Government Bond Swaptions and How They Might Work

Opinion: Samy Ben Aoun, Christophe Viard and Pascal Ardelet

Government bond swaptions and how they might work

Payoffs based on bond yields instead of swap rates could offer new hedging tool, argue Crédit Agricole CIB execs

allable bonds are typically based on bank or agency debt. But options based on European government bonds are attracting investor interest, too. They can offer greater yield because the sovereign curve is steeper than the Euribor equivalent. CTaking this concept a step further, it’s possible to create cash-settled swaptions using French government bond (OAT) yields. The payoffs would be calculated using the French constant maturity treasury rate, known as TEC, instead of constant maturity swap rates. As an alternative to the standard Euribor swaption, these instruments could provide a new asset-liability management or trading tool. A government bond yield curve or swap curve depends on three main factors: rate expectations, risk premium, and convexity. The risk premium causes the yield to increase according to the maturity of a bond. The longer the maturity of a bond, the higher the interest rate risk, and the higher the yield must be to compensate for this risk. By construction, bond yields are convex, and this convexity increases with maturity. Long-dated bonds offer the holder interest rate protection thanks to their implicit convexity. This convexity takes the form of protection and/or opportunity, which implies that investors are ready to pay by giving up yield to acquire this convexity. Forward yield curve and link with repo rate Thus the risk premium factor is opposed to the convexity factor, A forward swap of zero value is a combination of spot swaps of zero value, resulting in a swap rate curve that peaks generally at around 20 years, for example the payer swap 10y10y corresponds to a payer swap 20y plus a beyond which convexity outweighs the risk premium. receiver swap 10y. This can also be expressed as borrowing at 20 years and However government bond yield curves increase from that point, which lending at 10 years. means that the risk premium linked to credit risk dominates the convexity From the relationship between forward swap and spot swap, we can Yield SwapRate CreditSpread contribution. deduce the relationship between forwardC swap rate and spot swap rate. A T T1 1 T2 T 2 f.TECT / f YieldT YieldT The government bond yield depends on rates and credit: forward swap rate is a linear combination of a long swap spot rate and a D T2 T1 1 C T2 T1 2 Â Ã short swap spot rate, for example: T T1 T2 T f g.P 1 / g.P 2 / Yield SwapRate CreditSpread T1 T2 C T T1 T2 T D T2 T1 C T2 T1 1 102 Â Ã f.TECT / f YieldT Yieldlevel T 10y10y .1 D!/20Ty2 !10T1 y; with1 C!T2 T1 2 D C Â D level20 levelÃ10 The level of the credit spread depends implicitly on the cost of protec- T T1 1 T2 T 2 tion of the bond. The cost of the protection that represents the credit risk f g.PT / g.PT / D T2 T1 1 C T2 T1 2 of a government bond increases with maturity and is reflectedlevel10 in the shape In the case of forward Âbond yields the logic is slightly different.Ã Unlike 10y10y .1 !/20y !10y; with ! of the yield curve:D theC higher the risk on a bond theD levelsteeper20 thelevel yield10 curve the calculation of forward rates, which uses short rates versus long rates of EQT ŒTECT !.k/EQT ŒLvlT .TECT /.k TECT / t B .0; t / repo D repo C will be. This effect will dominate the convexity contribution. the same curve, the calculationspotBond of0;T forwardCoupon yields usesi 1shortrepo repo spreadsi k6F0 fwdBondt;T D X The TEC fixings are deduced from French government bond prices versus long yields. TheD value of the forwardBrepo bond.0; t/ depends on the value of P !.k/EQT ŒLvlT .TECT /.k TECT / quoted on the secondary market. These fixings are calculated by linear the bond today and its refinancing cost for the forward period. For C repo C EQT ŒTECT !.k/EQT ŒLvlT .TECT /.k TECT / kX>F0 interpolation between the yields of the governmentt bondsB whose.0; tmaturity/ example,repo the valueD of a forward five-yearrepo OAT2041 can be derivedC from spotBond0;T Coupon i 1 repo i k6F0 frames fwdBondthe fixingt;T maturity (e.g. TEC10…). D the value of the bondX today and the repo cost of the bond for five years. D Brepo.0; t/ P T t !.k/EQT ŒLvlT .TECT /.k TECT / C Coupon repo 1 C k>F0 fwdBondt;T X k T t D .1 fwdYieldt;T / C .1 fwdYieldt;T / k 1 C 1 risk.net May 2021 XD C CapSpread TEC10 (1.5%,3%) T t maxŒ0 TEC10y 1:5% maxŒ0TEC10y 3% Coupon 1 D I fwdBondt;T k T t D .1 fwdYieldt;T / C .1 fwdYieldt;T / k 1 C t XD C CapSpread TEC10 (1.5%,3%) fwdYieldt;T .1 !/spotYield ! repoRate ; with ! Š C T t D T t maxŒ0 TEC10y 1:5% maxŒ0TEC10y 3% D I

t DurationAdjustedSwaption Payer (2%) fwdYieldt;T .1 !/spotYield ! repoRate ; with ! Š C T t D T t Lvl10.TEC10y/ 2 2 2 max 0 .TEC10y 2%/ Yield SwapRate CreditSpread SwapRate CreditSpread D I .2%/ D C C Ä Lvl10 DurationAdjustedSwaptionq Payer (2%) Lvl10.TEC10y/ 2 2 2 max 0 .TEC10y 2%/ Yield D SwapRate C CreditSpread C SwapRate CreditSpread D I Lvl10.2%/ Payoff maxŒ0; LvlÄ30.TEC30y/.TEC30y 1%/ q D

Payoff maxŒ0; Lvl30.TEC30y/.TEC30y 1%/ D 30 1 Lvl30.x/ D .1 x/i i 1 XD C 30 OAT5 5y30y payer 1% FP E 5y ŒLvl30.TEC30y/.TEC30y 1%/ 1 D Qrepo C Lvl30.x/ D .1 x/i i 1 XD C OAT5 5y30y payer 1% FP E 5y ŒLvl30.TEC30y/.TEC30y 1%/ D Qrepo C f.YieldT / '.PT / D F0 '.k/ ' .F0/.PT F0/ ' .k/.k PT / dk D C 0 C 00 Z0 f.YieldT / '.PT / D C1 ' .k/.PT k/dk F0 00 C F0 '.k/ ' .F0/.PT F0/ ' .k/.k PT / dk Z D C 0 C 00 Z0 C1 ' .k/.P k/dk C 00 T ZF0 F0 EQT ŒPT fwdBond0;T D repo D t P EQT ŒPT Ft T D repo j dP t F0 E T ŒPT fwdBond0;T T Qrepo EQ repo.t/ dt D D P t D t Ä T P EQT ŒPT Ft T D repo j t dPT EQ repo.t/ dt P t D 2 risk.net Ä T

2 risk.net Opinion: Samy Ben Aoun, Christophe Viard and Pascal Ardelet

swap rate, the volatilityYield of theSwapRate credit spreadCreditSpread and the correlation between 1 Spot and forward curve (as of March 12, 2021) C T T1 T2 T them. The credit component,Yield SwapRate whose levelCreditSpread reflects the credit risk and/or the f. / f 1 2 TECT T T YieldT1 T T YieldT2 C D T2 1T1 1 C T22 T1 2 cost of the protection, can act as an accelerator of bond yield volatility or, f.TECT / f Yield Yield D ÂT T T1 C T T T2 Ã 2T T11 2T2 1T on the contrary, slow down bond yield volatility. Indeed, there are market fÂ g.P 1 / g.P 2Ã/ T T1 T1 T2 T T2 scenarios where the volatility of government yields is higher than the Df T2 Tg.P1 1 / C T2 Tg.P1 2 / level10 D ÂT T T1 C T T T2 Ã volatility10y of10 swapy .1 rates !/20and vicey versa,!10y ;dependingwith ! on the correlation Â 2 1 2 1 Ã D C D levellevel20 10level10 10y10y .1 !/20y !10y; with ! between creditD andC interest rates. D level20 level10 Let us assume that the credit risk increases, then two opposite scenarios can be envisaged, theYield first beingSwapRate the searchCreditSpread for duration and convexity C EQT ŒTECT !.k/ET TQ1T ŒLvl1 T .TECT2T /.kT TEC2 T / (e.g.: 2008, Covid) the second being the excess andt therefore the sale of repo f. D/ f repo C spotBond Coupon Brepo.0; ti / E T Œ TEC T !.k/E T YieldŒ T1. /.k YieldT2 / 0;T t i 1 Q TECT kD6F0 T2 QT1 LvlT TECC T2T T1 TECT durationfwdBond (e.g.: 2015).t;T DB .0; t / repo D X Â repo ÃC spotBond0;T Coupon i 1 repo i k6F0 d D Brepo.0; t/ X T T1 T2 T fwdBondt;T PD !.k/EQT1 ŒLvlT .TECT /.k 2 TECT / D Brepo.0; t/ Cf g.PrepoT / g.PT/ C P !.k/E T 1Œ . /.k 2 / A orrdYield cre SwapRate s orrdCreditSpread cre sot cre OAT swaption: definition and construction D k>TF20 T1 Q LvlCTTTEC2 TT1 TECT C T T1 1 T2 T level2 10 C ÂX repo Ã C A sot cre 10f.10 /.1 f!/20 !10 ; ! k>F0 yTECy T y Yieldy T1with YieldT2 X Yield SwapRate CreditSpread Based on the valueD D ofC spotT 2governmentT1 bonds,C T a2 repoD Tlevel1 curve20 andlevel an10 Source: Authors Â Ã C assumption of the implied volatilityT T1 of government1 T2 yieldsT compared2 to the f.TECT /T tf T T1 Yield1 T T2 T Yield2 T Yield SwapRate CreditSpread Df T2 CouponTg.P1 / 1 C T2 Tg.P1 1 / 2 implied volatility Tof swapt rates,T itT is possibleT1 to valueT bondT optionsT2 and C fwdBondt;T D ÂT2 CouponT11 1 Ck T2 2 T1 1 2 Ã T t level10 f.TECTD/ fÂ.1 TfwdYieldT1 Yieldt;TT/ C T.12 TfwdYieldYieldTÃt;T / 10 10 .1 !/20 !10 ; ! OATfwdBond swaptions.t;T k 1 T T 1k1 T CT 22 T t y y y y with D XDD.1f C2 1 g.PT/ /CC .12 1 g.PT // E TCapSpreadŒTECT TEC10 (1.5%,3%)!.k/E T ŒLvlT .TECT /.k TECT / D C D Â fwdYield t;T 1 t fwdYieldt;T2 Ã Qrepo Qrepo This relationship can also be written as a loan to a levelgovernment20 level for10 20 years The OAT swaptionk D1 usesC Tthe2 sameT1 cash settlementC T2C Tmethod1B .0;as a t Euribor/ CapSpread TEC10D (1.5%,3%) C level 10 XD spotBondÂ T T0;T1 Coupon1 T2 i T1 repo2 Ãi k6F0 10 10 .1 !/20 !10 ; ! fwdBondt;T f g.P / D g.P / maxXŒ0 TEC10y 1:5% maxŒ0TEC10y 3% and a cashy borrowingy for fivey years securedy with by the government price. Thus swaption, where the payoff is based on Tan1 annuity which representsT2 the D I D C D level20 level10 DD T2 T1 BrepoC.0;TP2 t/ T1 maxŒ0 TEC10y 1:5% maxŒ0TEC10y 3% we can write the value in t of a bond of maturity T as beinglevel equal10 to: cashflows that would haveÂ been received if the swap had been deliveredÃ at D I !.k/E QT ŒLvlT .TECT /.k TECT / 10y10y .1 !/20y !10y; with ! t C repo C k>F0 D C D level20 level10 Eexpiry.fwdYieldT Œ TECThet;T Tannuity .1 represents!/!.k/EspotYield theT differenceŒLvl!TrepoRate.TEC betweenT /.k; thewithTEC amount!T / that is X t Qrepo QrepoT t t B .0; t / ŠD.1 C!/ ! ; ! DCT t spotBond0;T Coupon i 1 repo i fwdYieldpayable byt;T the fixed-ratek6F0 spotYield payer usingT therepoRate fixed ratet setwith when the swaption fwdBondt;T D Š CX D T t B .0; t/ EQT ŒTECT !.k/EQT ŒLvlT .TECT /.k TECT / D repo P t was struck,repo and theD Tamountt payablerepo by the fixed-rate payer using theC spotBond0;T Coupon i 1 Brepo.0; ti / k F !.k/EQT ŒLvlT .TECT /.k 1 TECT / 6 0 Couponrepo C fwdBondt;T D settlementE T Œ rate at theC Xtime!.k/E of expiry.T Œ . /.k / DurationAdjustedSwaption Payer (2%) QfwdBondTECt;TT k>F0 Q LvlTk TECT TECT T t D Brepo.0; t/t repo DD X.1 fwdYieldrepo t;T / C .1 fwdYield t;T /C DurationAdjustedSwaption Payer (2%) From the forward bondspotBond prices,0;T we canCoupon deduce Pthei forward1 Brepo bond.0; ti /yield In Euribor swaptions,kk 1F the settlement!.k/EQ rateT isŒLvl theT Ice.TEC swapT /.k rate, whichTECT / . / XX6 0 C repo C CapSpread TEC10 (1.5%,3%)Lvl10 TEC10y fwdBondt;T D 2 DC 2 2 C max 0 .TEC10y 2%/ associated with a security.D We define theB forwardrepo.0; t/yield in t of maturity T as representsYield the marketSwapRate mid-pricek>FCreditSpread0 for the fixed leg ofSwapRate a new interestCreditSpread rate D LvlI 10.TEC10y.2%// P D 2 CX2 !.k/E T CŒLvlT .TECT /.k TECT / Œ0max 0 Lvl1:510 .TEC10yŒ0 2%/3 T t Yield Qrepo 2SwapRateCreditSpread max TEC10yÄ % max TEC10y % follows: 1 swap. DAn OATqSwapRate swaptionCC insteadCreditSpread uses the TECC as the settlement rate. C D D I I Lvl 10.2%/ Coupon k>F0 Ä fwdBondt;T Yield SwapRate CreditSpreadk T t An OATq swaption 5y30yX payer 1% pays the following cashflow at D T .1t fwdYieldt;TC/ C .1 fwdYieldt;T / T T1 T2 T t k 1 C Coupon C 1 maturityCapSpreadf. in five TEC10 years:.1/ (1.5%,3%)f!/ !1 ; 2! XD fwdYieldt;TTECT spotYieldTYieldT1repoRatet Yieldwith T2 fwdBondt;T k T t Š DC T2 T1 C T2 T1 D T t D T t .1 fwdYieldt;T / C .1 fwdYieldt;T / ÂŒ0; . 30 /. 30 1 /Ã k1 Coupon C 1 PayoffmaxmaxŒ0 TEC10yLvl 30 TEC1:5% y maxTECŒ0TEC10yy % 3% XD C CapSpreadD TEC10D I (1.5%,3%)T T1 1 T2 T 2 fwdBondt;T k T t Payoff maxf Œ0; Lvl30.TECg.P30y//.TEC30y g.P1%/ / D .1 fwdYieldt;T / C .1 fwdYieldt;T / D T1 T2 We observe that thek 1forwardC bond yield depends Cmainly on the spot maxD Œ0 TTEC10y2 T1 1:5%C Tmax2 Œ0T1TEC10y 3% DurationAdjustedSwaption Payer (2%) XD level10 t CapSpreadD TEC10 (1.5%,3%)ÂI Ã valuefwdYield 10of ythet;T10 ybond.1.1 and!/ !/20thespotYield repoy rate.!10T yIn; ! therepoRatewith case! of ta; zero-couponwith ! bond, we With ŠD CC D level20 levelD T10 t maxŒ0 TEC10y 1:5% maxŒ0TEC10y 3% Lvl10.TEC10y/ t 2D I 2 2 max 0 .TEC10y 2%/ can derive the following relationship: Yield SwapRate CreditSpread30 SwapRate CreditSpread D I .2%/ fwdYieldt;T .1 !/spotYieldT ! repoRatet ; with ! D C C1 Ä Lvl10 Š C D T t q .x/ 30 t Lvl30 1 i fwdYieldt;T .1 !/spotYield ! repoRate ; with ! DurationAdjustedSwaption.x/ PayerD (2%).1 x/ Š C T t D T t Lvl30 i 1 C i EQT ŒTECT !.k/EDQT XDŒLvl.1 T .TECx/ T /.k TECT / t repo D Lvlrepoi 101.TEC10yC / C 2 2 B .0; ti / DurationAdjustedSwaption0E Payer5Xy Œ (2%)30. . /. 2 / 1%/ spotBond 0;T Coupon2 i 1 repo OAT5 5y30y payerk 1%6Fmax FP0 Q D Lvl TEC30yTEC10yTEC30y% Yield SwapRate CreditSpread SwapRateD CreditSpread XD D I repo 10.2%/ C TheDfwdBond forward t;Tyield is Cproportional to theC spot yield and inversely OAT5 5y30yPayoff payer 1%max FP Œ0;ELvl530y .LvlTECŒLvl3030.yTEC30y/.TEC30/.yTEC30y1%/ 1%/ D Brepo.0; t/ Ä QrepoLvl10.TEC10y/ q 2 2 P DurationAdjustedSwaptionD D !.k/E0 Payer (2%)T ŒLvlT .TEC. T/.k TEC2 /T /C proportional to the repo rate. The dependence2 of the forward yield on the max Qrepo TEC10y % Yield D SwapRate C CreditSpread C SwapRate CreditSpread CD I Lvl10.2%/ C repo rate is greater the longer the forward period. The shape of the forward We use the Q-Repo-T-forwardkX>F0 Ä probabilityLvl10.TEC10y assuming/ the trade is q2 2 2 max 0 .TEC10y 2%/ Yield SwapRate CreditSpread SwapRate CreditSpread D I .2%/ yield curveD therefore increasesC (repo rates areC generally lower than bond collateralised by bonds. Ä Lvl10 Payoffq maxŒ0; Lvl30.TEC30y/.TEC30y 1%/ yields) whereas theD Tswapt rate curve admits a maximum. f.TheYield OATT / swaption'.PT /(payer/receiver)30 with strikek can be seen as the cash Coupon 1 f. / D'.P / 1 We can deduce that forward bonds’ carry will be higher than the settlementYieldT version ofT the option30.x/ (put/call) on a syntheticF0 bond with fwdBondPayofft;T maxŒ0; Lvl30.TEC30ky/.TEC30y 1%/ T t D Lvl i D D .1 fwdYieldt;T / C .1 fwdYield t;T / D .1 x/ equivalent forward kswaps’1 carry. The differential betweenC repo spreads and coupon k and strike'.k/ 100%.'0.F0/.PT i 1F0/ C F0 '00.k/.k PT / dk XD C CapSpreadD TEC10C (1.5%,3%) XD C 0 bond spreads,Payoff which createsmaxŒ0; steepeningLvl30.TEC in30 they/. forwardTEC30 ycurves,1% /accentuates From the bijective'.k/ relationship'0.F0/.P TbetweenF0/ the priceZ ' of00 .k/.ka governmentPT / d bondk D OAT5 5y30yD payer 1%C FP E 5y ŒLvl30C.TEC30y0 /.TEC30y 1%/ 30 maxŒ0C1TEC10yD Qrepo1:5% ZmaxŒ0TEC10y 3% C this relationship. 1 and the governmentD yieldI (Yield'00.k/.PT = gT(PT)),k/ onedk can evaluate any payoff .x/ C C1F Lvl30 i that is written as a functionZ 0 ' of00.k/.P the governmentT k/dk yield from the bond price, D 30.1 x/ t C F Volatility of bond yield vs volatilityi 1 C 1of swap rates Z 0 fwdYieldt;T .1 !/spotYield.x/ TXD ! repoRatet ; with ! formally: Š C Lvl30 i D T t OAT5From 5y30ythe formulation payer 1% FPof the Ebond5yD yieldŒLvl30 30as.1 .aTEC30y functionx/ /. ofTEC30y the swap rate1% /and D Qrepo i 1 C1 C .x/ XD the credit spread, we canLvl derive30 a relationship betweeni the volatility of f.YieldT / '.PT / E D5 Œ .1 . x/ /. 1%/ OAT5 5y30y payer 1% FP Q y i Lvl1 30 TEC30y TEC30y D bond yields and the volatilityD of swaprepoX rates. C C DurationAdjustedSwaptionF0 EQT PayerŒPT (2%)fwdBondF00;T D D repo D E 5 Œ . /. 1%/ '.k/F0 E'Q.FT 0/.PŒPTT FfwdBond0/ 0;T' .k/.k PT / dk OAT5 5y30y payer 1% FP Q y Lvl30 TEC30y TEC30y D 0t repo Lvl10D.TEC10y/ 00 2 D2 repo C D CmaxPT 0EQT ŒPT CFt0 . 2%/ Yield 2SwapRateCreditSpread t D repo j Z TEC10y SwapRate CreditSpread DP E I T LvlŒP10T .2F%t/ f. D / '.P / C C T Ä Qtrepo YieldTq T C1D'dP.k/.P k/j k D 00 t T T d F C FEQ P repo.t/ dt We observe that the volatility of a bond depends0 on the volatility of the Z 0 d PTt D f.YieldT / '.PT / EQ T repo.t/ dt D'.k/ '0.F0/.PT F0/ '00.k/.k PT / dk Ä t D C C 0 PT D Z F0 Ä f.YieldT / '.PT / Payoff'.k/max'Œ0;.FLvl0/.P30.TECF300/y/.TEC30'y .k/.k1%/ P / dk D D C1 0 T 00 T D C '00.k/.PT k/dCk 0F0 risk.net 2 C F Z 2 risk.net '.k/Z 0 '0.F0/.PT F0/ '00.k/.k PT / dk D CC1 C 2 risk.net F0 EQT ŒPT fwdBond0;T '00.k/.PT k/dZk0 D repo D C F0 t Z C1 P E T ŒPT Ft ' .k/.P k/dk T Qrepo C 00 30T D j ZF0 1 P t Lvl30.x/ d T F0 E T ŒPDT fwdBondi0;T EQ repo.t/ dt Qrepo .1 x/ t D i D1 C PT D t XD Ä F0 P E TE 5ŒPT ŒP T Ft OAT5 5y30y payer 1% FPT QE Qrepoy TŒLvl30fwdBond.TEC30y0;T/.TEC30y 1%/ D DD repoQrepo D j C tdP t F0 PETQT TEŒPQT ŒPfwdBondT Ft 0;T EDQ Drepot reporepoD .t/j dt 2 risk.net t PT t D PÄT dPETQT ŒPT Ft EQ D repo repoj.t/ dt P t D dPTt f. / '.P / Ä T YieldT T EQ t repo.t/ dt 2 risk.netD P D Ä T F0 '.k/ '0.F0/.PT F0/ '00.k/.k PT / dk 2 risk.netD C C Z0 2 risk.net C1 ' .k/.P k/dk C 00 T ZF0

F0 EQT ŒPT fwdBond0;T D repo D t P EQT ŒPT Ft T D repo j t dPT EQ repo.t/ dt P t D Ä T

2 risk.net Yield SwapRate CreditSpread C T T1 1 T2 T 2 f.TECT / f YieldT YieldT D T2 T1 1 C T2 T1 2 Â Ã T T1 1 T2 T 2 f g.PT / g.PT / D T2 T1 1 C T2 T1 2 level10 Â Ã 10y10y .1 !/20y !10y; with ! D C D level20 level10

EQT ŒTECT !.k/EQT ŒLvlT .TECT /.k TECT / t B .0; t / repo D repo C spotBond0;T Coupon i 1 repo i k6F0 fwdBondt;T D X D Brepo.0; t/ P !.k/EQT ŒLvlT .TECT /.k TECT / C repo C kX>F0

T t Coupon 1 fwdBondt;T k T t D .1 fwdYieldt;T / C .1 fwdYieldt;T / k 1 C XD C CapSpread TEC10 (1.5%,3%) maxŒ0 TEC10y 1:5% maxŒ0TEC10y 3% D I t fwdYieldt;T .1 !/spotYield ! repoRate ; with ! Š C T t D T t

DurationAdjustedSwaption Payer (2%) Lvl10.TEC10y/ 2 2 2 max 0 .TEC10y 2%/ Yield SwapRate CreditSpread SwapRate CreditSpread D I .2%/ D C C Ä Lvl10 q

Payoff maxŒ0; Lvl30.TEC30y/.TEC30y 1%/ D

30 1 Lvl30.x/ D .1 x/i i 1 XD C OAT5 5y30y payer 1% FP E 5y ŒLvl30.TEC30y/.TEC30y 1%/ D Qrepo C

Opinion: Samy Ben Aoun, Christophe Viard and Pascal Ardelet

f.YieldT / '.PT / D F0 '.k/ ' .F0/.PT F0/ ' .k/.k PT / dk D C 0 C 00 Z0 C1 ' .k/.P k/dk C 00 T ZF0

With 2 Payoff for 10y

F0 EQT ŒPT fwdBond0;T istorical realisation since 2010 D repo D t P EQT ŒPT Ft T D repo j t dPT EQ repo.t/ dt P t D Ä T Therefore, we can determine the value of any TEC-indexed payoff from 2the bond risk.net option price using the following formulation. Yield SwapRate CreditSpread C T T1 1 T2 T 2 f.TECT / f YieldT YieldT D T2 T1 1 C T2 T1 2 Â Ã T T1 1 T2 T 2 f g.PT / g.PT / C sred C C sred C D T2 T1 1 C T2 T1 2 level10 Â Ã 10y10y .1 !/20y !10y; with ! Source: Authors D C D level20 level10 It is possible to implement a forward grid and volatility matrix in the same way as for Euribor swaptions. Thus, OAT swaptions are constructed from the information contained in the repo market and in the bond EQT ŒTECT !.k/EQT ŒLvlT .TECT /.k TECT / 3 Payoff difference t optionsrepo market. D repo C YieldspotBondSwapRate0;T CouponCreditSpreadi 1 Brepo.0; ti / k F C X6 0 T T1 T2 T fwdBondt;T D f. / f 1 2 DYield SwapRate BrepoCreditSpread.0; t/ TECT YieldT1 YieldT2 C P OAT swaption: ALMD applicationT2!.k/ETT11 T andŒLvlC TTtactical.22TECTT1 /.k strategyTECT / Â Qrepo1 2 Ã f.TECT / Cf YieldT YieldT C A first application consistsD k>T FinT0 simplyT1 valuing1 C a TcashflowT2 TT that 2pays the TEC. f X2 1 g.P 1 / 2 1 g.P 2 / Â T1 T2 Ã To do so, it is sufficientD toT 2applyTT 11the same logicC T 22andT mechanics1 of Yield SwapRate CreditSpread level10 f Â g.P 1 / g.P 2 / Ã construction as the one used to value constantT1 maturity swapsT2 (CMSs). 10y10y .1 !/20 y !10y;C with ! D TT2 TT11 1 C TT22 TT1 2 D TCt D 20 10 f. / f Coupon levellevel1 10level TECT Â YieldT1 YieldT2 Ã 10y10y .1 !/20y !10y; with ! We therefore evaluateD the TECT2 byT1 static replicationC T2 fromT1 the OAT payer fwdBondt;T k T t Â Ã DD C .1 fwdYield t;T / C .1D levelfwdYield20 levelt;T /10 and receiver swaptions: k 1 C T T1 1 T2 T 2 XD C CapSpread TEC10f (1.5%,3%) g.P / g.P / D T T T1 C T T T2 Â 2 1 2 1 Ã level10 EQT ŒTECT maxŒ0 TEC10y!.k/E QT 1:5ŒLvl%T .TECmaxTŒ0/.kTEC10yTECT3/% 10y10y .1 !/20y !10y; with ! t repo D D I repo C D C spotBond 0;T CouponD i 1 Brepo.0; ti / level20 level10 E T Œ k6F0!.k/E T Œ . /.k / fwdBondt;T D Q TECT X Q LvlT TECT TECT D B .0; t/t B .0; t / t repo D repo C spotBond0;T Couponrepo Pi 1 repo i k6F0 fwdYieldt;T .1 !/spotYield ! repoRate ;D with ! X !.k/EQT ŒLvlT .TECT /.k TECT / fwdBondt;T T t C repo C Š CD Brepo.0; t/ D T t k F P X> 0!.k/EQT ŒLvlT .TECT /.k TECT / C repo C EQT ŒTECT k!.k/EF QT ŒLvlT .TECT /.k TECT / t B .0; t / repo D X> 0 repo C C strddle s C strddle spotBond0;T Coupon i 1 repo i Two popular payoffsk6F 0in the 2000s indexed to CMS, the CMS10 cap fwdBondt;T T t D DurationAdjustedSwaptionX Payer (2%) D Coupon Brepo.0; t/ 1 spread and the duration-adjusted swaption, can be considered with a TEC Source: Authors P !.k/ELvl10Q.TTEC10yŒLvlT /.TECT /.k TECT / fwdBondt;T2 T t 2 k T t C max 0 repo . 2%/ C Yield D .1 CouponfwdYieldt;T / 2C .1SwapRatefwdYield1CreditSpreadt;T / indexation. k F TEC10y D SwapRatek 1 C CCreditSpread C C D X> 0 I Lvl10.2%/ fwdBondt;T XD k T t TheCapSpread TEC cap TEC10 spread (1.5%,3%)(staticallyÄ replicated by OAT swaptions) and the q D .1 fwdYieldt;T / C .1 fwdYieldt;T / k 1 C XD C duration-adjustedCapSpread TEC10 swaptionmaxŒ0 (1.5%,3%)TEC10y payer are used1:5% in anmax asset-liabilityŒ0TEC10y manage3%- Finally, there is a tactical application that takes advantage of levels of T t D I Coupon 1 ment framework tomax hedgeŒ0 TEC10ythe risk of lapse.1:5% maxŒ0TEC10y 3% OAT volatility and Euribor volatility. The trade consists of buying/selling D I fwdBondt;T k T tt an OAT straddle swaption against a Euribor straddle swaption, which PayoffD max.1Œ0; LvlfwdYield30.TECt;T30/y/.CTEC.130yfwdYield1%/t;T / fwdYieldt;T .1k 1!/spotYieldT ! repoRateCt ; with ! Š DXDC C D Tt t CapSpread TEC10 (1.5%,3%) makes it possible to take a volatility spread option while limiting the rate fwdYieldt;T .1 !/spotYieldT ! repoRatet ; with ! Š C D T t maxŒ0 TEC10y 1:5% maxŒ0TEC10y 3% exposure. Here, a straddle is a payer swaption combined with a receiver D I swaption. DurationAdjustedSwaption Payer (2%) Over a 10-year history, the difference between the payoff of a 10y30y 30 t fwdYieldt;T .1 !/spotYieldT ! repoRate1 t ; with ! DurationAdjustedSwaption PayerLvl10 (2%).TEC10y/ Š2 C .x/2 2 D T t max 0 .TEC10y 2%/ TEC straddle 0% versus CMS is above 7%. Yield SwapRateLvl30 CreditSpread i SwapRate CreditSpread D I .2%/ D C D .1 C x/ Ä Lvl10Lvl.10TEC10y/ In summary, OAT swaptions using TEC fixings can offer investors an q2 2 i 1 C2 max 0 .TEC10y 2%/ Yield SwapRate CreditSpreadXD SwapRate CreditSpread D I .2%/ D C C Ä Lvl10 alternative to Euribor cash settlement swaptions. TEC-indexed payoffs OAT5 5y30yq payer 1% FP E 5y ŒLvl30.TEC30y/.TEC30y 1%/ DurationAdjustedSwaption Payer (2%) D Qrepo C such as the TEC10 cap spread can provide an asset-liability management Over a 10-year history, the differenceLvl10. TEC10ybetween /the payoff of a 10y cap tool, constructed by static replication from OAT swaptions. 2 Œ0;2 . 30 /.2 30 1 / max 0 .TEC10y 2%/ Yield PayoffSwapRatemax CreditSpreadLvl30 TEC y TECSwapRatey %CreditSpread D I .2%/ D D C C spread on TEC10y 1%/1.5% Äversus CMS10yLvl10 1%/1.5% is above 1%. The concept could also be applied beyond the French markets, to other qPayoff maxŒ0; Lvl30.TEC30y/.TEC30y 1%/ D Another structural application concerns the downside risk (or duration swaption fixings and government bond yield curves.■ requirement). The steeper shape of the government bond yield curve f.YieldT / '.PT / D implies cheaper receiver swaptions on the OAT curve than on the Euribor Samy Ben Aoun is global head of structured trading, Christophe Viard is head F0 Payoff maxŒ0; Lvl30.TEC30 30y/.TEC30y 1%/ '.k/D '0.F0/.PT F0/ 1 '00.k/.k PT / dk curve, i.e. the downward indexation on the OAT curve has a positive carry. of structuring and product development in Paris, and Pascal Ardelet is head of D C Lvl30.x/ 30 C D .1 1Zx/0 i This assumes a comparable level of volatility between the curves. France and Benelux financial institution sales, all at Crédit Agricole CIB. .x/ i 1 C C1Lvl30 XD i '00.k/.PD T .1k/dkx/ OAT5 5y30y payerC 1% FP E 5y i ŒLvl1 30C.TEC30y/.TEC30y 1%/ ZF0 D QrepoXD C OAT5 5y30y payer 1% FP E 5y 30ŒLvl30.TEC30y/.TEC30y 1%/ 3 risk.net May 2021 D Qrepo 1 C Lvl30.x/ D .1 x/i i 1 XD C OAT5 5y30y payer 1% FP E 5y ŒLvl30.TEC30y/.TEC30y 1%/ F0 E T QŒPrepoT fwdBond0;T f.YieldT / '.PT / DQrepo C D D D t F0 f.YieldT / '.PT / P EQT ŒPT Ft D '.k/ 'T .FD /.Prepo F /j ' .k/.k P / k 0 0 T 0 F 00 T d D C P t C 0 0 '.k/ ' .Fd/.PT F / Z ' .k/.k P / dk EQ0 0 T repo0 .t/ dt 00 T D C C1 P t D C 0 f.YieldT / '.PT / Ä'00T.k/.P T k/dZk D C FC10 Z ' .k/.P k/dk F0 C 00 T '.k/ F0' .F0/.PT F0/ ' .k/.k PT / dk D ZC 0 C 00 2 risk.net Z0 C1 '00.k/.PT k/dk CF0F0 EQT ŒPT fwdBond0;T Z D repo D F0 EtQT ŒPT fwdBond0;T P repoE T ŒPT Ft D T QrepoD t D j PT EQt T ŒPT Ft DdPT repo j EQ t repo.t/ dt F0 E TdP ŒPT D fwdBond0;T QÄrepoT DEQ t Drepo.t/ dt t PT D P Ä EQT ŒPT Ft T D repo j t 2 risk.net dPT EQ repo.t/ dt P t D 2 risk.net Ä T

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