Hull & Whie onvexiy Adjusmens for redi Riskless Ineres Rae Swaps Under SA Denis Papaioannou Senior Quaniaive onsulan, Hiram inance, e-mail: denis@hiram-finance.com Meriem houqi Junior Quaniaive onsulan, MargOconseil Benjamin Giardina Junior Quaniaive onsulan, Hiram inance Absrac Using a mulicurve pricing framework has become sandard marke pracice for invesmen banks. A new IBOR curve boosrapping procedure is now in use, consising of discouning using an OIS curve. The curve obained allows us o recover marke forwards corresponding o a SA world which is he inerbank world. urrenly, his curve is being used o calculae forwards for non-sa rades, which represens an imporan approximaion since i does no ake ino accoun he convexiy adjusmen implied when changing from SA o non-sa probabiliy measure. By reducing counerpary risk using SA conracs, new risk facors have arisen ha are lef unhandled. We show in his aricle how o calculae non-sa forwards convexiy adjusmen. This adjusmen depends on collaeral and funding raes volailiies as well as correlaion beween boh. We ake ino accoun hese parameers under one-facor Gaussian shor-rae models, and give a deailed developmen for he Hull & Whie specific case. These closed formulae allow in urn fas boosrapping procedures and herefore poenial risk managemen for non-sa swaps. Keywords SA onvexiy, IRS, OIS-IBOR dynamics, Hull-Whie, SA Unmanaged Risk ramework The leers and, when used as subscrips, will refer respecively o OLLATERAL and UNDING. Noe as well ha we will use he erms funding rae and IBOR rae indifferenly, as well as he erms collaeral rae and OIS rae. inancial noaion r r shor rae a which collaeral grows shor rae corresponding o unsecured funding rae P,T value a ime of a zero coupon mauring a ime T, associaed wih cos of collaeral P,T value a ime of a zero coupon mauring a ime T, associaed wih cos of funding f,t insananeous forward collaeral rae defined by f,t = ln P,T T f,t insananeous forward funding rae defined by f,t = ln P,T T LT IBOR rae fixing a ime T and whose enor is T T,T value a ime of he SA forward rae associaed wih IBOR rae LT,T value a ime of he non-sa forward rae associaed wih IBOR rae LT Mahemaical noaion The mahemaical noaion used is summarized in he following able. or he sake of simpliciy, we used he same noaion for Brownian moions and filraions under risk-neural and forward measures. Monocurve unding SA Risk-neural measure Q Q Q Numéraires associaed B B B Brownian moions W W W ilraions associaed Expecaions E Q E Q E Q T-forward measure Q T T Q T Q Numéraires associaed P.,T P.,T P.,T Brownian moions W W W ilraions associaed Expecaions E Q T E QT E QT 50 WILMOTT magazine
TEHNIAL PAPER We use subscrip for condiional expecaions. or insance, E Q denoes expecaion condiional on. To disinguish Brownian moions under he same measure, we use he subscrip linked o he asse. or insance, W S will denoe asse S s Brownian moion. igure : Impac of fuures convexiy adjusmen on zero-coupon raes Moivaion Before SA swaps became a marke sandard, fuures were already subjec o margin calls implying a convexiy adjusmen. The laer can be seen as he difference beween risk-neural and forward-neural expecaions of an IBOR rae see Andersen and Pierbarg, 00, secion 6.8 for a deailed demonsraion: VX = E Q [LT, T ] E Q T [LT, T ] alculaing E Q [LT ] is no rivial and currenly invesmen banks use a modeldependen approach. A commonly used adjusmen is implied by a Hull & Whie model on shor rae, leading o VX = σ BT, T [ BT, T e at + ab0, T ] 4 a τt, T where B,T = a [ e at ], a denoes shor-rae mean reversion, and σ is volailiy as shown in Kirikos and Novak 997. I is ypically he adjusmen one can find on Bloomberg using IVS. Now he quesion we have o ask ourselves is, How does his adjusmen impac he yield curve? igure answers our quesion quickly: using a fixed volailiy a %, we obain an adjusmen magniude a abou a basis poin for a 3-year mauriy. Alhough his impac is no neglecable and affecs forwards, i does no affec significanly he overall yield curve shape. Le us now focus on collaeralized swaps, which are used for he long erm of he yield curve. Jus like fuures, collaeralized swaps are subjec o margin calls and herefore heir pricing involves a convexiy adjusmen. urrenly, banks ake ino accoun his adjusmen by separaing discouning using an OIS curve corresponding o he rae a which collaeral grows from forwards calculaion. However, he forwards obained his way correspond o he SA-forward measure, ha is,, T, T = E QT [LT, T ]. These are he forwards we can rerieve from he inerbank marke, since swaps are collaeralized. In order o price non-collaeralized deals, we need forwards under he non-sa-forward measure, ha is,, T, T = E QT [LT, T ]. These are no direcly observed in he marke, and herefore have o be deduced from,t. Alhough i is clear in he lieraure ha his convexiy adjusmen exiss see Mercurio, 00; Pierbarg, 00; Bianchei and arlicchi, 0, for insance, here are currenly no sudies of is impac on he long end of he yield curve. The approach we presen in his aricle consiss of modeling OIS and IBOR shor raes separaely using correlaed Hull & Whie dynamics. Under his framework we obain closed formulae linking he SA forward,t o he non-sa forward,t. Our aim is no o specify a new yield curve boosrap procedure since he adjusmen depends on OIS volailiy and OIS IBOR correlaion which are difficul o observe, bu raher o quanify he risk implied by hese parameers which is currenly being underesimaed. By reducing counerpary risk using SA conracs, new risk facors have arisen ha are lef unhandled. Mos imporanly, a ypical seup involves a bank having a non-collaeralized rade wih a clien, hedged wih a collaeralized rade see igure. Therefore, in order o hedge properly, he non-collaeralized rade sensiiviies have o be moniored as precisely as possible. Our work shows how imporan i is o ake ino accoun OIS IBOR correlaion, as well as heir respecive volailiies. igure : Non-SA swap hedged by a SA swap ^ WILMOTT magazine 5
Theoreical SA convexiy adjusmen and marke pracice Our developmens are based on he collaeralized derivaives valuaion framework inroduced by ujii e al. 009 and generalized o he case of parial collaeralizaion by Pierbarg 00. Noe ha, for he sake of simpliciy, we do no ake ino accoun differen ypes of collaeralizaion. When we menion a collaeralized deal we mean a bilaeral fully cash collaeralized deal wih daily margin calls. Moreover, we focus on he specific case of mono-currency swaps collaeralized in heir own currency. We herefore do no incorporae he general case of collaeral currency differen from he swap s currency, nor he opion of collaeral currency choice. While his generalizaion is imporan as i concerns many SA conracs, i would considerably complicae hings. onsider now an IBOR rae LT paid a T. If his cashflow is par of a collaeralized deal is presen value wries as V = E Q T [e r sds LT, T ], r being he rae a which collaeral grows. Expressing he previous resul under he SA-forward measure Q T associaed wih he numéraire P.,T leads o V = P, T E QT [LT, T ]. Similarly, he value of he same cashflow no being collaeralized wries as, under he non-sa-forward measure Q T associaed wih he numéraire P.,T, V = P, T E QT [LT, T ]. As inerbank swaps are collaeralized, non-sa forwards are no direcly observed in he marke and have o be deduced from SA forwards using a convexiy adjusmen, which wries as VX SA = E QT [LT, T ] E QT [LT, T ]. urrenly, marke praciioners neglec his impac and use SA forwards direcly for non-sa swaps pricing. This is due o he difficuly of calculaing he adjusmen, and also o he fac ha he main concern for non-sa swaps is counerpary risk. While counerpary risk for non-sa swaps is fundamenal, we demonsrae hereafer ha non-sa forwards convexiy adjusmen should no be negleced. To measure purely he impac of his adjusmen, we do no accoun for counerpary risk. In he nex secion we se up a framework using one-facor Gaussian shor-rae models for collaeral and funding raes. Our framework is similar o hose inroduced in Pierbarg 00 and Kenyon 00, bu we focus specifically on calculaing non-sa forwards for which we obain closed formulae. These formulae allow, in urn, fas boosrapping procedures and poenial risk managemen via sensiiviies compuaions. Evaluaing SA adjusmen using Gaussian shor-rae models Noe ha we are assuming banks raise funds a IBOR rae and herefore we use indifferenly funding rae and IBOR rae. In pracice, banks raise funds a IBOR plus a funding spread which mus be aken ino accoun, as explained in Whiall 00. Moreover, he funding spread is sochasic and herefore a proper framework would imply modeling his spread separaely and correlaing i o IBOR and OIS raes. However, our firs aim is o measure he impac of OIS IBOR join disribuion on non-sa forwards. The nex sep would be o exend he model by incorporaing a sochasic funding spread. Modeling framework In or der o model he funding shor rae r and collaeral shor rae r, we use onefacor Gaussian shor-rae models. The r and r dynamics under Q wrie as dr = μ, r d + σ dw Q dr = μ, r d + σ dw Q where σ, σ, μ, and μ are deerminisic funcions. The μ funcions deermine he ype of Gaussian model considered. or insance, μ, r = f 0, + σ defines a Ho & Lee model for r. Le ρ, = dw Q, dwq /d denoe he insananeous correlaion beween r and r Brownian moions. Under his framework, we can express he zero-coupon dynamics as dp, T P, T = r d + Ɣ, T dw Q dp, T P, T = r d + Ɣ, T dw Q Γ and Γ being deerminisic. Moreover, zero coupons can be expressed as follows: P, T = A, T e B,T r P, T = A, T e B,T r wih A, A, B, and B being deerminisic as well. Adjusmen calculaion Using he previous resuls w e obain he following dynamics for r under Q T and QT : dr = [μ, r ρ, σ Ɣ, T ] d + σ dw QT dr = [μ, r σ Ɣ, T ] d + σ dw QT In order o link he r disribuion under Q T wih is disribuion under QT, we inroduce ˆr T = r T cvx,t wih cvx such ha he ˆr disribuion under Q T maches he r disribuion under QT. The idea of inroducing ˆr is based on he fac ha r has deerminisic drifs under boh measures Q T and QT, and herefore we can define a deerminisic funcion cvx linking hese dynamics. This allows, in urn, a sraighforward calculaion of he adjusmen: LT, T = T T P T, T = T T A T, T eb T r T = e B T cvx,t T T A T, T eb T r T e B T cvx,t [ = e B T cvx,t T T A T, T eb T r T + ] e B T cvx,t T T [ = e B T cvx,t LT ], T + e B T cvx,t T T where we inroduced he noaion LT, T = T T A T, T eb T r T. 5 WILMOTT magazine
TEHNIAL PAPER As he rˆ disribuion under QT maches he r disribuion under QT, we have ha LT,T under QT is equally disribued wih L T,T under QT. Therefore: T Q E [LT, T ] = eb T,T cvx,t,t T Q E [LT, T ] + e B T,T cvx,t,t T T This finally allows us o link he SA forward o he non-sa forward :, T, T = eb T,T cvx,t,t, T, T + eb T,T cvx,t,t T T Now, he ask ha is lef is o specify a one-facor Gaussian model in order o calculae he cvx erm. We show hereafer how o calculae his erm using Hull & Whie models. Using Hull & Whie models onsidering he Hull & Whie dynamics for r, we have μ, r = θ a r wih he θ funcion allowing us o recover he iniial ineres rae erm srucure: θ = σ f 0, + a f 0, + e a a Γ, B, and A wrie as e a T a B T, T = e a T T a P 0, T exp B T, T f, T A T, T = P 0, T T a T e σ d B T, T 0, T = σ Replacing he subscrip wih allows us o wrie he corresponding funcions for he collaeral shor rae r. Afer calculaion see Appendix B. for more deails, we obain T cvx, T, T = e a T s s, T σ s ρ, s s, T σ s ds which can be analyically inegraed. Assuming consan volailiies leads o see Appendix B. for calculaion deails σ σ σ cvx, T, T = ρ, e a T a a a σ e a T +T ea T ea a σ σ e a T a T ea +a T ea +a + ρ, a a + a igure 3: EONIA 6M swap vs. EURIBOR 6M from 004 o 0 WILMOTT magazine 50-56_Papaioannou_TP_March_03_inal.indd 53 ^ Source: Bloomberg 53 06/03/03 03:3 AM
Now we have obained a closed formula for he cvx erm, we have explicily linked he SA forward,t o he non-sa forward,t and we are able o sudy he impac of SA adjusmen on he yield curve. Impac measuremen under Hull & Whie framework Decorrelaion impac Decorrelaion beween OIS and IBOR ra es is he key o undersanding he impac of he adju smen. The decorrelaion ha appeared in 007 was an imporan facor ha led banks o separae OIS discouning from forwards calculaion. Nohing can make i more obvious han he hisorical char of he OIS IBOR spread presened as igure 3. More precisely, he EURIBOR 6-monh rae is ploed agains a 6-monh EONIA swap from 004 o 0. The graphs in igure 4 show he impac on he yield curve obained when aking ino accoun decorrelaion beween OIS and IBOR raes. The blue curve corresponds o radiional monocurve boosrapping of a 6-monh EURIBOR swap curve. The red curve is obained using OIS discouning. We see ha hey have similar levels; he difference can be seen if we look a he zero-coupon raes shown in Table A., in which we noe a basis poin difference for he 30-year mauriy. All he oher curves are obained aking ino accoun convexiy wih consan % volailiy for OIS and IBOR raes and for differen correlaion levels varying from 00% o 70%. We can noe he imporance of he impac as decorrelaion becomes more imporan. The viole curve is obained aking ino accoun convexiy wih consan % volailiy for OIS and IBOR raes and for a correlaion level of 80%. 00% correlaion would correspond o a pre-crisis scenario before 007 and 80% would correspond o a decorrelaion ha could be observed under sressed marke condiions. We noice an imporan downward shif on boh he yield curve and he forward curve as decorrelaion becomes more imporan. Even wih 90% correlaion we obained for he 50Y mauriy a rae 6 basis poins lower han wha we obained wih a ypical OIS boosrap and a 34 basis poins impac on he forward rae absolue difference. Numerical resuls are presened in Tables A. and A.. This alers us o he imporance of aking ino accoun he adjusmen for risk managemen purposes: all non-collaeralized swaps are highly sensiive o his adjusmen. Therefore, in order o manage risks on non-sa swaps, one should monior as precisely as possible OIS and IBOR correlaion, as well as heir respecive volailiies. igure 4: OIS IBOR decorrelaion impac on spo raes and forward raes igure 5: Adjusmen impac wih σ < σ 54 WILMOTT magazine
TEHNIAL PAPER Assuming OIS volailiy lower han IBOR volailiy In he previous secion we analyzed he correlaion s impac using collaeral rae volailiy which is unobservable equal o funding rae volailiy a a % level. In his secion we ake a look a he behavior of he adjusmen under he assumpion of collaeral rae volailiy lower han funding rae volailiy. Alhough i is a difficul ask o define a level of collaeral rae volailiy σ, i seems logical o keep i lower han he funding rae volailiy σ. The reason is ha he collaeral rae is driven by cenral bank raes ha are less volaile han inerbank raes. In his case he downward shif in spo raes and forward raes observed in he previous secion is increased. In he curren example we used as previously σ = % and ρ = 90%, bu we ook σ = 50% see igure 5. rom he cvx formula obained in Secion 3.3, we would expec he downward shif o be more significan since ρ and σ are inerchangeable. This is indeed wha we observe: he 50Y spo rae is 89 basis poins lower han a ypical OIS boosrap and he forward rae is.83% lower absolue difference, as shown in Tables A.3 and A.4. onclusion By modeling OIS and IBOR raes separaely and under he assumpion of no funding spread, we managed o calculae he forwards convexiy adjusmen impacing non-collaeralized swaps. We have shown how crucial i is o ake ino accoun OIS IBOR decorrelaion in erms of yield curve risk managemen. The nex imporan seps consis of: rying o quanify he OIS IBOR correlaion erm srucure as well as he OIS volailiy erm srucure, boh being difficul o observe; modeling a sochasic funding rae correlaed o IBOR and OIS raes. Appendix A: Numerical resuls Table A.: Spo raes decorrelaion impac. Adjused Mauriy Monocurve OIS discouning ρ, = 90% ρ, = 80% ρ, = 70% Y 0.40% 0.40% 0.4% 0.4% 0.4% 5Y 0.93% 0.93% 0.9% 0.9% 0.9% 0Y.74%.74%.7%.7%.69% 0Y.3%.3%.6%.%.7% 30Y.35%.34%.6%.8%.09% 40Y.48%.47%.35%.%.0% 50Y.6%.60%.44%.8%.% Table A.3: Spo raes impac wih σ < σ. Mauriy Monocurve OIS discouning Adjused ρ, = 90% σ = 0.5%, σ = % Y 0.40% 0.40% 0.4% 5Y 0.93% 0.93% 0.9% 0Y.74%.74%.66% 0Y.3%.3%.05% 30Y.35%.34%.88% 40Y.48%.47%.79% 50Y.6%.60%.7% Table A.4: orward raes impac wih σ < σ. Mauriy Monocurve OIS discouning Adjused ρ, = 90% σ = 0.5%, σ = % Y 0.5% 0.5% 0.5% 5Y.4%.4%.06% 0Y 3.% 3.0%.86% 0Y.45%.45%.67% 30Y.80%.79%.55% 40Y 3.06% 3.04%.40% 50Y 3.6% 3.4%.4% Appendix B: Adjusmen calculaion under Hull & Whie framework B. alculaing he cvx,t erm Sep : r dynamics under Q T Le us apply he change of measure from Q o Q T derivaive: dq T dq = Z = K P, T B using he Radon Nikodym where K is a consan erm K = B 0/P 0,T. The Radon Nikodym derivaive Z has he same volailiy erm as P : dz/z = Ɣ, T dw Q Table A.: orward raes decorrelaion impac. Adjused Mauriy Monocurve OIS discouning ρ, = 90% ρ, = 80% ρ, = 70% 5Y.4%.4%.%.%.09% 0Y 3.% 3.0% 3.06% 3.0%.97% 0Y.45%.45%.3%.7%.03% 30Y.80%.79%.56%.34%.% 40Y 3.06% 3.04%.74%.45%.5% 50Y 3.6% 3.4%.90%.57%.4% Applying he Girsanov heorem we can now express he funding shor-rae r dynamics under Q T as dr = [ ρ, Ɣ, T σ + θ a r ] d + σ dw QT Solving his equaion yields r T = r e a T + + e a T s [ ρ, sɣ s, T σ s + θ s ] ds σ e a T s dw QT ds ^ WILMOTT magazine 55
Sep : r dynamics under Q T hanging measure from Q o Q T leads o he following dynamics for r under Q T : or equivalenly: dr = [ Ɣ, T σ + θ a r ] d + σ dw QT r T = r e a T + + This leads us o define cvx, T, T = e a T s [ Ɣ, T σ + θ s ] ds σ e a T s dw QT ds e a T s [ Ɣ s, T σ s ρ, sɣ s, T σ s ] ds r T = r T cvx HW, T, T which makes ˆr T under Q T equally disribued wih r under QT. B. alculaing he inegral Le us now calcu lae he cvx,t inegral cvx, T, T = where he Γ funcions are given by cvx, T, T = = e a T s [ Ɣ s, T σ s ρ, sɣ s, T σ s ] ds Γ = σ e a T s a Γ = σ e a T s a [ e a T s σ e a T s a σ σ ] ρ, e a T s ds a σ σ σ ρ, e a T s ds a a }{{} I σ e a T +T s ds a }{{} I σ σ + ρ, e a T a T e a +a s ds a }{{} I 3 We finally obain I = σ e a T +T s ds = σ a a e a T +T e a T e a σ σ I 3 = ρ, e a T a T e a +a s ds a σ σ = ρ, a a + a e a T a T e a +a T e a +a cvx, T, T = σ a σ σ ρ, e a T a a σ a e a T +T e a T e a σ σ + ρ, a a + a e a T a T e a +a T e a +a Denis Papaioannou is a Senior Quaniaive onsulan a Hiram inance, which he joined in 009. He has worked for major invesmen banks and asse managers on quaniaive and risk managemen projecs. Meriem houqi is a Junior Quaniaive onsulan working a MargOconseil in Paris. Benjamin Giardina Junior Quaniaive onsulan working a Hiram inance in Paris. ENDNOTES. The calculaion can easily be exended o ime-dependen volailiies bu his leads o heavier formulae we chose no o expose for he sake of clariy.. Soluion of an Orsein Uhlenbeck process, simply obained by calculaing dr e a and inegraing. REERENES Andersen, L.B.G. and Pierbarg, V.V. 00. Ineres Rae Modeling. Vol. 3: Producs and Risk Managemen. Alanic inancial Press. Bianchei, M. and arlicchi, M. 0. Ineres raes afer he credi crunch: Markes and models evoluion. The apco Insiue Journal of inancial Transformaion 3, 35 48. ujii, M., Shimada, Y., and Takahashi, A. 009. A noe on consrucion of muliple swap curves wih and wihou collaeral. SSRN. hp://ssrn.com/absrac=440633. Kenyon,. 00. Pos-shock shor-rae pricing. Risk 3:, 83 87. Kirikos, G. and Novak, D. 997. onvexiy conundrums. Risk 0:3, 60 6. Mercurio,. 00. LIBOR marke models wih sochasic basis. SSRN. hp://ssrn.com/ absrac=563685. Pierbarg, V.V. 00. unding beyond discouning: ollaeral agreemens and derivaives pricing. Risk 3:, 97 0. Whiall,. 00. Dealing wih funding on uncollaeralized swaps. Risk.ne. hp:// www.risk.ne/risk-magazine/feaure/687538/dealing-funding-uncollaeralisedswaps. W Le us now calculae he hree inegrals above: I = σ σ σ ρ, e a T s ds a a = a σ a ρ, σ σ a e a T 56 WILMOTT magazine