A Note on Lando s Formula and Conditional Independence

Transcription

1 A Noe on Lando s Formula and Condiional Independence Xin Guo Rober A. Jarrow Chrisian Menn May 29, 2007 Absrac We exend Lando s formula for pricing credi risky derivaives o models where a firm s characerisics and is defaul poin process need no be condiionally independen. his resul is presened under a simple filraion expansion framework wih basic probabiliy echniques. Inroducion Credi risk refers o he risk ha he erms of a financial agreemen may no be honored. As such, credi risk modeling sudies he probabiliy of such a failure or defaul, and he loss given defaul. An ofen used formula for pricing credi derivaives is conained in [28], where defaul is formulaed as he firs jump ime of a Cox process (i.e., a doubly sochasic Poisson process) wih a random inensiy (λ ) 0 depending upon a finie dimensional sochasic process (X ) 0 characerizing he firm s economic condiion. Lando shows ha P (τ > (X s ) 0 s ) = e R 0 λ(xs) ds, () for any (X ) 0 ha is righ-coninuous wih lef limis, where P is a maringale probabiliy measure, E is a uni exponenial random variable independen of (X ) 0, and τ = inf { 0 λ(x s ) ds E }. (2) Furhermore, he price of a defaulable, zero-coupon bond wih a zero recovery rae is given by [ {τ>} E P e R ] (r s+λ s ) ds (X s ) 0 s, (3) where (r ) 0 is he defaul-free spo rae of ineres. his equaion (3) is someimes referred o as Lando s formula, and is imporance is due o he fac ha is form is he same as is defaul-free counerpar excep for a modified spo rae We are graeful o M. Jeanblanc for her generous help and insighful commens. 473 Echeverry Hall, Deparmen of Indusrial Engineering and Operaions Research. xinguo@ieor.berkeley.edu. el: UC a Berkeley, CA Sage Hall, Johnson School of Managemen. raj5@cornell.edu. el: Cornell Universiy, Ihaca, NY 4853 School of Operaions Research and Indusrial Engineering. cm328@cornell.edu. Cornell Universiy, Ihaca, NY 4853

2 (r s +λ s ). his simple modificaion enables he usage of numerous heorems available from he erm srucure of ineres rae lieraure o price credi risky derivaives. he usefulness of his approach has led o various generalizaions, see for example [2, 4, 5, 5, 7, 24, 30]. Our approach akes as given a (defaul ime, filraion) pair (τ, F) where τ is a non-negaive random variable, no necessarily an F-sopping ime, and F = (F ) 0 is a filraion on a probabiliy space (Ω, F, P ). Nex, we define G o be any filraion expansion of F such ha τ is a G-sopping ime wih G { < τ} = F { < τ}. (4) Under his filraion expansion, we re-derive Lando s formula using an inensiy based reduced-form model. We hen exend Lando s formula for a class of srucural models where he firm s asse value process is observed discreely. In his class, Lando s formula holds wihou he usual condiional independence assumpion beween he firm s defaul poin process and sae variables. And he proof uses only simple probabiliy ools and he local deerminisic propery of he inensiy process. A word of cauion here: his paper is by no means any aemp o review he vas exising lieraure on probabiliy ools and models in credi risk. Raher, we aim a unwinding he relaion of condiional independence assumpion wih he Lando s formula wih he minimal probabiliy background possible, and wih a genle ouch on he filraion expansion heory. More mahemaically sophisicaed and curious minds are direced o [7, 24, 6], and o [5, 6, 36, 29] for comprehensive reviews. An ouline of our paper is as follows. Secion 2 presens he mahemaical seup. Lando s formula and is generalizaion are presened in Secion 3. For compleeness, he equivalence beween wo differen definiions of an inensiy process for any oally inaccessible sopping ime τ for a given filraion G is provided in he Appendix. 2 he Mahemaical Seup 2. Defaul ime τ, Filraion (F ) 0, and Inensiy of τ Le us sar wih (Ω, F, F, P ), a complee filered probabiliy space wih F = (F ) 0 an arbirary filraion ha conains he null ses and is righ-coninuous. Le τ be a non-negaive random variable wih (N ) 0 is associaed poin process, i.e. N = {τ }. For simpliciy, we assume ha P (τ = ) = 0 for any. Now define Z as he F-opional projecion of {τ>}, i.e., Z = E[ {τ>} F ] = P (τ > F ). (5) hen, (Z ) 0 is clearly an F-supermaringale. In he conex of credi risk modeling, τ represens he defaul ime of a firm, wih F being he informaion available o invesors a ime, N he defaul indicaor, and Z = P (τ > F ) he condiional survival probabiliy a he ime given he informaion F. Usually, he defaul ime τ and he survival probabiliy Z are he focus of economic analysis. For any arbirary non-negaive random variable τ and a given filraion F, τ may no be an F -sopping ime. For example, consider a reduced-form model where F = σ(x s, s ) and τ is he firs jump ime of a Cox process wih a random inensiy depending upon X. Here τ is no an F-sopping ime. In his case, one can enlarge he filraion F o a bigger filraion G o make τ a G-sopping ime. hese wo condiions are someimes referred o as he usual condiions. 2

3 Moreover, given such a riple (τ, F, G), (N ) 0 is a G-submaringale. herefore, by he Doob Meyer decomposiion, here exiss a unique increasing G-predicable process (A G ) 0 wih A G 0 = 0 such ha (N A G ) 0 is a G-maringale. Here (A G ) 0 is called he G-compensaor of τ. 2 Finally, if (A G ) 0 is a.s. absoluely coninuous wih respec o Lebesgue measure, hen he Radon Nikodym derivaive (da G /d) 0 is called he inensiy process (λ ) 0 of τ (see Brémaud [7, Chaper II, D7, 2, and 3]). In he Appendix, we idenify under proper inegrabiliy condiions he inensiy process wih is more inuiive counerpar, known as he Meyer s Laplacian approximaion. 2.2 he Filraion Expansion G of F For a given pair (τ, F) for which τ is no an F-sopping ime, i is well known ha here are many ways o expand he filraion (F ) 0 in order o make τ a sopping ime. wo sandard approaches are he progressive filraion G and he minimal filraion F(τ). he progressive filraion G is [25, 37] obained by defining G = F σ(τ) wih F = 0 F, and G := {B G B F : B { < τ} = B { < τ}}. (6) hen G = (G ) 0 saisfies he usual condiions and i makes τ ino a sopping ime. Noe ha F {τ } G, implying ha afer ime τ, he progressively expanded filraion a ime includes all he informaion from he original filraion F up o. In some economic seings, his could make he progressive filraion expansion unnaural for credi risk modeling. he minimal filraion F(τ) is obained by enlarging F in he minimal way so as o include τ as a sopping ime, i.e. F (τ) = F σ( τ) = F σ({τ s} s ). (7) his ype of filraion expansion is explored, for example, in [24], [27], [7], [4], [9], and [6]. Noe ha F (τ) G from expressions (6) and (7). And, boh G and he righ-coninuous augmenaion of F (τ) coincide wih F on { < τ}. Furhermore, many quaniies associaed wih a defaul ime, such as he inensiy process, are well-defined only up o τ (see [24] for more discussions.) In view of hese observaions, i suffices o specify he filraion expansion G = (G ) 0 o be any filraion expansion of F such ha τ is a G-sopping ime wih G { < τ} = F { < τ}. (8) Given expression (8), we see F(τ) G. his is inuiive since a any give ime, one should know wheher defaul happens or no by ime. Clearly boh F(τ) and G are special cases. (Ineresed readers are referred o [3], where his consrucion seems o firs appear, for more discussions on his ype of filraion). hroughou he main secion of his paper, unless oherwise specified, we consider only a nonnegaive random variable τ, he couning process of defaul (N ) 0, an arbirary filraion F, and is filraion enlargemen G saisfying expression (8). his framework makes sense due o he following lemma (see [20]). 2 Noe ha for purpose of exposiion, he superscrip G is used here for A o indicae he dependence of A on G. 3

4 Lemma. Compensaors of τ under he expansion of class (8) are idenical (up o τ). In paricular, compensaors of τ under is progressive expansion of F and under is minimal expansion of F are idenical. 3 A Generalized Lando s Formula his secion derives wo exensions of Lando s formula using G from expression (8). he firs exension is using he sandard assumpion of condiional independence in an inensiy based model, and he second exension is wihou condiional independence for a class of srucural models. 3. Wih Condiional Independence his secions re-derives Lando s formula [28], assuming condiional independence, bu under G of (8). Firs, as in [7, Proposiion 3.]: heorem. For any F -measurable inegrable random variable X, [ E[X {τ> } G ] = {τ>} E X Z ] F for <. (9) Z Proof. Since X is F measurable, by properies of condiional expecaions and he Bayes lemma (see for insance [7]), {τ>} E[X Z [ F ] = Z {τ>} E X P (τ > F ] ) P (τ > F ) F = {τ>} E[E[X {τ> } F ] F ] E[ {τ>} F ] = {τ>} E[X {τ> } F ] E[ {τ>} F ] = E[X {τ> } G ]. Nex, leing τ be he firs jump ime of a Cox process so ha expression () holds, seing X and F = σ(x s, s ), we ge 3 [ ] P (τ > F ) P (τ > G ) = {τ>} E P (τ > F ) F [e R ] 0 λ(x s) ds = {τ>} E e R 0 λ(xs) ds F = {τ>} E[e R λ(x s ) ds F ]. Las, leing (r ) 0 be he F-adaped defaul-free spo rae of ineres, hen under suiable inegrabiliy condiions as in [28, Proposiion 3.], and by argumens as in Proposiion and in [28], we obain he following exensions of Lando s formulas [28] under G. 3 his is he essence of Proposiion 3. in Lando [28]. 4

5 Corollary. Assume he condiional independence in he sense of expression (). Assume ha X is F -measurable, and Y and V are F-adaped where F = σ(x s, s ), so ha E [e R ] r s ds X, [ E e R ] [ s r u du Y s ds and E e R ] s (r u+λ u ) du V s λ s ds are all finie. hen, and E [e R ] r s ds X {τ> } G = {τ>} E [e R ] (rs+λs) ds X F, ( E Y s {τ>s} e R ) [ s r u du ds G = {τ>} E Y s e R ] s (r u+λ u ) du ds F, E [e R ] [ τ r s ds V τ G = {τ>} E V s λ s e R ] s (r u+λ u ) du ds F. 3.2 Wihou Condiional Independence In his secion we derive Lando s formula for a naurally occurring class of filraions F ha do no a priori assume condiional independence. As an imporan special case, his class includes he siuaion where a firm s coninuous ime asse value process is only observed a discree ime inervals. heorem 2 (Generalized Lando s Formula under Discree Observaions). Le (X ) 0 be a onedimensional, ime-homogeneous Markov process wih a coninuous sample pah. Le ( n ) n N be a deerminisic and sricly increasing sequence. If τ = inf{ > 0 X D} for a Borel subse of he sae space D has a coninuous probabiliy densiy funcion f(x, ) where f(x, )d = P x (τ d), and if τ > and n < n+, hen we have e R n+ λ s ds {τ>} = P (τ > n+ G ), (0) where F = {σ(x i ), i < }, G = (G ) 0 is any enlarged filraion of F from expression (8), and λ s = f(x X n,s n) P Xn (τ>s n ) for s [, n+). he proof relies on he following resul (see [33, Pages 34-35]). Lemma 2. Le (X ) 0 be a one-dimensional, ime-homogeneous Markov process wih a coninuous sample pah. Le F = {σ(x i ), i < } be a deerminisic delayed filraion. If τ = inf{ > 0 X D} for a Borel subse of he sae space D has a coninuous densiy funcion f(x, ) where f(x, )d = P x (τ d), hen on τ > and n < n+, he F(τ)- sopping ime τ has a defaul inensiy f(x X n, n) P X n (τ> n). Proof. Clearly, for s = = n, λ s = 0, so wihou loss of generaliy we consider only n < < n+ and F = {σ(x i ), i < }. In his case, λ s = f(x X n,s n) P X (τ>s n) for τ > and s [, n+ ). hus n 5

6 denoing Z s = P X n (τ > s n), we see e R n+ λ s ds {τ>} = e = e R n+ R n+ Z (s) Z(s) ds {τ>} dz(s) Z(s) {τ>} = P X n (τ > n+ n ) P (τ > {τ>} X n n) = {τ> n }P (τ > X n n+ n ) {τ>n}p Xn (τ > n ) {τ>} = P (τ > n+ F n ) P (τ > F n ) = P [τ > n+ G ]. {τ>} he Markov propery is applied o ge he fifh equaion, while he las equaion is due o he Bayes formula. Discussion: A comparison wih Lando [28]. Recall ha in an inensiy based reduced-form model as in [28], he defaul ime represens he firs jump ime of a Cox process, wih inensiy λ = (λ(x )) 0. And expression () is derived using he condiional independence assumpion beween (X ) 0 and E, which also yields he following relaion [28, Page 04, Eq. (3.4)]: E [ {τ> } σ((x s ) 0 s ) σ((n s ) 0 s ) ] ( ) = {τ>} exp λ s ds. () he filraion in expression () involves σ((x s ) 0 s ), he informaion up o ime. In conras, our resul holds boh wih and wihou he condiional independence assumpion. Moreover, in he laer case where he filraion is generaed by he discree observaions of X, we see ha P (τ > n+ G ) = E[ {τ>n+ } G ] = {τ>} e R n+ λ s ds (2) where G akes he form of σ((x i, i ) σ((n s ) 0 s ) according o heorem 2. Noe also ha τ in expression (2) is no he firs jump ime of a Cox process. Insead, τ := inf{ > 0, X D} for some Borel subse of he sae space. his laer formulaion is he sandard defaul ime definiion used in srucural models for credi risk. Finally, we noe ha [24] proposes an idea similar o ha in heorem 2. References [] M. Ammann. Credi Risk Valuaion, Springer, Heidelberg, (200). [2] P. Arzner and F. Delbaen. Defaul risk insurance and incomplee marke. Mahemaical Finance, 5, 87 95, (995). [3]. Aven. A heorem for deermining he compensaor of a couning process. Scandinavian Journal of Saisics, 2, 69 72, (985). 6

7 [4] A. Bélanger, S. Shreve, and D. Wong. A general framework for pricing credi risk. Mahemaical Finance, 4(3), , (2004). [5]. Bielecki and M. Rukowski. Credi Risk: Modeling, Valuaion, and Hedging, Springer, Heidelberg, (2002). [6]. Bielecki, M. Jeanblanc, and M. Rukowski. Credi Risk, Lecure of M. Jeanblanc, Lisbonn, June [7] P. Brémaud. Poin Processes and Queues: Maringale Dynamics, Springer, Heidelberg, (98). [8] P. Brémaud and M. Yor. Change of filraion and of probabiliy measure. Z. Wahr. Verw. Gebiee, 45: , (978). [9] U. Çein, R. Jarrow, P. Proer, and Y. Yildirim. Modeling credi risk wih parial informaion. Annals of Applied Probabiliy, 4, 67 78, (2004). [0] P. Collin-Dufresne, R. Goldsein, and J. Helwege. Is credi even risk priced? Modeling conagion via he updaing of beliefs. Working paper, Carnegie Mellon Universiy, (2003). [] M. Davis. Hazard raes in he credi grades model, unpublished manuscrip, (2002). [2] C. Dellacherie and P. A. Meyer. Probabiliies and Poenial, Chaper I-IV, Norh-Holland Publishing Co., Amserdam, (978). [3] C. Dellacherie, B. Maisooneuve, and P. A. Meyer. Probabiliés e Poeniel, Chapires XVII- XXIV, Processus de Markov (fin). Complémens de calcul sochasique, Hermann, Paris, (992). [4] D. Duffie and D. Lando. erm srucures and credi spreads wih incomplee accouning informaion. Economerica, 69, , (200). [5] D. Duffie and K. Singleon. Modeling erm srucure of defaulable bonds. Review of Financial Sudies, 2, , (999). [6] D. Duffie and K. Singleon. Credi Risk, Princeon Universiy Press, Princeon (2003). [7] R. Ellio, M. Jeanblanc, and M. Yor. On models of defaul risk. Mahemaical Finance, 0, 79 95, (2000). [8] K. Giesecke. Defaul and Informaion, Journal of Economic Dynamics and Conrol, (2006). (o appear) [9] X. Guo, R. Jarrow and Y. Zeng. Credi risk wih incomplee informaion. (Earlier version under he ile Informaion reducion in credi risk models (2005)). Revised, (2006). [20] X. Guo and Y. Zeng. Compensaors and inensiy processes: a new filraion expansion wih Jeulin Yor formula. Annals of Applied Pribabiliy, o appear, (2006). [2] R. Jarrow and P. Proer. Srucural versus reduced-form models: A new informaion based perspecive. Journal of Invesmen Managemen, 2(2), 34 43, (2004). 7

8 [22] R. Jarrow and S. urnbull. Credi Risk: Drawing he analogy. Risk Magazine, 5 (9), (992). [23] M. Jeanblanc and S. Valchev: Parial informaion and hazard process. Preprin, (2003). [24] M. Jeanblanc and M. Rukowski. Modeling of defaul risk: an overview, Shanghai summer school Augus 999. Mahemaical finance: heory and pracice, J. Yong and R. Con R. (eds), Higher Educaion Press, (999). [25]. Jeulin and M. Yor. Grossissemen d une filraion e semi-maringales: formules explicies. Séminaire de Probabiliés XII, 78 97, Springer, Heidelberg, (978). [26] J. P. Klein and M. L. Moeschberger Survival Analysis echniques for Censored and runcaed Daa, Springer, Heidelberg, (997). [27] S. Kusuoka. A remark on defaul risk models, Advances in Mahemaical Economics,, 69 82, (999). [28] D. Lando. Cox processes and credi-risky securiies. Review of Derivaive Research, 2, 99 20, (998). [29] D. Lando. Credi Risk Modeling: heory and Applicaions, Princeon Universiy Press, Princeon, (2004). [30] D. Madan and H. Unal. Pricing he risk of defaul. Review of Derivaive Research, 2, 2 60, (995). [3] R. Mansuy and M. Yor. Random imes and Enlargemens of Filraions in a Brownian Seing, Lecure Noes in Mahemaics 873, Springer, Heidelberg, (2006). [32] A. Nikeghbali and M. Yor. A definiion and some properies of pseudo-sopping imes. Annals of Probabiliy, 33, , (2005). [33] J-L. Prigen. Weak Convergence of Financial Markes, Springer Finance, Berlin, (2003). [34] P. Proer. Sochasic Inegraion and Differenial Equaions, 2nd ediion, Springer, (2004). [35] D. Revuz and M. Yor. Coninuous Maringales and Brownian Moion. Springer, Heidelberg, (99). [36] P. J. Schönbucher. Credi Derivaives Pricing Models: Model, Pricing and Implemenaion, Wiley (2003). [37] M. Yor. Grossissemen d une filraion e semi-maringales: héorèmes généraux. Séminaire de Probabiliés, XII, 6 69, Lecure Noes in Mah., 649, Springer, Berlin, Appendix: he G-inensiy and Meyer s Laplacian Approximaion In his Appendix, we idenify he inensiy process wih he Meyer s Laplacian approximaion, under proper inegrabiliy condiions. I is imporan o emphasize ha here we assume G o be any appropriae filraion expansion of F, and no necessarily of expression (8). 8

9 4. (λ ) 0 from Meyer s Laplacian Approximaion Duffie and Lando [4] showed ha if he hypohesis for using dominaing convergence is saisfied as required by Aven s lemma [3], hen he G-inensiy can be calculaed via Meyer s Laplacian approximaion. 4 ha is, le λ = lim h 0 h P ( < τ + h G ). (3) so ha λ is he insananeous likelihood of defaul a ime given G. hen, under Aven s lemma, i idenifies wih λ, he Radon Nikodym derivaive of he G-compensaor of τ. he converse is rue: if he Radon Nikodym derivaive of he G-compensaor saisfies proper inegrabiliy condiions, hen i idenifies wih he Laplacian approximaion λ. We provide several such condiions below. For simpliciy, we will assume wihou loss of generaliy ha 0 < Z < for any > 0, and impose he following wo assumpions. Assumpion A. he G-sopping ime τ has an inensiy process (λ ) 0 wih righ-coninuous sample pahs. Assumpion A2. he limi λ = lim h 0 h P ( < τ + h G ) exiss almos surely and is a.s. righ-coninuous. Proposiion. Given any G-sopping ime τ, ogeher wih assumpions A and A2. If for any given > 0 here exiss an h 0 () so ha ( +h h λ s ) h>0 is uniformly inegrable for h (0, h 0 ], hen (λ ) 0 is indisinguishable from ( λ ) 0. Proof. By he righ-coninuiy of λ and λ, i suffices o show ha for any > 0, λ = λ a.s. Firs, ake he Doob Meyer decomposiion of (N ) 0 so ha N = M + A G, where (M ) 0 is an G-maringale and (A G ) 0 is an increasing G-predicable process. Since A G = 0 λ sds, hen for any fixed, h P ( < τ + h G ) = h E [N +h N G ] = [ ] h E A G +h AG G = [ +h ] h E λ s ds G. Nex, he righ-coninuiy of λ implies lim h 0 h λ s = λ. his ogeher wih he uniform inegrabiliy shows ha ( +h h λ s ) h>0 converges o λ in L. Hence ( [ +h ]) E λ s ds G λ in L. h herefore, +h h>0 [ λ = lim h 0 h P ( < τ + h G ) = lim E h 0 h +h λ s ds G ] = λ. here are various echnical condiions sufficien for obaining he uniform inegrabiliy condiion, see [35]. Here are a few possibiliies. 4 he link beween hazard rae (or defaul inensiy) and survival probabiliy has long before been sudied in he survival analysis lieraure (see e.g., [26] for an overview) bu he role of he filraion has no been examined. 9

10 Corollary 2. Given any G-sopping ime τ, ogeher wih assumpions A and A2. If for any given > 0 here exiss p = p() > and h 0 = h 0 () > 0 so ha +h h E[λ p s] is bounded for h (0, h 0 ], hen λ is indisinguishable from λ. Proof. Le q be he number conjugae o p so ha p + q =, hen for h sufficienly small, [ ( +h ) p ] E λ s E +h p +h p λ p s q h h [ +h ] = E h p h p λ p s = +h E[λ p h s]. So ( h +h λ s is uniformly inegrable, and saemen now follows from Proposiion. )h>0 Corollary 3. Given a G-sopping ime τ, ogeher wih assumpions A and A2. If E{λ p } is bounded on any finie inerval, hen λ is indisinguishable from λ. Remark. o our bes knowledge, he only oher exising reference concerning he equivalence of an inensiy process of a sopping ime and is Meyer s Laplacian approximaion is Proposiion 5.0 of [8]. We give a more general characerizaion by specifying weaker condiions of λ. 0