Risk Inernaional 0
An inroducion o credi risk modelling Abukar Ali from YieldCurve.com provides essenial background on he wo main models of credi defaul risk Models of credi risks have long exised in he insurance and corporae finance lieraure. Those models concenrae on defaul raes, credi raings and credi risk premiums. These radiional models focus on diversificaion and assume ha since he credi risk for he individual asses in he porfolio are unique; hese can be diversified away in a large porfolio. Models of his kind are along he line of porfolio heory ha employs he capial asse pricing model (CAPM). In he CAPM, only he sysemaic risk or marke risk maers. For single, isolaed credis, he models calculae risk premiums as mark-ups ono he risk-free rae. Since he defaul risk is no diversified away, a similar model o he CAPM called he capial marke line is used o compue he correc markup for bearing he defaul risk. The Sharpe raio is commonly used o measure how credi risks are priced Modern credi derivaive models can be pariioned ino wo groups known as srucural models and reduced form models. Srucural models were pioneered by Black and Scholes and Meron. The basic idea, common o all srucural-ype models, is ha a company defauls on is deb if he value of he asses of he company falls below a cerain defaul poin. For his reason, hese models are also known as firm-value models. In hese models i has been demonsraed ha defaul can be modelled as an opion and, as a resul, researchers were able o apply he same principles used for opion pricing o he valuaion of risky corporae securiies. The applicaion of opion pricing avoids he use of risk premium and ries o use oher markeable securiies o price he opion. The use of opion pricing heory se forh by Black-Scholes-Meron (BSM) provides significan improvemen over radiional mehods for valuing defaul risky bonds. I also offers much more accurae prices, bu provides informaion abou how o hedge ou he defaul risk, which was no obainable from radiional mehods. Subsequen o he work of BSM, here have been many exensions. The second group of credi models, known as reduced form models, are more recen. These models, mos noably he Jarrow-Tunbull and Duffie and Singleon models, do no look inside he firm. Insead, hey model direcly he likelihood of defaul or downgrade. No only is he curren probabiliy of defaul modelled, some researchers aemp o 00 Risk Inernaional Yearbook 2004 model a forward curve of defaul probabiliies. This can be used o price insrumens of varying mauriies. Modelling a probabiliy has he effec of making defaul a surprise he defaul even is a random even ha can suddenly occur a any ime. All we know is is probabiliy. There is no sandard model for credi. Par of he reason why his is so is ha each of he models has is own se of advanages and disadvanages, making he choice of which o use depend heavily on wha he model is o be used for. I is ulimaely down o wha suis he user s requiremens bes. Pricing credi derivaives and credi risk in general, is quie similar in echnique o pricing radiional derivaives, such as ineres rae swaps or sock opions. This paper inroduces he concep behind wo general frameworks for valuing defaul risk claims and exending hese models o valuaion of credi derivaives, in paricular defaul swap or credi defaul swap conracs (CDS). The models or approaches invesigaed are he srucural and reduced form models. We will examine he suiabiliy of hese models o he pricing of credi proecion in rapidly growing credi defaul swap marke by idenifying some of he key advanages and drawback. The following are some of he key quesions ha marke praciioners mus address. How is credi defaul swap priced? Which model is mos appropriae model o use for pricing implemenaions? In a subsequen paper he auhors use reduced or inensiy based model o implemen pricing defaul swaps using corporae bond yields and solve for he defaul swap premium hey imply. In pracice, we see ha when comparing he implied credi defaul swap premium o acual marke CDS prices, implied premiums end o be much higher han he CDS prices quoed in he marke. Wha accouns for hese differences? The differences are relaed o measures of Treasury specialness, corporae bond illiquidiy, and coupon raes of he underlying bonds, suggesing he presence of imporan ax-relaed and liquidiy componens in corporae spreads. Also, boh credi derivaives and equiy markes end o lead he corporae bond marke. In his paper, we inroduce he concep behind credi risk models. Srucural models Srucural credi pricing models are based on modelling he sochasic evoluion of he balance shee of he issuer, wih
defaul when he issuer is unable or unwilling o mee is obligaions. In his model, he asse value of he firm is assumed o follow a diffusion process and defaul is modelled as he firs ime he firm s value his a pre-specified boundary. Because of he coninuiy of he process used, he ime of defaul is a predicable sopping ime. The models of Meron (1974), Black and Cox (1976), Geske (1977), Longsaff and Schwarz (1993) and Das (1995) are represenaives of his approach. Reduced-form models/inensiy models In he inensiy models he ime of defaul is modelled direcly as he ime of he firs jump of a Poisson process wih random inensiy. The firs models of his ype were developed by Jarrow and Thurnbull (1995), Madal and Unal (1998), and Duffie and Singleon (1997). Jarrow and Turnbull assume defaul is driven by a Poisson process wih consan inensiy and known payoff a defaul. The Duffie and Singleon (1997) model assumes he payoff when defaul occurs as cash, bu denoed as a fracion (1-q) of he value of defaulable securiy jus before defaul. This model was applied o a variey of problems, including swap credi risk, wo-sided credi risk and pricing credi defaul swap, binary credi defaul swap and credi defaul swap opion. Srucural credi models The basic of srucural approach, which goes back o Black and Scholes (1973) and Meron (1974), is ha corporae liabiliies are coningen claims on he asse of a firm. The marke value of he firm is he fundamenal source of uncerainy deriving credi risk. Basic assumpions Consider a fix finie forward or horizon dae T* > 0, and we suppose ha he underlying probabiliy space (Ω, F, P), endowed wih some (reference) filraion F= (f ) 0 T*, is sufficienly rich o suppor he following objecs. The shor-erm ineres rae process r, and hus also a defaul-free erm srucure model. The firms value process V, which is inerpreed as a model for he oal value of he firm s asses. There is a barrier process v, which can be used in he specificaion of he defaul ime τ. The promised coningen claim X represens he firm s liabiliies o be redeemed a mauriy dae T T*. The process C, which models he promised dividends, ie he liabiliies sream ha is redeemed coninuously or discreely over ime o he holder of a defaulable claim. The recovery claim X represens he recovery pay-off received a he ime T, if defaul occurs prior o or a he mauriy dae of he claim X. The recovery process Z, specifies he recovery pay-off a he ime of defaul, if i occurs prior o or a he mauriy dae T. Defaulable claims Technical assumpions The processes V, Z, C and v are progressively measurable wih respec o he filraion F, and ha he random variables X and X are F T measurable. In addiion, C is assumed o be a process of finie variaion, wih C = 0. I is assumed ha all random objecs inroduced above saisfy suiable inegrabiliy condiions. Probabiliies The probabiliy P is assumed o represen he real-world (or saisical) probabiliy, as opposed o he maringale measure (also known as he risk-neural probabiliy). The laer probabiliy is denoed by P*. Defaul ime/sopping imes In order o be able o model he arrival risk of a credi even, one needs o model a known, random poin in ime R +. This can also be exended o he possible se of realisaions of τ o include for evens ha may never occur. Thus, τ is a random variable wih values in R + { } Bu one may need o link τ o he way informaion is revealed in he filraion (F ) 0. In paricular, if τ is he ime of some even, we wan ha a he ime of defaul even i is known ha his even has occurred. Formally, his shows ha a every ime we know if τ has already occurred or no: (2.1) This propery defines he random ime τ as a sopping ime. Equaion 2.1 says ha we can observe he even a he ime i occurs. Bu i does no require ha he even comes as a surprise. The value of he sopping ime may be known a long ime before ime = τ. The maximum and minimum of a se of sopping imes is again a sopping ime, and he sum of he wo sopping imes. In order o represen sopping ime wih a sochasic process, we define is indicaor process ha jumps from zero o one a he sopping ime: (2.2) Nτ(): = II{ τ } In paricular, depending on he model and he purpose we may have ha τ={t} meaning ha he defaul even may only ake place a he mauriy of he company s ousanding deb as in he classical Meron model, or ha τ={t 1, T 2,...T N } if defaul can only happen (or raher, can be declared) a some discree ime insans, such as he coupon paymen daes. For defaul risk modelling, defaul indicaor funcions can be used (he indicaor funcion of he defaul even) and he survival indicaor funcions (one minus he defaul indicaor funcion). Anoher concep in conjuncion wih he defaul indicaor is also he idea of predicable sopping ime τ. The indicaor process of a predicable sopping ime is a predicable process. A predicable sopping ime has an announcing sequence of sopping imes. τ 1 2... wih τ < τand τ = τfor all ω { τ()> ω } n { τ } lim Ω 0 n n F 0 This means ha here is a sequence of early warning signals τ n ha occurs before τ and ha announce he predicable sopping ime. An example of predicable sopping ime is he firs hiing ime of a coninuous sochasic process X(), ie he firs ime when X() his a barrier, K. If Risk Inernaional 00
he process sars above he barrier X()>K, hen one possible announcing is given by he following imes: The announcing sequence gives he imes when X() his barriers ha are closer o he final barrier K. Recovery rules If defaul does no occur before or a ime T, he promised claim X is paid in full a ime T. Oherwise, depending on he marke convenion, eiher; (1) he amoun X is paid a mauriy dae T; or (2) he amoun Z τ is paid a ime τ. In his paper we assume ha he recovery paymen of X is paid in he even of defaul a mauriy, i.e. on he even τ=t. Risk neural valuaion formula We consider a financial marke model ha is arbirage free, in he sense ha here exiss a maringale measure (risk neural probabiliy) P*, meaning ha he price process of any radable securiy, which pays no coupon or dividends, becomes an F-maringale under measure P*, when discouned by he saving accoun B given as: We inroduce he jump process H =1 τ>t, and we denoe D as he process ha models all cash flows received by he owner of a defaulable claim. Le us denoe The above equaion shows he payoff of a defaulable claim if defaul does no happen a he mauriy of he conrac and he even ha defaul akes place before or a he mauriy of he conrac. The dividend process D of a defaulable coningen claim (X, C, X, Z, τ), which seles a ime T, equals D = X ( T ) II + ( 1 H ) dc + Z dh τ n = inf{ X( ) K+ 1/ n} d d B = exp( 0 rudu) X ( T) = XII + X II T 0. u u 0, u u D is a process of finie variaion, and ( 1 Hu) dcu = II > udcu = C II + CII >. 0. τ τ τ τ 0, In principal, he promised payoff X could be incorporaed ino he promised dividends process C. However, his would be inconvenien, since in pracice he recovery rules concerning he promised dividends C and he promised claim X are differen, in general. For insance, in he case of a defaulable coupon bond, i is frequenly posulaed ha, in case of defaul, he fuure coupons are los, bu a sricly posiive fracion of he face value is usually received by he bondholder. Le us denoe S as he ex-dividend price of a defaulable claim. A any ime, he random variable S represens he curren value of all fuure cash flows associaed wih a given defaulable claim. For any dae [0, ] he ex-dividend price of he defaulable claim (X, C, X, Z, τ) is given as: T, u u S = B E ( B dd F ) I is common o use he above equaion, bu wih he probabiliy measure P* subsiued wih Q*. τ> T τ T Table 1 Payoffs a mauriy in he classical approach Asses Bonds Equiy No defaul V T K K V T K Defaul V T <K V T 0 Defaulable zero-coupon bond Assume ha C=0, Z=0 and X=L for some posiive consan L>0. Then he value process S represens he arbirage price of defaulable zero-coupon bond wih face value of L and recovery a mauriy only. In general, he price D(,T) of such a bond equals: The above formula can also be rewrien as follows: where he random variable δ(t)=x /L represens he recovery rae upon defaul. I is also naural o assume beween 0% and 100% of he bond s face value. This can be wrien as: where DT (, ) = BE ( B ( LII + XII ) F) DT (, ) = LBE ( B ( II + δ( T) II ) F) 0 X L so ha δ() T saisfies 0 δ() T 1 Alernaively, one can re-express he bond price as follows: DT (, ) = LBT ( (, ) BE ( B wt ( ) II ) F)) is he price of a uni defaul free zero coupon bond and w(t)=1 δ(t) is he wrie-down rae upon defaul. Generally, he ime- value of a corporae bond depends on he join probabiliy disribuion of under measure P* of he hreedimensional random variable (B T, δ(t), τ). Classical approach If we consider a firm wih a marke value V, and le V represen he presen value of fuure he firm s cash flows. Le K represen he value of face value of he firm s deb and T he mauriy dae of hese debs. As saed above, le s define he defaul ime τ as a discree random variable given by: To calculae he probabiliy of defaul, we make assumpions abou he disribuion of asses a deb mauriy under he probabiliy measure P*. The change of he asse prices over ime follows geomeric Brownian moion: dv = ud + σ dw, V0 = 0, V T τ> T τ T T τ> T τ T T BT (, ) = BE ( B F) τ= T if V < K τ= if else dv = uvdr +σvdw T τ T where u is he drif, σ is he volailiy parameer and W is a T 00 Risk Inernaional Yearbook 2004
sandard Brownian moion. Via Io s lemma, he soluion of he above equaion can be wrien as 1 : V = V0 exp( m + σ W ) Since W is N~(0, T), he defaul probabiliies P(T) are given by pt ( ) = PV ( T < K) = P( σwt < log L mt) log L mt = Φ T σ where L=K/V 0 is he iniial leverage raio and Φ is he sandard normal disribuion funcion. If we assume ha he firm canno repurchase is shares or issue new deb, he payoffs o he firms liabiliies a deb mauriy Τ can be summarized in Table 1. If he asse value V T is equal or greaer han he face value of he firm s deb K, he bondholder will receive he face value back, while he equiy holders will ge he difference beween he value of he firm and deb value. In he even he firm value is below he firm s deb value, he equiy will be worhless and he deb holders will assume ownership of he firm. Summary and conclusions Credi risk is he disribuion of financial losses due o unexpeced changes in he credi qualiy of counerpary in a financial agreemen. Examples range from agency downgrades o failure o service debs o liquidaion. Credi risks exis in virually all financial ransacions. The disribuion of credi losses is complex. A is cenre is he probabiliy of defaul or he likelihood of failure o honour a financial agreemen. To esimae hese probabiliies of defaul, one needs o specify a model of invesor uncerainy, a model of he available informaion and is evoluion over ime, and a model definiion of he defaul even. However, defaul probabiliies alone are no sufficien o price credi sensiive securiies. One needs, in addiion, a model for he risk free ineres rae, a model of recovery upon defaul and a model of he premium invesors require as a compensaion for bearing sysemaic credi risk. The credi premium maps acual defaul probabiliies o he marke-implied probabiliies ha are embedded in marke prices. To price securiies ha are sensiive o he credi risk of muliple issuers and o measure aggregaed porfolio credi risk, we also need o specify a model ha links defauls of several eniies 2. There are hree main quaniaive approaches o analysing credi. In he srucural approach, we ake explici assumpions abou he dynamics of a firm s asses, is capial srucure, and is deb and shareholders. A firm defauls if is asses are insufficien according o some measure. In his siuaion, a corporae liabiliy can be characerized as an opion 1 For he derivaion of his soluion see Hull (1995) or any inroducory derivaive pricing book 2 Pricing credi derivaive conracs, such as firs-o-defaul credi defaul swaps (CDS), is an example of his. The Copula echnique, which incorporaes defaul correlaions of he reference eniies in he baske, has become a common approach now o model hese ypes of conracs on he firm s asses. The reduced form approach is silen abou why a firm defauls. Insead, he dynamics of defaul are exogenously given hrough a defaul rae, or inensiy. In his approach, prices of credi sensiive securiies can be calculaed as if hey were defaul free using an ineres rae ha is he risk-free rae adjused by he inensiy. The incomplee informaion approach combines he srucural and reduced form models. While avoiding he difficulies, i picks he bes feaures of boh approaches: he economic and inuiive appeal of he srucural approach and he racabiliy and empirical fi of he reduced form approach. Abukar Ali is a research parner wih YieldCurve.com. Conac Moorad Choudhry T: +44 1483 280 217 E: amali@blueyonder.co.uk info@yieldcurve.com References 1) Black, F. and M. Scholes, The Pricing of Opions and Corporae Liabiliies Journal of Poliical Economy 81, 1973, 637-654. 2) Black, Fischer and John C. Cox, Valuing Corporae Securiies: Some Effecs of Bond Indenure Provisions, Journal of Finance, Vol. XXXI, No. 2, (May 1976), pp. 351-367. 3) Das (1995) 4) Duffie, D., K. Singleon, An economeric model of he erm srucure of ineres rae swap yields, (1997), Journal of Finance 52, pp.1287-1321 5) Geske, Rober, The Valuaion of Corporae Liabiliies as Compound Opions, Journal of Financial and Quaniaive Analysis, Vol. 12, No. 4, UCLA, (November 1977), pp. 541-552. 6) Hull, J., Inroducion o Fuures and Opions Markes, Englewood Cliffs, New Jersey: Prenice-Hall, Second ediion, 1995 7) Jarrow, Rober A. and Suar M. Turnbull. Pricing Derivaives on Financial Securiies Subjec o Credi Risk, Journal of Finance, Vol. L, No. 1, Cornell Universiy, and Queen s Universiy (Canada) (Mar- 1995), pp. 53-85 8) Longsaff and Schwarz (1993) 9) Madal, D., and Unal, H., Pricing he Risk of Ddefaul, Working paper, The Wharon School 1998 10) Meron, Rober C. On he Pricing of Corporae Deb: The Risk Srucure of Ineres Raes, Journal of Finance, Vol. 29, MIT (1974), pp. 449-470 11) F.A Longsaff and E.S. Schwarz (1995) A simple approach o valuing risky fixed and floaing. Rae deb. Journal of Finance 51, 987-1019 Risk Inernaional 0