Pricing Single Name Credi Derivaives Vladimir Finkelsein 7h Annual CAP Workshop on Mahemaical Finance Columbia Universiy, New York December 1, 2
Ouline Realiies of he CDS marke Pricing Credi Defaul Swaps Generaing Clean Risky Discouning Curve Effec of Recovery Value Hedging CDS Pricing Defaul in Foreign Currency 1
Credi Derivaives Size of he Marke Noional ($ billion) 1,8 1,6 1,4 1,2 1, 8 6 4 2 4 BBA esimaed BBA forecas 17 35 586 893 1237 1581 199-96 1997 1998 1999 2 21 22 Year Approximaely 4% of he marke noional come from Credi Defaul Swaps 2
Realiies of CDS Marke Sandardized ISDA Credi Derivaives Definiions (1999) provides indusry-wide sandards and ease of execuion Two-way Credi Defaul Swap marke in Invesmen Grade and Emerging Markes, nascen HY CDS marke High spread volailiy: from 4% up o 3% Risk managemen wih a lack of liquidiy: Shor end of he yield curve Vs. long end Gap risk Wide range of spreads: from 3 bp o he sky is he limi Defaul is no a heoreical possibiliy bu a fac of life (Russia, Ecuador, Laidlaw, ec) Reasonably deep cash marke wih a variey of bonds Traded volailiy in EM (mosly shor mauriies) Illiquid longer erm volailiy hrough opions on CDS and Asse Swaps Increase in acive risk managemen and more raional credi pricing Widespread opporuniies o exploi pricing anomalies 3
Benchmark Curves for a Given Name Defaul-free Discouning Curve (PV of $1 paid wih cerainy) D(, ) = E = exp r d rˆ τ τ exp τ dτ Clean Risky Discouning Curve [CRDC] (PV of $1 paid coningen on no defaul ill mauriy, oherwise zero) λ τ dτ Z (, ) = E exp ( r τ + λ τ ) dτ has a meaning of defaul probabiliy a ime over ime period Z (, ) = D(, ) Q(, ) Q(, ) = exp ˆ λ τ dτ Q(, ) where is survival probabiliy ill ime, and is usually inerpreed λˆτ as forward (no expeced! ) probabiliy of defaul per uni ime τ dτ 4
Credi Defaul Swap Goldman, Sachs & Co. A basic credi derivaives insrumen: ABC is long defaul proecion ABC par S D wih frequency condiional on survival of a reference name XYZ ABC 1 REC S par accr condiional on defaul of a reference name XYZ REC is a recovery value of a reference bond Reference bond: no guaranied cash flows cheapes-o-deliver cross-defaul (cross-acceleraion) Assume same recovery value REC for all CDS of he same senioriy on a given name 5
Pricing CDS For corporae and EM coupon bonds a defaul claim is (Principal + Accrued Ineres) Recovery value has very lile sensiiviy o a srucure of bond cash flows For his Face Value Claim, REC = R, and PV of CDS is given by PV CDS T τ τ = S T E + (1 e d E R ( r + λ ) dτ ( r + λ ) T dτ R = R, R = E( R) Assume, and no correlaion of R wih spreads and ineres raes As Eq (1) is linear in R, CRDC jus depends on expeced value R, no on disribuion of R Pu PV of CDS =, and boosrapping allows us o generae a clean risky discouning curve 6 ) λ e τ τ d (1)
Generaing CRDC E A erm srucure of par credi spreads T par is given by he marke To generae CRDC we need o price boh legs of a swap No Defaul (fee) leg T ( r ) dτ T Defaul leg S + λ τ T, pare ST, par Z (, ) τ e d = ( r + λ ) dτ T ( r + λ ) T τ τ τ τ dτ T ~ ( 1 R ) λe d = (1 R) E λe d = (1- R) λz (, ) If correlaion beween credi spreads and ineres raes is no zero, ~ λ ˆ τ λ τ S, d d 7
Correlaion Adjusmen 8 Goldman, Sachs & Co. Need o ake ino accoun correlaion beween spreads and ineres raes o calculae adjused forward defaul probabiliy λ λ + aλ Defaul-free rae r~ condiional Defaul Probabiliy and Rae Adjusmens Vs. Mauriy on no defaul also needs o be adjused as ~ r = rˆ a λ.5.4.3.2.1 -.1 -.2 -.3 -.4 -.5 Vols=8%,Volr=2%,MRs=.5,MRr=.5,Corr=.3, =6% r=5% 2. 5 7. 1 12 15 17 2 22 25 27 3 Rae Adj DefProb Adj ~ = ˆ dz(, ) d = E ( r + λ )exp For high spreads and high volailiies an adjusmen is no negligible For given par spreads forward defaul probabiliy decreases wih increasing volailiy, correlaion and level of ineres raes and par spreads and ~ s + ~ r = sˆ + rˆ + λ τ dτ ( rτ )
Recovery Value For he Face Value Claim he price of a generic coupon bond can be approximaed prey accuraely as ~ L n B (, N ) = C n Z (, n ) + Z (, N n N ~ + R 1 nln 1Z (, n) Z (, N ) n where 1 is forward defaul-free floaing rae for period n Bond price goes o recovery value in defaul For he same defaul risk and recovery value, high coupon bond should rade a higher credi spread han a low coupon bond There are no generic risky zero coupon bonds wih non zero recovery Using CRDC and given recovery value srucure one can creae any synheic insrumen ) 9
More on Recovery Value Oher ways o model recovery value: Recovery of he Risky Price (Duffie-Singelon) : Defaul claim is a raded price jus before he defaul even Recovery of he Riskless Price: Defaul claim is given by defaul-free PV of he bond cash flows a he momen of defaul For a zero coupon bond his defaul claim corresponds o he claim on a face value a mauriy Boh mehods operae wih risky zero coupon bonds wih embedded recovery values. One can use convenional bond mah for risky bonds Boh mehods are no applicable in he real markes 1
Effec of Recovery Value Assumpions on Relaive Value Implicaions for pricing off-marke deals, synheic insrumens, risk managemen Example: Relaive bond value Same name, senioriy and mauriy, differen coupons For Recovery of Face Value N For Recovery of Risky/Riskless Price B C B C B = ( C C n= 1 C B = ( C C (, Risky zero coupon bonds wih embedded recovery value are given by n C n N n= 1 ) D(, n n ) Q(, n ) Z R n ) n ) Z R (, ) = E exp τ λ τ τ (1 R ) ( r + (1 R ) ) d D (, ) Q (, ) [(1 R ) Q (, R ] Z R (, ) = D (, ) ) + for Risky Claim for Riskless Claim 11
Defaul Probabiliy and Recovery For a given par credi spread curve defaul probabiliies depend on recovery value definiion Defaul Probabiliy Curve for Various Claim Types Par Spreads = 6% Volailiy = 4% R =.4 5.% 45.% 4.% 35.% 3.% 25.% 2.% 15.% 1.% FV Claim Risky Claim Riskless Claim 5.%.% 1.25 2.5 3.75 5 6.25 7.5 8.75 1 Mauriy (yrs) 11.3 12.5 13.8 15 12
Hedging CDS books Two ypes of exposures: credi spread risk, defaul risk Goldman, Sachs & Co. Using N hedging insrumens (bonds or CDS) on can hedge a CDS porfolio agains (N-1) predeermined facors for spread moves + defaul Differen Recovery Value definiions resul in differen hedging posiions Robusness of hedging depends on spread curve inerpolaion mehod Transacion cos may be significan: need o opimize hedging sraegy When hedging wih bonds, bond/cds basis risk can be an issue In EM cheapes-o-deliver opion is equivalen o firs-o-defaul feaure For HY CDS equiy opions/shares should be considered as possible hedging insrumen 13
Pricing Defaul in Foreign Currency Assume ha one needs o sell defaul proecion in foreign currency and hedge i by buying proecion in $. Q: A wha level o sell? If here is no inerdependence beween credi spreads and forward FX, implied defaul probabiliies should say he same in foreign currency Due o he correlaion beween defaul spread and each of FX, dollar ineres raes, and foreign ineres raes, he defaul probabiliy in a foreign currency will differ from ha in dollars Two sources for he adjusmen: - Devaluaion condiional on defaul - Day-o-day spread/fx/ir correlaion 14
Adjusmen for FX jump condiional on Defaul FX rae jumps by - % when defaul occurs (e.g. devaluaion) As probabiliy of defaul (and FX jump) is given by λ, under no defaul condiions he foreign currency (FC) should have an excessive reurn in erms of USD (DC) given by λ o compensae for a possible loss of value α Consider a FC clean risky zero coupon bond (R=) wih an excessive F no defaul reurn λ ha compensaes for a possible defaul The posiion value in DC = (Bond Price in FC) * (Price of FC in DC) F An excessive reurn of he posiion in DC is λ α + λ The posiion should have he same excessive reurn as any oher risky bond in DC which is given by λ To avoid arbirage he FC credi spread should be F λ = λ ( 1 α ) An adjusmen can be subsanial 15
Quano Spread Adjusmen In he no defaul sae, correlaion beween FX rae and ineres raes on one side and he credi spread on anoher resuls in a quano adjusmen o he credi spread curve used o price a synheic noe in FC Consider hedges for a shor in synheic risky noe in FC sell defaul proecion in DC long FC, shor DC If DC srenghens as spreads widen, in order o hedge he noe we would need o buy back some defaul proecion and sell he foreign currency ha depreciaed. Our P&L would suffer and we would need o pass his addiional expense o a couner pary in a form of a negaive credi spread adjusmen For high correlaion he adjusmen can be significan 16
Quano Adjusmen (con d) Adjusmen for a DC fla spread curve of 6 bp. Spread MR is imporan Effec of Mean Reversion on Log-Normal Spread Spr ead Adj us me n (bp ) -5-1 -15-2 -25 Mean Reversion.2.5 1-3 -35 1 2 3 4 5 6 7 8 9 1 11 12 Year Spread adjusmen decreases wih increasing mean reversion and consan spo volailiy. =.2, S=6%, r$=5%, rf=2%, σs=8%, σ$=12.5%, β$=, σf=4%, βs=.5, σx=2%, ρ$s=, ρfs=.5, ρxs=.7. All curves are fla. 17
Quano Adjusmen (con d) Assumpions on spread disribuion are imporan Difference of Normal and Log-Normal Spread Adjusmens Adjusmen Difference (N-LN) (bp) -5-1 -15-2 -25-3 1 2 3 4 5 6 7 8 9 1 11 12 Mean Reversion.2.5 1 Year Difference beween normal and log-normal adjusmen decreases as mean reversion is increased for consan spo volailiy 18
Conclusions For single-name insrumens pricing is well undersood Recovery value definiion can have an significan effec on pricing and hedging Hedging CDS wih bonds: basis risk canno be ignored The disincion beween EM and FI credi derivaives gradually disappears Consisency of pricing and hedging mehods becomes more and more imporan 19