Jarrow-Lando-Turnbull model

Transcription

1 Jarrow-Lando-urnbull model

2 Characerisics Credi raing dynamics is represened by a Markov chain. Defaul is modelled as he firs ime a coninuous ime Markov chain wih K saes hiing he absorbing sae K defaul sae. LGD is characerized as a fracion of an oherwise similar defaul-free claim.

3 Markov chain model o describe he dynamics of bond credi raings Le X represen he credi raing a ime of a bond, and X {X, 0,, 2, } is a ime-homogeneous Markov chain on he sae space N {, 2,, K, K + }, K + designaes defaul absorbing sae ,,! " " "! K K KK k K k Q [ ]! 0,,2,,,, Pr + N i i X X i

4 Assumpions on he risk premia he credi raing process X ~ afer risk neuralizaion is no necessarily Markovian ~, + π i where i is he acual ransiional probabiliies of he observed imehomogeneous Markov chain X, π i are he risk premium adusmens. JL model assumes π i π i for i, and hey are deerminisic funcions of. Some srucure is imposed o improve analyical racabiliy. he assumpion is imposed o faciliae saisical informaion since he hisorical i can be used in he inference process. i i

5 Risk neuralized ransiion marix Risk neuralized process X ~ becomes a non-homogeneous Markov chain wih ransiional probabiliies ~ i, + π i i π i i i i X ~ and {r} spo rae process are assumed o be muually independen under he risk neural measure accuracy may deeriorae for speculaive-grade bonds.

6 Price of risky discoun bonds Risky discoun bond in he h credi raing class { } { } { } { } { } [ ] P v E e E e E v ds s r ds s r > > > τ δ δ δ δ τ τ τ τ ~, ~ ~ ~, 0 where τ is he absorpion defaul ime of when X ~. ~ X { }., ~, ~ ~,, P K K k k + > τ independence

7 Numerical implemenaion of Jarrow-Lamdo-urnbull model modelling defaul and credi migraion in preference o modelling recovery rae I d J D I invesmen grade, J unk grade and D defaul absorbing r r , 0.09 s s I,0 I,02 0.0, 0.05 s s J,0 J, assume ha here is no correlaion beween raing migraion and ineres rae

8 Pricing of risky deb of mauriy one and wo periods are BI 0,, BI 0,2, B 0,, 0,2 2 J BJ Assume recovery rae φ 0.40 so ha payoff vecor. φ Suppose currenly a sae I, d I ransform d I ino he risk neural vecor I wih adusmen π I 0.0π I I 0.05π I π I

9 We find π I by making he expeced value of discouned cash flow eual o he raded price of bond B.09 Similarly, I B J,2.0, r C I r C J 0.4 Risky neural ransiion marix I Q J D π 0.05π I 0.05π I 0.0π 0.20π JI 0.0π JI 0 I J giving π I giving π J

10 Duffie-Singleon model

11 Formulaion in Duffie-Singleon model rea defaul as an unpredicable even involving a sudden loss in marke value. Defaul is assumed o occur a a risk-neural hazard rae h ; defaul over ime, given no defaul before ime, is approximaely h. V risky ~ E exp 0 Rd X 0 where E ~ denoes risk-neural expecaion, R is he defaul-adused shor-rae process, X is he value of coningen claim a mauriy.

12 Defaul-adused shor-rae process R r + h L + # where r is he defaul-free shor rae, L is he fracional loss given defaul, # represens he fracional carrying coss of he defaulable claim liuidiy premium can be included, h is he arrival inensiy a ime under risk neural process of a Poisson process whose firs ump occurs a defaul.

13 Difficulies in parameer esimaion In order o disenangle he separae conribuions of h and L, one would need addiional daa on i defaul recovery values, ii freuency of defaul of bonds of a given class. he exogeneiy of h and L can misspecify some conracual feaures in some cases. he expeced loss rae can swich from one regime o anoher in swap conracs depending on he reciprocal moneyness and non-moneyness of boh counerparies hrough ime.

14 Defaul probabiliy densiy and hazard rae τ * random variable represening he defaul ime of a credi even Q P[τ * ] probabiliy disribuion funcion of τ * dq * P[ < τ < + d d]. is no he same as he hazard defaul inensiy rae,h. Indeed, h d is he probabiliy of defaul beween ime and + d as seen as ime, assuming no defaul beween ime zero and ime.

15 . ] [ Q d Q Q d Q d P h + > + < τ τ Define G survival funcion P[τ > ] - Q, hen and G0. ' G G h Solving for G, we obain. exp 0 ds s h G Defaul probabiliy densiy and hazard rae funcion are relaed by. exp 0 ' ds s h h G

16 Valuaion of risky bonds v i, value of a defaulable zero-coupon bond of a firm ha currenly has credi raing i a ime, mauring a where φ recovery raio v i, P, [φ + φ i, ] i, probabiliy of a defaul occuring afer, given ha he deb has credi raing i as of ime. P, value of a defaul-free zero-coupon bond

17 Madan-Unal model

18 Formulaion of Madan-Unal model Decomposes risky deb ino wo embedded securiies: survival securiy: paying a dollar if here is no defaul and nohing oherwise defaul securiy: paying he rae of recovery in bankrupcy if defaul occurs and nohing oherwise

19 Evaluaion of payoffs Defaul occurs wih probabiliy φ. Recovery condiional on defaul is high H or low L wih probabiliies H and L. If here is no defaul in he firs period, hen second period oucomes depend on he evoluion of firm specific informaion x and ineres raes r. up x 2up, r 2up V φ un dp dn x 2un, r 2un x 2dn, r 2dn x 2dn, r 2dn φ H H L L

20 Risk of recovery in defaul Use he opion componens of unior and senior deb o exrac informaion on he defaul payou disribuion from he marke prices of hese deb insrumens. Meron model predeermined disribuion given by he shorfall of firm value relaive o he promised paymen. Mos oher models assume a consan payou rae condiional on defaul.

21 Srucural models reuire Reduced form models reuire. Issuer's asse value process and. Issuer's defaul bankrupcy process issuer's capial srucure 2. Loss given defaul; 2. Loss given defaul can be specified erms and condiions of he deb issue as a sochasic process 3. Defaul-risk-free ineres rae process 3. Defaul-risk-free ineres rae process 4. Correlaion beween he defaul-risk- 4. Correlaion beween he defaul-riskfree ineres rae and he asse price free ineres rae and he asse price Differen models measure he same risk bu impose differen resricions and disribuional assumpions differen echniues for calibraion and soluion.