must be true for Rswp0 , hence for Rswp0 0, leading to (5).Secondly, from the linearity and martingale property

+ URswp0 = D0U/A0

Proposition 1 (Linear Swap ASW Convexity Correction) Underthe assumptions above

CC =

1

A0

D0Tp

EN0

RaswT0 R

swpT0

Rswp0

Rasw0

(7)

with given by equation (5).

Proof. From equalities (4) and (6)

CC =

A0

D0Tp 1

Rasw0 +

A0TpD0Tp

EN0

RaswT0 R

swpT0

=

A0

D0Tp 1

Rasw0 +

A0D0Tp/A0

Rswp0

D0TpEN0

RaswT0 R

swpT0

=

A0

D0Tp 1

Rasw0 +

D0Tp A0

Rswp0 D0TpEN0

RaswT0 R

swpT0

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which yields the result.

If we model our underlyings, under the swaption measure, as dis-placed lognormal (see [9])

dRswptRswpt + aswp

= swpdWswpt (8)

dRaswtRaswt + aasw

= aswdWaswt (9)

with instantaneous correlation

= d Wswp, Waswt /dt

we obtain

EN0 R

swpT0

RaswT0 = (Rswp0 + aswp) (R

asw0 + aasw)exp(swpasw T0)

aswp (Rasw0 + aasw) aasw (R

swp0 + aswp)

+ aswpaasw

so that formula (7) can be readily implemented.

Remark 2 Care is required when using the displaced diffusion setting.According to [9], the following assumptions need be met: Either

Rasw0 > aasw

or0 < Rasw0 < aasw

Likewise for the forward swap rate,

Rswp0 > aswp

or0 < Rswp0 < aswp

For each underlying, the former is the DL lognormal assumption, thelatter the DL anti-lognormal. The points of attainability are respec-tively, (a, ) and (, a).

We have already touched on the possibility for the ASW spreadof changing sign. This suggests that displaced diffusion may be areasonable choice.

For easy reference we point out the lognormal case:

Corollary 3 (Lognormal ASW Convexity Correction) Under theassumptions yielding formula (7), lognormal dynamics for the both theforward asset swap rate and the swap rate (i.e. equations (8), (9) withaswp = aasw = 0) and instantaneous correlation

CC = Rasw0

1

A0

D0Tp

(exp[swpasw T0] 1)

with given by equation (5).

Note, equation (7) may be generalized to a multi-curve frameworkwhere the forward swap rate is specified independently of the discount-ing/funding factors.

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6 Hedging Considerations

Hedging a short position in an ASW-let requires trading in the under-lying forward asset swap package, which in turn embeds a forwardtrading on the underlying bond. Forward repo markets on bonds arerarely liquid but for issuer of a sizeable and articulate debt structuresuch as main sovereign entities (but also for top-tier financial institu-tions), forward positions on bonds may be synthesized via zero curveestimation or bootstrapping. Extended literature is available on thesubject ([2], [10], [1]). Moreover, for relevant issuers, such as France,Italy, Belgium and Germany a viable solution is the replication bymeans of coupon stripping.

7 Numerical ExampleGiven the ASW-let with schedule as in table 4 (and Tp = 08/02/2017),we report in table 2 the convexity corrections for a grid of forwardASW volatilities and correlations. The forward rates are shown intable 1. While the forward swap rate is consistent with the discountcurve table 5, the reported forward ASW level could well be that ofan Italian treasury at the present time. The volatility of the forward

Table 1: Spread levels

ASW 490bp

SWP 4.29%

swap rate is set to 30%, in line with current average levels. We seethat the convexity correction is positive with positive correlation dueto the gamma cost effect inherent in a replication strategy based ontrading the asset swap package. As for the impact of smile, a result

Table 2: Convexity Corrections

-90.00% -50.00% 0.00% 50.00% 90.00%

asw

20.00% -0.09% -0.05% 0.00% 0.06% 0.12%30.00% -0.13% -0.08% 0.00% 0.10% 0.19%50.00% -0.19% -0.12% 0.00% 0.18% 0.37%

in [9] allows us to translate displaced diffusion coefficients into At-The-Money level and slope. We report in table 3 the sensitivity of theconvexity corrections to the ATM slope of ASW.

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8 Model Limitations

We point out some of the issues that may affect the pricing quality ofour approach:

The Linear Approximation. Assumption (2) may not be realisticfor bonds with long residual life. Other functional forms areavailable (see [12], formulas A.3, A.8 A.12a and A.15a) and a2-factor model could be required instead. Yet, such modelingassumptions will normally require numerical computation and,besides, the specification of extra parameters. Being an ASW-let a correlation product, these may hardly be calibrated to anyliquid market instruments at the present time.

Calibration of Volatility: The implied volatility of the forward

swap rate is listed. The specification of asw is both crucial(see table 2) and not straightforward at the present time as theare no liquid instruments. Historical estimation may be com-plemented via derivation of ASW volatility through bond optionquotes where available.

Skew/Smile Risk. Table 3 proves the impact of skew/smile tobe relevant. We suggest analyzing skew/smile scenarios via thedisplaced diffusion approximation. Note, the static replicationmethodology that is popular for CMS does not trivially translateto ASW-lets due to the presence, in the swaption measure pricingformula (3), of both RaswT0 and R

swpT0

.

Calibration of Correlation. Table 2 also shows the impact of

correlation. Unfortunately correlation is hardly listed for mostasset classes. In our case resorting to historical estimation isunavoidable.

9 Conclusions

Motivated by the need for market participants to hedge their exposureto sovereign asset swap levels, we have shown how to derive an ap-proximated closed-form solution for the pricing of constant maturityASW-lets taking into account the (cross-)gamma cost inherent in areplication strategy based on dynamically buying/selling the forwardasset swap package. The use of a displaced diffusion may come in

Table 3: Sensitivity to ASW Slope

-90.00% -50.00% 0.00% 50.00% 90.00%

asw20.00% 0.02% 0.01% 0.00% 0.01% 0.03%30.00% 0.05% 0.02% 0.00% 0.02% 0.08%50.00% 0.11% 0.04% 0.00% 0.07% 0.28%

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handy in a conservative evaluation of an ASW-let in the absence of

smile information on the forward ASW volatility.

References

[1] Estimating the term structure of interest rates. In Deutsche Bun-desbank Monthly Report. October 1997.

[2] Zero-coupon yield curves: technical documentation. In BIS Pa-pers, volume No 25. Bank For international Settlements, October2005.

[3] Li Anlong. Valuation of Swaps and Option on Costant MaturityCDS Spreads. The IUP Journal of Derivatives Market, April 2009.

[4] Pelsser Antoon. Mathematical Foundation of Convexity Correc-tion. Quantitative Finance, (3), 2001.

[5] W Boenkost and W Schmidt. Notes on convexity and quantoadjustments for interest rates and related options. volume No 47.Hochschule fur Bankwirtschaft, October 2003.

[6] Moorad Choudhry. The credit default swap basis: illustratingpositive and negative basis arbitrage trades. YieldCurve.com, July2006.

[7] Mark Davies and Dmitry Pugachevsky. Bond spreads as a proxyfor credit default swap spreads. Risk Magazine, 2005.

[8] P.J. Hunt and J.E. Kennedy. Financial Derivatives in Theory and

Practice. WILEY SERIES IN PROBABILITY AND STATIS-TICS, 2004.

[9] Roger Lee and Dan Wang. Displaced Lognormal Volatility Skews:Analysis and Applications to Stochastic Volatility Simulations.Annals of Finance, 2009.

[10] Charles R Nelson and Andrew F Siegel. Parsinomious modelingof yield curves. Journal of Business, 60, 1987.

[11] Zhou Richard. Bond Implied CDS Spread and CDS-Bond Basis.2008.

[12] Hagan Patrick S. Convexity Conundrums: Pricing CMS Swaps,Caps, and Floors. Wilmott Magazine, March 2003.

[13] Wendong Zheng and Yue Kuen Kwok. Convexity meets repli-cation: Hedging of swap derivatives and annuity options. TheJournal of Futures Markets, No 31, July 2010.

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Table 4: forward ASW schedule

start date 09/02/2016 coveragepayment date 08/02/2017 1payment date 08/02/2018 1payment date 08/02/2019 1payment date 08/02/2020 1payment date 07/02/2021 1

Table 5: Funding Curve

10/02/2011 1

11/02/2011 0.99997222315/08/2011 0.992911489

14/09/2011 0.99142324621/09/2011 0.99103460821/12/2011 0.98684803421/03/2012 0.98206684221/06/2012 0.97671729620/09/2012 0.97096015819/12/2012 0.96492610119/03/2013 0.95855982614/02/2014 0.93027163916/02/2015 0.8986979215/02/2016 0.865507577

14/02/2017 0.8316288614/02/2018 0.79802302714/02/2019 0.76497266314/02/2020 0.73267850515/02/2021 0.7008920414/02/2022 0.66974476414/02/2023 0.639325914

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