Credit Index Options: the no-armageddon pricing measure and the role of correlation after the subprime crisis

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1 Second Conference on The Mahemaics of Credi Risk, Princeon May 23-24, 2008 Credi Index Opions: he no-armageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo - Join work wih Massimo Morini Fich Soluions and Dep. of Mahemaics, Imperial College, London

2 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions Ouline The Forward Credi Index and Credi Index Opions The flaws in he marke approach Pricing wihou Armageddon A consisen Index Spread Arbirage-free pricing of Credi Opions The role of correlaion Credi Crisis and Mispricing beween Marke and No-Arbirage Formulas 1

3 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions The Credi Index Porfolio of n names wih iniial noional N 0 = 1. Each name has 1 noional n. τ i defaul ime of name i, wih associaed loss 1 Rec i 1 n. Toal porfolio loss L = 1 n n 1 Rec i = L = 1 n 1 R n 1 {τi <} 1 {τi <} i=1 i=1 where he second equaliy holds for a fla recovery rae R. A ime, he Ousanding Noional is N = 1 L / 1 R. The Proecion Leg pays, a defaul of one name in he index, he corresponding loss. A τ i : dl τ i = 1 R 1/n from sar dae o T M or unil all names defaul. 2

4 Copyrigh 2008 Damiano Brigo and Massimo Morini The Index paymens No-armageddon arbirage-free measure for credi index opions The Premium Leg, a paymen imes T j, j = A + 1,..., M or unil all names have defauled, wih year fracions α j, pays a premium K on he daily average ˆN T j of he ousanding noional N for T j 1, T j. A T j : ˆN Tj α j K The discouned payoff of he Proecion Leg is P ro,t M = TM D, u dl u M j=a+1 D, T j L T j L T j 1 By D, T we indicae he discoun facor from T o. Is expecaion is he corresponding bond price, P, T = E Q D, T 1 F. 3

5 Copyrigh 2008 Damiano Brigo and Massimo Morini The Index paymens No-armageddon arbirage-free measure for credi index opions The discouned payoff of he Premium Leg is P rem,t M K = M j=a+1 M j=a+1 D, T j D, T j α j N T j K = N d T j 1 K Tj M j=a+1 D, T j α j 1 L T j 1 R K The quaniy in curly brackes is called. 4

6 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions Evaluaing Premium and Proecion Legs The value of he wo legs is compued by expecaion under he risk neural probabiliy measure Q. We use he noaion Π X, s = E X F s. When s =, we omi he argumen s wriing simply Π X. The Payer Forward Index saring a and lasing unil T M has a price Π I,T M K := E I,T M K F := E P ro,t M P rem,t M Subsiuing he payoffs, one sees ha he Index value does no depend on he Loss disribuion bu only on he Expeced Loss E L T a differen mauriies. K F 5

7 Copyrigh 2008 Damiano Brigo and Massimo Morini The Index Spread No-armageddon arbirage-free measure for credi index opions In he simples definiion, he equilibrium spread a ime is he value of he spread K ha ses he value of he forward index o zero a ime : S,T M = Π P ro,t M. Π allowing o wrie he index value as Π I,T M K = Π S,T M K 6

8 Copyrigh 2008 Damiano Brigo and Massimo Morini Index Opions No-armageddon arbirage-free measure for credi index opions A payer Index Opion wih incepion 0, srike K and exercise, wrien on an index wih mauriy T M, gives he righ bu no obligaion o ener a ino he running Index wih final paymen a T M as proecion buyer paying a fixed rae K. However, he opion buyer would give away proecion from incepion 0 o mauriy. In order o arac invesors, sandard Credi Index Opion payoff includes he paymen of he losses from he opion incepion o as well fron end proecion F = D, L Π F = E D, L F. 7

9 Copyrigh 2008 Damiano Brigo and Massimo Morini The rough approach No-armageddon arbirage-free measure for credi index opions In he roughes approach, he opion payoff in case of exercise Π I,T M K + F = Π S,T M K + F is improperly evaluaed as + should include all pars + E D, Π S,T M K F +E F and hen expressing he 1s comp. hrough a sd Black formula Π Black, K, σ,t M TA +Π S,T M F F Pedersen 2003 properly includes he fron-end proecion ino he opion and hrough numerical inegraion ges o an improved Black formula., 8

10 Copyrigh 2008 Damiano Brigo and Massimo Morini Ĩ,T M The marke approach K = P ro,t M No-armageddon arbirage-free measure for credi index opions P rem,t M K + F. I is naural o give a new spread definiion, seing o zero Π Ĩ,T M This leads o he Loss-Adjused Marke Index Spread S,T M = Π P ro,t M + Π F /Π K. ha allows o wrie he opion price as E D, Π ST A,T M + K F. Numeraire Π Π and lognormal S,T M : Mk Index Opion Form.: Black ST A,T M, K, σ,t M TA. 1 9

11 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions The hree flaws of he Sandard Formula 1. The definiion of S,T M Π = is valid only when he denominaor M j=a+1 is differen from zero. Since Π defaulable asses, i can go o zero. 2. When Π E D, T j α j 1 L T j F 1 R is he price of a porfolio of = 0 he pricing formula 1 is undefined, and in his scenario S,T M does no se he value of he adjused index o zero. 3. If we worked condiionally on Π > 0, he resuling survival measure would no be equivalen o he sandard risk-neural measure. 10

12 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions Subfilraion Pricing wihou armageddon We adap a echnique used earlier by Jamshidian 2004 and Brigo 2005 for single name CDS opions, making use of a subfilraion srucure, separaing defaul free informaion from informaion on he defaul even. This is based on he Jeanblanc-Rukowski J-R filraion-swiching formula. To effecively use J-R in index opions conex, define ˆτ = max τ 1, τ 2,..., τ n and define a new filraion Ĥ such ha F = ˆ J Ĥ, ˆ J = σ {ˆτ > u}, u, so ha Ĥ excludes informaion on he porfolio armageddon even. 11

13 Copyrigh 2008 Damiano Brigo and Massimo Morini Define No-armageddon arbirage-free measure for credi index opions A DV01 γ wihou Armageddon ˆΠ := E Ĥ. Exploiing = 1 {ˆτ>}, which is necessary o be able o apply he J-R-ype formula wr he filraion H Π = E = Q F = 1 {ˆτ>} ˆΠ ˆτ > Ĥ Q 1 {ˆτ>} E ˆτ > Ĥ Ĥ. 2 The quaniy ˆΠ is never null, and we will see ha i is wha we need for an effecive definiion of he Index Spread and of an equivalen pricing measure for Index Opions. 12

14 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions The Arbirage-free Index Spread Applying a J-R ype formula o he loss adjused index does no work, since he necessary condiion does no hold: 1 {ˆτ>} Ĩ,T M Ĩ,T M Indeed, he Loss-Adjused index, differenly from, is never null even in case ˆτ we receive fron end proecion. Wrie hen E Y T F = E 1 {ˆτ } Y T F + E 1{ˆτ>} Y T F = 1 {ˆτ } E Y T 1 {ˆτ>} σ ˆτ Ĥ + E Q ˆτ > Ĥ 1 {ˆτ>} Y T Ĥ The 1s componen corresponds o he value when we know ha all defauls have already happened, and we know he exac armageddon ime. 13

15 Copyrigh 2008 Damiano Brigo and Massimo Morini = Π Ĩ,T M Q 1 {ˆτ>} ˆτ > Ĥ No-armageddon arbirage-free measure for credi index opions The Arbirage-free Index Spread K {ˆΠ + 1 {ˆτ>} 1 R E Q ˆτ > Ĥ = ΠP ro,t M P ro,t M +1 {ˆτ } 1 R P, P rem,t M K + F = 3 } P rem,t M K + E 1 {ˆτ>TA }F Ĥ ˆΠ 1 {<ˆτ TA }D, Ĥ This shows he acual componens of Loss-Adjused index, and will lead us o a consisen valuaion of he Index Opion. In a Loss-Adjused porfolio we canno define in all scenarios he equilibrium spread as he value of he spread zeroing he index value, as here are always scenarios where he index is non zero, regardless of he spread. + 14

16 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions The Arbirage-free Index Spread The financially meaningful definiion of he Index Spread considers he level of K seing he Index value o 0 in all scenarios where some names survive unil mauriy. Only in such scenarios he payoff acually depends on K. Se o 0 only he firs componen of he index value 3, Q 1 {ˆτ>} {ˆΠ ˆτ > Ĥ P ro,t M ˆΠ P rem,t M K } + E 1 {ˆτ>TA }F Ĥ, which is he price of an armageddon-knock ou radable asse. We obain he following definiion of he equilibrium Arbirage-free Index Spread Ŝ,T M = ˆΠ P ro,t M + E 1 {ˆτ>TA }F Ĥ / ˆΠ 4 This definiion of he index spread is boh regular, since ˆΠ is bounded away from zero, and has a reasonable financial meaning. 15

17 Copyrigh 2008 Damiano Brigo and Massimo Morini Π Q + Q No-armageddon arbirage-free measure for credi index opions The Opion Formula Componens Opion,T M K 1 {ˆτ>} ˆτ > Ĥ 1 {ˆτ>} E ˆτ > Ĥ = E D, E D, Π Ĩ,T M 1 {ˆτ>TA } ˆΠ Q ˆτ > ĤTA + K F = 5 D, 1 {<ˆτ TA } 1 R Ĥ Ŝ,T M + K Ĥ +1 {ˆτ } 1 R P, =: Ô1 + Ô2 + Ô3 follows easily. Ô 3 is he opion value when armageddon is before. Ô 2 akes ino accoun he probabiliy of such an even beween now and he opion expiry. For Ô 1, ha was he only one considered in he simpler formula 1, we develop a sandard opion formula. 16

18 Copyrigh 2008 Damiano Brigo and Massimo Morini The numeraire No-armageddon arbirage-free measure for credi index opions For ha we need a change of measure, solving also Problem 3. echnically demanding, bu he preceding analysis helps. Ô 1 = Q 1 {ˆτ>} E ˆτ > Ĥ D, ˆΠ Ŝ,T M This is + K Ĥ Now i is naural o ake he quaniy ˆΠ = E Ĥ = E M j=a+1 D, T j α j 1 L T j 1 R o define our probabiliy measure ˆQ,T M. Differenly from Π he sub-filraion based ˆΠ is sricly posiive. Ĥ, 17

19 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions The No-Armageddon Pricing Measure We define he, T M -no-armageddon pricing measure ˆQ,T M definiion of he Radon-Nykodim derivaive wih respec o Q Z TA = d ˆQ,T M B 0 ˆΠ =. dq ˆΠ B ĤTA TA 0 hrough and we can compue Z = E Z TA Ĥ = E d ˆQ,T M dq ĤTA Ĥ = B 0 ˆΠ ˆΠ 0 B Thus also he Radon-Nykodim derivaive resriced o all Ĥ,, can be expressed in closed form hrough marke quaniies. This is sufficien o apply he Bayes rule for condiional change of measure. 18

20 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions Bayes rule for condiional change of measure: Consider a sub σ-algebra N of σ-algebra M and an M-measurable X, inegrable under he measures P 1 and P 2, P 1 P 2. Then E P 1 X EP 1 dp 2 dp 1 M E P 1 dp 2 dp 1 N N = E P 2 X N. Ô 1 = = Q Q = Π 1 {ˆτ>} ˆΠ ˆτ > Ĥ 1 {ˆτ>} ˆΠ ˆτ > Ĥ E ÊT A,T M ŜT A,T M E E dˆq,t M dq Ĥ dˆq,t M dq Ĥ ÊT A,T M ŜT A,T M + K Ĥ Ŝ,T M + K Ĥ + K Ĥ 19

21 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions Arbirage-free Credi Index Opion Formula Ô 1 = Π ÊT A,T M ŜT A,T M + K Ĥ We have also ha Ŝ,T M is a Ĥ-maringale under ˆQ,T M. Assuming lognormaliy no necessary, can use any smile maringale dynamics = ˆσ,T M Ŝ,T M dv,t M, T a we have he following dŝ,t M Arbirage-free Credi Index Opion formula Π Opion,T M + Q 1 {ˆτ>} ˆτ > Ĥ K = Π E +1 {ˆτ } 1 R P, Black Ŝ,T M 0, K, ˆσ,T M TA D, 1 {<ˆτ TA } 1 R Ĥ 6 20

22 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions Implemening he Arbirage-free Formula How is he formula pracically implemened? Π Opion,T M 0 K = Π 0 Black Ŝ,T M 0, K, ˆσ,T M TA + 1 R P 0, Q ˆτ = Π 0 Bl S 1 R P 0, Q ˆτ,T M 0, K, ˆσ,T M TA + 1 R P 0, Q ˆτ Now we apply he formula in pracice. Π γ,t M 0 21

23 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions Credi Index Opions before and afer 2007 subprime crisis For compuing ˆτ probabiliies, saying close o correlaion coming from liquid CDO ranches on he same pool, one needs he correlaions associaed o he mos senior ranche n 1 n 1 R, 1 R very hin, a a shor mauriy. Marke agrees correlaion increases wih senioriy and decreases wih mauriy. We expec a correlaion higher han he highes level quoed by he CDX marke, ρ 30%. However we consider a range of equally spaced correlaions in-beween i-traxx and CDX mos seniors, as his choice ends o underesimae he probabiliy of ˆτ, compared o he sandard marke approach of exrapolaing correlaions. Thus he relevance of he new formula will be underesimaed, and if we find a relevan impac his implies he acual impac will be even larger. 22

24 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions Opions on i-traxx Europe Main vs 2008 In he nex able we repor he marke inpus. opions in March 08 was in he range 5-8 bps. The bid-offer spread for March March Spo Spread 5y: S 9m,5y bp bp Forward Spread Adjused 9m-5y: S9m,5y bp bp 9m,5y Implied Volailiy, K = S % 108% 9m,5y Implied Volailiy, K = S % 113% Correlaion 22% I-Traxx Main: ρ I Correlaion 30% CDX IG: ρ C Annuiy 9m-5y: Π γ 9m,5y 0 Marke Inpus: : March lef, March righ 23

25 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions Opions on i-traxx Europe Main - March 2007 Srike Call Marke Formula No-Arb. Form. ρ = No-Arb. Form. ρ = No-Arb. Form. ρ = No-Arb. Form. ρ = Srike Pu Marke Formula No-Arb. Form. ρ = No-Arb. Form. ρ = No-Arb. Form. ρ = No-Arb. Form. ρ = March Opions on i-traxx 5y, Mauriy 9m 24

26 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions Opions on i-traxx Europe Main - March 2008 Srike Call Marke Formula No-Arb. Form. ρ = Difference No-Arb. Form. ρ = Difference No-Arb. Form. ρ = Difference No-Arb. Form. ρ = Difference March Opions on i-traxx 5y, Mauriy 9m 25

27 Copyrigh 2008 Damiano Brigo and Massimo Morini No-armageddon arbirage-free measure for credi index opions x 10 3 Armageddon Probabiliy in T= 9 monhs Index Spread =154.5%, March Index Spread =22.5%, March Prτ<T ρ Figure 26

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