Credit risk. T. Bielecki, M. Jeanblanc and M. Rutkowski. Lecture of M. Jeanblanc. Preliminary Version LISBONN JUNE 2006

Transcription

1 i Credi risk T. Bielecki, M. Jeanblanc and M. Rukowski Lecure of M. Jeanblanc Preliminary Version LISBONN JUNE 26

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3 Conens Noaion vii 1 Srucural Approach Basic Assumpions Defaulable Claims Risk-Neural Valuaion Formula Defaulable Zero-Coupon Bond Classic Srucural Models Meron s Model Black and Cox Model Furher Developmens Opimal Capial Srucure Sochasic Ineres Raes Random Barrier Independen barrier Commens on Srucural Models Hazard Funcion Approach The Toy Model Defaulable Zero-coupon wih Paymen a Mauriy Defaulable Zero-coupon wih Paymen a Hi Implied probabiliies Spreads Toy Model and Maringales Key Lemma Some Maringales Represenaion Theorem Change of a Probabiliy Measure Incompleeness of he Toy model Risk Neural Probabiliy Measures Parial informaion: Duffie and Lando s model Pricing and Trading Defaulable Claims iii

4 iv CONTENTS Recovery a mauriy Recovery a defaul ime Pricing and Trading a CDS Valuaion of a Credi Defaul Swap Marke CDS Rae Price Dynamics of a CDS Hedging of Defaulable Claims Successive defaul imes Two imes Poisson Jumps Cox Processes and Exensions Consrucion of Cox Processes wih a given sochasic inensiy Condiional Expecaions Choice of filraion Key lemma Condiional Expecaion of F -Measurable Random Variables Exension Dynamics of prices Defaulable Zero-Coupon Bond Recovery wih Paymen a mauriy Recovery wih Paymen a Defaul ime Price and Hedging a Defaulable Call Corporae bond Term Srucure Models Jarrow and Turnbull s model Vacicek Model The CIR model Analysis of Several Random Times Hazard process Approach General case The model Key lemma Maringales Inerpreaion of he inensiy Resricing he informaion Enlargemen of filraion H Hypohesis Complee model case Definiion and Properies of H Hypohesis

5 CONTENTS v Change of a probabiliy measure Sochasic Barrier Represenaion heorem Generic Defaulable Claims Buy-and-hold Sraegy Spo Maringale Measure Self-Financing Trading Sraegies Maringale Properies of Prices of a Defaulable Claim Parial informaion Informaion a discree imes Delayed informaion Inensiy approach Aven s Lemma Ordered Random Times Properies of he Minimum of Several Random Times Change of a Probabiliy Measure Kusuoka s Example Hedging Semimaringale Model wih a Common Defaul Dynamics of asse prices Trading Sraegies in a Semimaringale Se-up Unconsrained sraegies Consrained sraegies Maringale Approach o Valuaion and Hedging Defaulable asse wih oal defaul Defaulable asse wih non-zero recovery Two defaulable asses wih oal defaul PDE Approach o Valuaion and Hedging Defaulable asse wih oal defaul Defaulable asse wih non-zero recovery Two defaulable asses wih oal defaul Indifference pricing Defaulable Claims Hodges Indifference Price Hodges prices relaive o he reference filraion Soluion of Problem PF X Exponenial Uiliy: Explici Compuaion of he Hodges Price Risk-Neural Spread Versus Hodges Spreads Recovery paid a ime of defaul

6 vi CONTENTS 6.3 Opimizaion Problems and BSDEs Opimizaion Problem Hodges Buying and Selling Prices Quadraic Hedging Quadraic Hedging wih F-Adaped Sraegies Quadraic Hedging wih G-Adaped Sraegies Jump-Dynamics of Price MeanVariance Hedging Quanile Hedging Dependen Defauls and Credi Migraions Baske Credi Derivaives Differen Filraions The i h -o-defaul Coningen Claims Case of Two Eniies Role of H hypohesis Condiionally Independen Defauls Independen Defaul Times Signed Inensiies Valuaion of FDC and LDC Copula-Based Approaches Direc Applicaion Indirec Applicaion Lauren and Gregory s model Two defauls, rivial reference filraion Applicaion of Norros lemma for wo defauls Jarrow and Yu Model Consrucion and Properies of he Model Exension of Jarrow and Yu Model Kusuoka s Consrucion Inerpreaion of Inensiies Bond Valuaion Defaulable Term Srucure Sanding Assumpions Credi Migraion Process Defaulable Term Srucure Premia for Ineres Rae and Credi Even Risks Defaulable Coupon Bond Examples of Credi Derivaives Markovian Marke Model Descripion of some credi baske producs

7 Noaion vii Valuaion of Baske Credi Derivaives in he Markovian Framework Appendix Hiing imes Hiing imes of a level and law of he maximum for Brownian moion Hiing imes for a Drifed Brownian moion Hiing Times for Geomeric Brownian Moion Oher processes Non-consan Barrier Fokker-Planck equaion Copulas Poisson processes Sandard Poisson process Inhomogeneous Poisson Processes General heory Semimaringales Inegraion by pars formula for finie variaion processes Inegraion by pars formula for mixed processes Doléans-Dade exponenial Iô s formula Sopping imes Enlargemens of Filraions Progressive Enlargemen Markov Chains Dividend paying asses Discouned Cum-dividend Prices Ornsein-Uhlenbeck processes Vacisek model Cox-Ingersoll-Ross Processes CIR Processes and BESQ Transiion Probabiliies for a CIR Process CIR Processes as Spo Rae Models Zero-coupon Bond Parisian Opions The Law of G,l D W, W G ,l D Valuaion of a Down and In Parisian Opion Index 228 Noaion B, T Price a ime of a zero coupon wih mauriy T, 19

8 viii Noaion D R, T Price a ime of a corporae bond which pays R a defaul ime and 1 a mauriy, if he defaul has no occurred before mauriy, 22 D R,T Price of a defaulable bond, which pays 1 a mauriy if no defaul and R a mauriy T if he defaul has occurred before mauriy, 2 F : condiional probabiliy, 63 Γ: hazard funcion, 21 Γ : hazard process, 63 g b X: las ime before a which he process X is a he level b, 226 T : rivial filraion, 172

9 Inroducion These noes are mainly based on he papers of Bielecki, Jeanblanc and Rukowski: Hedging of defaulable claims, Lecure Noes in Mahemaics, 1847, pages 1 132, Paris-Princeon, Springer-Verlag, 24, R.A. Carmona, E. Cinlar, I. Ekeland, E. Jouini, J.E. Scheinkman, N. Touzi, eds. Sochasic Mehods In Credi Risk Modelling, Valuaion And Hedging, Lecure Noes in Mahemaics, Frielli, M. ed, CIME-EMS Summer School on Sochasic Mehods in Finance, Bressanone, Springer, 24. Hedging of credi derivaives in models wih oally unexpeced defaul, in Sochasic processes and applicaions o mahemaical finance, Akahori, J. Ogawa, S. and Waanabe S. Ed, p Proceedings of he 5h Risumeikan Inernaional conference, World Scienific Pricing and hedging Credi defaul Swaps Work in progress. and on he book of T.R. Bielecki and M. Rukowski: Springer Verlag, 21. Credi risk : Modelling valuaion and Hedging, The reader can find oher ineresing informaion on he web sies quoed a he end of he bibliography of his documen. The goal of his lecure is o presen a survey of recen developmens in he area of mahemaical modeling of credi risk and credi derivaives. Credi risk embedded in a financial ransacion is he risk ha a leas one of he paries involved in he ransacion will suffer a financial loss due o decline in he crediworhiness of he couner-pary o he ransacion, or perhaps of some hird pary. For example: A holder of a corporae bond bears a risk ha he marke value of he bond will decline due o decline in credi raing of he issuer. A bank may suffer a loss if a bank s debor defauls on paymen of he ineres due and or he principal amoun of he loan. A pary involved in a rade of a credi derivaive, such as a credi defaul swap CDS, may suffer a loss if a reference credi even occurs. The marke value of individual ranches consiuing a collaeralized deb obligaion CDO may decline as a resul of changes in he correlaion beween he defaul imes of he underlying defaulable securiies i.e., of he collaeral. The mos exensively sudied form of credi risk is he defaul risk ha is, he risk ha a counerpary in a financial conrac will no fulfil a conracual commimen o mee her/his obligaions saed in he conrac. For his reason, he main ool in he area of credi risk modeling is a judicious specificaion of he random ime of defaul. A large par of he presen ex will be devoed o his issue. 1

10 2 CHAPTER. INTRODUCTION Our main goal is o presen he mos imporan mahemaical ools ha are used for he arbirage valuaion of defaulable claims, which are also known under he name of credi derivaives. We also examine he imporan issue of hedging hese claims. In Chaper 1, we provide a concise summary of he main developmens wihin he so-called srucural approach o modeling and valuaion of credi risk. We also sudy he random barrier case. Chaper 2 is devoed o he sudy of a oy model wihin he hazard process framework. Chaper 3 sudies he case of Cox processes. Chaper 4 is devoed o he reduced-form approach. This approach is purely probabilisic in naure and, echnically speaking, i has a lo in common wih he reliabiliy heory. Chaper 5 sudies hedging sraegies under assumpion ha a defaulable asse is raded. Chaper 6 sudies differen ways o give a price in incomplee marke seing. Chaper 7 provides an inroducion o he area of modeling dependen credi migraions and defauls. An appendix recalls some noion of sochasic calculus and probabiliy heory. Le us only menion ha he proofs of mos resuls can be found in Bielecki and Rukowski [23], Bielecki e al. [15, 18, 2] and Jeanblanc and Rukowski [119]. We quoe some of he seminal papers; he reader can also refer o books of Bruyère [39], Bluhm e al. [28], Bielecki and Rukowski [23], Cossin and Piroe [52], Duffie and Singleon [74], Frey, McNeil and Embrechs [89], Lando [143], Schönbucher [17] for more informaion. A he end of he bibliography, we also menion some web addresses where aricles can be downloaded. Finally, i should be acknowledged ha some resuls especially wihin he reduced form approach were obained independenly by various auhors, who worked under differen se of assumpions and wihin disinc seups, and hus we decided o omi deailed credenials in mos cases. We hope our colleagues will accep our apologies for his deficiency, and we sress ha his by no means signifies ha hese resuls ha are no explicily aribued are ours. Begin a he beginning, and go on ill you come o he end. Then, sop. L. Carroll, Alice s Advenures in Wonderland

11 Chaper 1 Srucural Approach In his chaper, we presen he srucural approach o modeling credi risk i is also known as he value-of-he-firm approach. This mehodology direcly refers o economic fundamenals, such as he capial srucure of a company, in order o model credi evens a defaul even, in paricular. As we shall see in wha follows, he wo major driving conceps in he srucural modeling are: he oal value of he firm s asses and he defaul riggering barrier. This was hisorically he firs approach used in his area, and i goes back o he fundamenal papers by Black and Scholes [26] and Meron [159]. 1.1 Basic Assumpions We fix a finie horizon dae T >, and we suppose ha he underlying probabiliy space Ω, F, P, endowed wih some reference filraion F = F 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 firm s value process V, which is inerpreed as a model for he oal value of he firm s asses. The barrier process v, which will be used in he specificaion of he defaul ime τ. The promised coningen claim X represening he firm s liabiliies o be redeemed a mauriy dae T T. The process C, which models he promised dividends, i.e., he liabiliies sream ha is redeemed coninuously or discreely over ime o he holder of a defaulable claim. The recovery claim X represening he recovery payoff received a ime T, if defaul occurs prior o or a he claim s mauriy dae T. The recovery process Z, which specifies he recovery payoff a ime of defaul, if i occurs prior o or a he mauriy dae T Defaulable Claims Technical Assumpions. We posulae ha he 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 =. We assume wihou menioning ha all random objecs inroduced above saisfy suiable inegrabiliy condiions. 3

12 4 CHAPTER 1. STRUCTURAL APPROACH Probabiliies P and Q. 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 Q in wha follows. Defaul Time. In he srucural approach, he defaul ime τ will be ypically defined in erms of he firm s value process V and he barrier process v. We se τ = inf { > : T and V v } wih he usual convenion ha he infimum over he empy se equals +. In main cases, he se T is an inerval [, T ] or [, in he case of perpeual claims. In firs passage srucural models, he defaul ime τ is usually given by he formula: τ = inf { > : [, T ] and V v}, where v : [, T ] IR + is some deerminisic funcion, ermed he barrier. Predicabiliy of Defaul Time. Since he underlying filraion F in mos srucural models is generaed by a sandard Brownian moion, τ will be an F-predicable sopping ime as any sopping ime wih respec o a Brownian filraion: here exiss a sequence of increasing sopping imes announcing he defaul ime. 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 he mauriy dae T, or 2 he amoun Z τ is paid a ime τ. In he case when defaul occurs a mauriy, i.e., on he even {τ = T }, we posulae ha only he recovery paymen X is paid. In a general seing, we consider simulaneously boh kinds of recovery payoff, and hus a generic defaulable claim is formally defined as a quinuple X, C, X, Z, τ Risk-Neural Valuaion Formula Suppose ha our financial marke model is arbirage-free, in he sense ha here exiss a maringale measure risk-neural probabiliy Q, meaning ha price process of any radeable securiy, which pays no coupons or dividends, becomes an F-maringale under Q, when discouned by he savings accoun B, given as B = exp r u du. We inroduce he jump process H = 1 {τ }, and we denoe by D he process ha models all cash flows received by he owner of a defaulable claim. Le us denoe X d T = X 1 {τ>t } + X 1 {τ T }. Definiion The dividend process D of a defaulable coningen claim X, C, X, Z, τ, which seles a ime T, equals D = X d T 1 { T } + 1 H u dc u + Z u dh u. ],] ],] I is apparen ha D is a process of finie variaion, and 1 H u dc u = 1 {τ>u} dc u = C τ 1 {τ } + C 1 {τ>}. ],] ],] Noe ha if defaul occurs a some dae, he promised dividend C C, which is due o be paid a his dae, is no received by he holder of a defaulable claim. Furhermore, if we se τ = min {τ, } hen Z u dh u = Z τ 1 {τ } = Z τ 1 {τ }. ],]

13 1.1. BASIC ASSUMPTIONS 5 Remark In principle, 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. We are in he posiion o define he ex-dividend price S 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. Definiion For any dae [, T [, he ex-dividend price of he defaulable claim X, C, X, Z, τ is given as S = B E Q B 1 u dd u F. 1.1 ],T ] In addiion, we always se S T = X d T. The discouned ex-dividend price S, [, T ], saisfies S = S B 1 Bu 1 dd u, [, T ], ],] and hus i follows a supermaringale under Q if and only if he dividend process D is increasing. The process S + B ],] B 1 u dd u is also called he cum-dividend process Defaulable Zero-Coupon Bond Assume ha C, Z and X = L for some posiive consan L >. Then he value process S represens he arbirage price of a defaulable zero-coupon bond also known as he corporae discoun bond wih he face value L and recovery a mauriy only. In general, he price D, T of such a bond equals D, T = B E Q B 1 T L 1 {τ>t } + X 1 {τ T } F. I is convenien o rewrie he las formula as follows: D, T = LB E Q B 1 T 1 {τ>t } + δt 1 {τ T } F, where he random variable δt = X/L represens he so-called recovery rae upon defaul. I is naural o assume ha X L so ha δt saisfies δt 1. Alernaively, we may re-express he bond price as follows: D, T = L B, T B E Q B 1 T wt 1 {τ T } F, where B, T = B E Q B 1 T F is he price of a uni defaul-free zero-coupon bond, and wt = 1 δt is he wriedown rae upon defaul. Generally speaking, he ime- value of a corporae bond depends on he join probabiliy disribuion under Q of he hree-dimensional random variable B T, δt, τ or, equivalenly, B T, wt, τ. Example Meron [159] posulaes ha he recovery payoff upon defaul I.E., when V T < L, equals X = V T, where he random variable V T is he firm s value a mauriy dae T of a corporae bond. Consequenly, he random recovery rae upon defaul equals δt = V T /L, and he wriedown rae upon defaul equals wt = 1 V T /L.

14 6 CHAPTER 1. STRUCTURAL APPROACH Expeced Wriedowns. For simpliciy, we assume ha he savings accoun B is non-random ha is, he shor-erm rae r is deerminisic. Then he price of a defaul-free zero-coupon bond equals B, T = B B 1 T, and he price of a zero-coupon corporae bond saisfies D, T = L 1 w, T, where L = LB, T is he presen value of fuure liabiliies, and w, T is he condiional expeced wriedown rae under Q. I is given by he following equaliy: w, T = E Q wt 1{τ T } F. The condiional expeced wriedown rae upon defaul equals, under Q, w = E Q wt 1{τ T } F = w, T Q{τ T F } p, where p = Q{τ T F } is he condiional risk-neural probabiliy of defaul. Finally, le δ = 1 w be he condiional expeced recovery rae upon defaul under Q. In erms of p, δ and p, we obain D, T = L 1 p + L p δ = L 1 p w. If he random variables wt and τ are condiionally independen wih respec o he σ-field F under Q, hen we have w = E Q wt F. Example In pracice, i is common o assume ha he recovery rae is non-random. Le he recovery rae δt be consan, specifically, δt = δ for some real number δ. In his case, he wriedown rae wt = w = 1 δ is non-random as well. Then w, T = wp and w = w for every T. Furhermore, he price of a defaulable bond has he following represenaion D, T = L 1 p + δl p = L 1 wp. We shall reurn o various recovery schemes laer in he ex. 1.2 Classic Srucural Models Classic srucural models are based on he assumpion ha he risk-neural dynamics of he value process of he asses of he firm V are given by he SDE: dv = V r κ d + σv dw, V >, where κ is he consan payou dividend raio, and he process W is a sandard Brownian moion under he maringale measure Q Meron s Model We presen here he classic model due o Meron [159]. Basic assumpions. A firm has a single liabiliy wih promised erminal payoff L, inerpreed as he zero-coupon bond wih mauriy T and face value L >. The abiliy of he firm o redeem is deb is deermined by he oal value V T of firm s asses a ime T. Defaul may occur a ime T only, and he defaul even corresponds o he even {V T < L}. Hence, he sopping ime τ equals τ = T 1 {VT <L} + 1 {VT L}. Moreover C =, Z =, and X d T = V T 1 {VT <L} + L 1 {VT L}

15 1.2. CLASSIC STRUCTURAL MODELS 7 so ha X = V T. In oher words, he payoff a mauriy equals D T = min V T, L = L max L V T, = L L V T +. The laer equaliy shows ha he valuaion of he corporae bond in Meron s seup is equivalen o he valuaion of a European pu opion wrien on he firm s value wih srike equal o he bond s face value. Le D, T be he price a ime < T of he corporae bond. I is clear ha he value DV of he firm s deb equals DV = D, T = L B, T P, where P is he price of a pu opion wih srike L and expiraion dae T. I is apparen ha he value EV of he firm s equiy a ime equals EV = V DV = V LB, T + P = C, where C sands for he price a ime of a call opion wrien on he firm s asses, wih srike price L and exercise dae T. To jusify he las equaliy above, we may also observe ha a ime T we have EV T = V T DV T = V T min V T, L = V T L +. We conclude ha he firm s shareholders are in some sense he holders of a call opion on he firm s asses. Meron s Formula. Using he opion-like feaures of a corporae bond, Meron [159] derived a closed-form expression for is arbirage price. Le N denoe he sandard Gaussian cumulaive disribuion funcion: N x = 1 x e u2 /2 du, x IR. 2π Proposiion For every < T he value D, T of a corporae bond equals D, T = V e κt N d + V, T + L B, T N d V, T where d ± V, T = lnv /L + r κ ± 1 2 σ2 V T. σ V T The unique replicaing sraegy for a defaulable bond involves holding a any ime < T : φ 1 V unis of cash invesed in he firm s value and φ 2 B, T unis of cash invesed in defaul-free bonds, where φ 1 = e κt N d + V, T and Credi Spreads φ 2 = D, T φ1 V B, T = LN d V, T. For noaional simpliciy, we se κ =. Then Meron s formula becomes: where we denoe Γ = V /LB, T and D, T = LB, T Γ N d + N d σ V T, d = dv, T = lnv /L + r + σv 2 /2T. σ V T Since LB, T represens he curren value of he face value of he firm s deb, he quaniy Γ can be seen as a proxy of he asse-o-deb raio V /D, T. I can be easily verified ha he inequaliy

16 8 CHAPTER 1. STRUCTURAL APPROACH D, T < LB, T is valid. This propery is equivalen o he posiiviy of he corresponding credi spread see below. Observe ha in he presen seup he coninuously compounded yield r, T a ime on he T -mauriy Treasury zero-coupon bond is consan, and equal o he shor-erm rae r. Indeed, we have B, T = e r,t T = e rt. Le us denoe by r d, T he coninuously compounded yield on he corporae bond a ime < T, so ha D, T = Le rd,t T. From he las equaliy, i follows ha r d, T = ln D, T ln L. T For < T he credi spread S, T is defined as he excess reurn on a defaulable bond: In Meron s model, we have S, T = r d, T r, T = 1 LB, T ln T D, T. S, T = ln N d σ V T + Γ N d T This agrees wih he well-known fac ha risky bonds have an expeced reurn in excess of he riskfree ineres rae. In oher words, he yields on corporae bonds are higher han yields on Treasury bonds wih maching noional amouns. Noice, however, when ends o T, he credi spread in Meron s model ends eiher o infiniy or o, depending on wheher V T < L or V T > L. Formally, if we define he forward shor spread a ime T as F SS T = lim T S, T >. hen F SS T ω = {, if ω {VT > L},, if ω {V T < L} Black and Cox Model By consrucion, Meron s model does no allow for a premaure defaul, in he sense ha he defaul may only occur a he mauriy of he claim. Several auhors pu forward srucural-ype models in which his resricive and unrealisic feaure is relaxed. In mos of hese models, he ime of defaul is given as he firs passage ime of he value process V o eiher a deerminisic or a random barrier. In principle, he bond s defaul may hus occur a any ime before or on he mauriy dae T. The challenge is o appropriaely specify he lower hreshold v, he recovery process Z, and o explicily evaluae he condiional expecaion ha appears on he righ-hand side of he risk-neural valuaion formula S = B E Q B 1 u dd u F, ],T ] which is valid for [, T [. As one migh easily guess, his is a non-rivial mahemaical problem, in general. In addiion, he pracical problem of he lack of direc observaions of he value process V largely limis he applicabiliy of he firs-passage-ime models based on he value of he firm process V. Corporae Zero-Coupon Bond Black and Cox [25] exend Meron s [159] research in several direcions, by aking ino accoun such specific feaures of real-life deb conracs as: safey covenans,

17 1.2. CLASSIC STRUCTURAL MODELS 9 deb subordinaion, and resricions on he sale of asses. Following Meron [159], hey assume ha he firm s sockholders receive coninuous dividend paymens, which are proporional o he curren value of firm s asses. Specifically, hey posulae ha dv = V r κ d + σv dw, V >, where W is a BM under he risk-neural probabiliy Q, he consan κ represens he payou raio, and σ V > is he consan volailiy. The shor-erm ineres rae r is assumed o be consan. Safey covenans. Safey covenans provide he firm s bondholders wih he righ o force he firm o bankrupcy or reorganizaion if he firm is doing poorly according o a se sandard. The sandard for a poor performance is se by Black and Cox in erms of a ime-dependen deerminisic barrier v = Ke γt, [, T [, for some consan K >. As soon as he value of firm s asses crosses his lower hreshold, he bondholders ake over he firm. Oherwise, defaul akes place a deb s mauriy or no depending on wheher V T < L or no. Defaul ime. Le us se v = { v, for < T, L, for = T. The defaul even occurs a he firs ime [, T ] a which he firm s value V falls below he level v, or he defaul even does no occur a all. The defaul ime equals inf = + τ = inf { [, T ] : V v }. The recovery process Z and he recovery payoff X are proporional o he value process: Z β 2 V and X = β 1 V T for some consans β 1, β 2 [, 1]. The case examined by Black and Cox [25] corresponds o β 1 = β 2 = 1. To summarize, we consider he following model: X = L, C, Z β 2 V, X = β1 V T, τ = τ τ, where he early defaul ime τ equals τ = inf { [, T : V v} and τ sands for Meron s defaul ime: τ = T 1 {VT <L} + 1 {VT L}. Bond Valuaion Similarly as in Meron s model, i is assumed ha he shor erm ineres rae is deerminisic and equal o a posiive consan r. We posulae, in addiion, ha v LB, T or, more explicily, Ke γt Le rt, [, T ], so ha, in paricular, K L. This condiion ensures ha he payoff o he bondholder a he defaul ime τ never exceeds he face value of deb, discouned a a risk-free rae. PDE approach. Since he model for he value process V is given in erms of a Markovian diffusion, a suiable parial differenial equaion can be used o characerize he value process of he corporae bond. Le us wrie D, T = uv,. Then he pricing funcion u = uv, of a defaulable bond saisfies he following PDE: on he domain wih he boundary condiion u v, + r κvu v v, σ2 V v 2 u vv v, ruv, = {v, IR + IR + : < < T, v > Ke γt }, uke γt γt, = β 2 Ke and he erminal condiion uv, T = min β 1 v, L.

18 1 CHAPTER 1. STRUCTURAL APPROACH Probabilisic approach. For any < T he price D, T = uv, of a defaulable bond has he following probabilisic represenaion, on he se {τ > } = { τ > } D, T = E Q Le rt 1 { τ T, VT L} F + E Q β 1 V T e rt 1 { τ T, VT <L} F + E Q Kβ 2 e γt τ e r τ 1 {< τ<t } F. Afer defaul ha is, on he se {τ } = { τ }, we clearly have D, T = β 2 vτb 1 τ, T B, T = Kβ 2 e γt τ e r τ. To compue he expeced values above, we observe ha: he firs wo condiional expecaions can be compued by using he formula for he condiional probabiliy Q{V s x, τ s F }, o evaluae he hird condiional expecaion, i suffices employ he condiional probabiliy law of he firs passage ime of he process V o he barrier v. Black and Cox Formula. Before we sae he bond valuaion resul due o Black and Cox [25], we find i convenien o inroduce some noaion. We denoe ν = r κ 1 2 σ2 V, m = ν γ = r κ γ 1 2 σ2 V b = mσ 2. For he sake of breviy, in he saemen of Proposiion we shall wrie σ insead of σ V. As already menioned, he probabilisic proof of his resul is based on he knowledge of he probabiliy law of he firs passage ime of he geomeric exponenial Brownian moion o an exponenial barrier see Appendix equaions 8.11 and Proposiion Assume ha m 2 + 2σ 2 r γ >. Prior o bond s defaul, ha is: on he se {τ > }, he price process D, T = uv, of a defaulable bond equals D, T = LB, T N h 1 V, T Z 2bσ 2 N h 2 V, T + β 1 V e κt N h 3 V, T N h 4 V, T + β 1 V e κt Z 2b+2 N h5 V, T N h 6 V, T + β 2 V Z θ+ζ N h 7 V, T + Z θ ζ N h 8 V, T, where Z = v/v, θ = b + 1, ζ = σ 2 m 2 + 2σ 2 r γ and h 1 V, T = ln V /L + νt σ, T h 2 V, T = ln v2 lnlv + νt σ, T h 3 V, T = ln L/V ν + σ 2 T σ, T h 4 V, T = ln K/V ν + σ 2 T σ, T

19 1.2. CLASSIC STRUCTURAL MODELS 11 h 5 V, T = ln v2 lnlv + ν + σ 2 T σ, T h 6 V, T = ln v2 lnkv + ν + σ 2 T σ, T h 7 V, T = ln v/v + ζσ 2 T σ, T h 8 V, T = ln v/v ζσ 2 T σ. T Special Cases Assume ha β 1 = β 2 = 1 and he barrier funcion v is such ha K = L. Then necessarily γ r. I can be checked ha for K = L we have D, T = D 1, T + D 3, T where: D 1, T = LB, T N h 1 V, T Z 2â N h 2 V, T D 3, T = V Z θ+ζ N h 7 V, T + Z θ ζ N h 8 V, T. Case γ = r. If we also assume ha γ = r hen ζ = σ 2ˆν, and hus I is also easy o see ha in his case while V Z θ+ζ = LB, T, V Z θ ζ = V Z 2â+1 = LB, T Z 2â. h 1 V, T = lnv /L + νt σ T h 2 V, T = ln v2 lnlv + νt σ T = h 7 V, T, = h 8 V, T. We conclude ha if v = Le rt = LB, T hen D, T = LB, T. This resul is quie inuiive. A corporae bond wih a safey covenan represened by he barrier funcion, which equals he discouned value of he bond s face value, is equivalen o a defaul-free bond wih he same face value and mauriy. Case γ > r. For K = L and γ > r, i is naural o expec ha D, T would be smaller han LB, T. I is also possible o show ha when γ ends o infiniy all oher parameers being fixed, hen he Black and Cox price converges o Meron s price Furher Developmens The Black and Cox firs-passage-ime approach was laer developed by, among ohers: Brennan and Schwarz [35, 36] an analysis of converible bonds, Kim e al. [134] a random barrier and random ineres raes, Nielsen e al. [161] a random barrier and random ineres raes, Leland [147], Leland and Tof [148] a sudy of an opimal capial srucure, bankrupcy coss and ax benefis, Longsaff and Schwarz [152] a consan barrier and random ineres raes, Brigo [37]. One can sudy he problem τ = inf{ : V L} where L is a deerminisic funcion and V a geomeric Brownian moion. However, here exiss few explici resuls. See he appendix for some references. Oher sopping imes Moraux suggess o chose, as defaul ime a Parisian sopping ime For a coninuous process V and a given >, we inroduce g b V, he las ime before a which he process V was a level b, i.e., g b V = sup{s : V s = b}.

20 12 CHAPTER 1. STRUCTURAL APPROACH The Parisian ime is he firs ime a which he process V is under b for a period greaer han D, i.e., G,b D V = inf{ > : gb V 1 {V<b} D} This ime is a sopping ime. Le τ = G,b D V. See Appendix for resuls on he join law of τ, V τ in he case of a Black-Scholes dynamics. Anoher defaul ime is he firs ime where he process V has spend more han D ime below a level, i.e., τ = inf{ : A V > D} where A V = 1 Vs >bds. The law of his ime is relaed wih cumulaive opions. Campi and Sbuelz [4] presen he case where he defaul ime is given by a firs hiing ime of a CEV process and sudy he difficul problem of pricing an equiy defaul swap. [4] More precisely, hey assume ha he dynamics of he firm is ds = S r κd + σs β dw dm where W is a BM and M he compensaed maringale of a Poisson process i.e., M = N λ, and hey define τ = inf{ : S } In oher erms, hey ake τ = τ β τ N where τ N is he firs jump of he Poisson process and τ β = inf{ : X } where dx = X r κ + λd + σx β dw. Using ha a CEV process can be expressed in erms of a Bessel process ime changed, and resuls on he hiing ime of for a Bessel process of dimension smaller han 2, hey obain closed from soluions. Zhou s model Zhou [18] sudies he case where he dynamics of he firm is dv = V µ λνd + σdw + dx where W is a Brownian moion, X a compound Poisson process X = N law 1 eyi 1 where ln Y i = N a, b 2 wih ν = expa+b 2 /2 1. This choice of parameers implies ha V e µ is a maringale. In a firs par, Zhou sudies Meron s problem in ha seing. In a second par, he gives an approximaion for he firs passage problem when he defaul ime is τ = inf{ : V L} Opimal Capial Srucure We consider a firm ha has an ineres paying bonds ousanding. We assume ha i is a consol bond, which pays coninuously coupon rae c. Assume ha r > and he payou rae κ is equal o zero. This condiion can be given a financial inerpreaion as he resricion on he sale of asses, as opposed o issuing of new equiy. Equivalenly, we may hink abou a siuaion in which he sockholders will make paymens o he firm o cover he ineres paymens. However, hey have he righ o sop making paymens a any ime and eiher urn he firm over o he bondholders or pay hem a lump paymen of c/r per uni of he bond s noional amoun. Recall ha we denoe by EV DV, resp. he value a ime of he firm equiy deb, resp., hence he oal value of he firm s asses saisfies V = EV + DV. Black and Cox [25] argue ha here is a criical level of he value of he firm, denoed as v, below which no more equiy can be sold. The criical value v will be chosen by sockholders, whose aim is o minimize he value of he bonds equivalenly, o maximize he value of he equiy. Le us

21 1.2. CLASSIC STRUCTURAL MODELS 13 observe ha v is nohing else han a consan defaul barrier in he problem under consideraion; he opimal defaul ime τ hus equals τ = inf { : V v }. To find he value of v, le us firs fix he bankrupcy level v. The ODE for he pricing funcion u = u V of a consol bond akes he following form recall ha σ = σ V 1 2 V 2 σ 2 u V V + rv u V + c ru =, subjec o he lower boundary condiion u v = min v, c/r and he upper boundary condiion lim V u V V =. For he las condiion, observe ha when he firm s value grows o infiniy, he possibiliy of defaul becomes meaningless, so ha he value of he defaulable consol bond ends o he value c/r of he defaul-free consol bond. The general soluion has he following form: u V = c r + K 1V + K 2 V α, where α = 2r/σ 2 and K 1, K 2 are some consans, o be deermined from boundary condiions. We find ha K 1 =, and { v K 2 = α+1 c/r v α, if v < c/r,, if v c/r. Hence, if v < c/r hen u V = c r + v α+1 c r vα V α or, equivalenly, u V = c α α v v 1 + v. r V V I is in he ineres of he sockholders o selec he bankrupcy level in such a way ha he value of he deb, DV = u V, is minimized, and hus he value of firm s equiy EV = V DV = V c r 1 q v q is maximized. I is easy o check ha he opimal level of he barrier does no depend on he curren value of he firm, and i equals v = c α r α + 1 = c r + σ 2 /2. Given he opimal sraegy of he sockholders, he price process of he firm s deb i.e., of a consol bond akes he form, on he se {τ > }, D V = c r 1 α+1 c αv α r + σ 2 /2 or, equivalenly, where Furher Developmens D V = c r 1 q + v q, v q = V α = 1 V α c r + σ 2 /2 We end his secion by remarking ha oher imporan developmens in he area of opimal capial srucure were presened in he papers by Leland [147], Leland and Tof[148], Chrisensen e al. [46]. Chen and Kou [43], Dao [56], Hilberink and Rogers [1], LeCourois and Quiard-Pinon [145] sudy he same problem modelling he firm value process as a diffusion wih jumps. The reason for his exension was o eliminae an undesirable feaure of previously examined models, in which shor spreads end o zero when a bond approaches mauriy dae. α.

22 14 CHAPTER 1. STRUCTURAL APPROACH 1.3 Sochasic Ineres Raes In his secion, we assume ha he underlying probabiliy space Ω, F, P, endowed wih he filraion F = F, suppors he shor-erm ineres rae process r and he value process V. The dynamics under he maringale measure Q of he firm s value and of he price of a defaul-free zero-coupon bond B, T are dv = V r κ d + σ dw and db, T = B, T r d + b, T dw respecively, where W is a d-dimensional sandard Q-Brownian moion. Furhermore, κ : [, T ] IR, σ : [, T ] IR d and b, T : [, T ] IR d are assumed o be bounded funcions. The forward value F V, T = V /B, T of he firm saisfies under he forward maringale measure P T where he process W T [, T ], we se Then df V, T = κf V, T d + F V, T σ b, T dw T = W bu, T du, [, T ], is a d-dimensional SBM under P T. For any F κ V, T = F V, T e T κu du. df κ V, T = F κ V, T σ b, T dw T. Furhermore, i is apparen ha F κ V T, T = F V T, T = V T. We consider he following modificaion of he Black and Cox approach: X = L, Z = β 2 V, X = β1 V T, τ = inf { [, T ] : V < v }, where β 2, β 1 [, 1] are consans, and he barrier v is given by he formula v = wih he consan K saisfying < K L. Le us denoe, for any T, κ, T = T { KB, T e T κu du for < T, L for = T, κu du, σ 2, T = T σu bu, T 2 du where is he Euclidean norm in IR d. For breviy, we wrie F = FV κ, T, and we denoe η +, T = κ, T σ2, T, η, T = κ, T 1 2 σ2, T. The following resul exends Black and Cox valuaion formula for a corporae bond o he case of random ineres raes. Proposiion For any < T, he forward price of a defaulable bond F D, T = D, T /B, T equals on he se {τ > } L N ĥ 1 F,, T F /Ke κ,t N ĥ 2 F,, T + β 1 F e κ,t N ĥ 3 F,, T N ĥ 4 F,, T + β 1 K N ĥ 5 F,, T N ĥ 6 F,, T + β 2 KJ + F,, T + β 2 F e κ,t J F,, T,

23 1.4. RANDOM BARRIER 15 where ĥ 1 F,, T = ln F /L η +, T, σ, T ĥ 2 F, T, = 2 ln K lnlf + η, T, σ, T ĥ 3 F,, T = ln L/F + η, T, σ, T ĥ 4 F,, T = ln K/F + η, T, σ, T ĥ 5 F,, T = 2 ln K lnlf + η +, T, σ, T ĥ 6 F,, T = lnk/f + η +, T, σ, T and for any fixed < T and F > we se T J ± F,, T = e κu,t lnk/f + κ, T ± 1 2 dn σ2, u. σ, u In he special case when κ, he formula of Proposiion covers as a special case he valuaion resul esablished by Briys and de Varenne [38]. In some oher recen sudies of firs passage ime models, in which he riggering barrier is assumed o be eiher a consan or an unspecified sochasic process, ypically no closed-form soluion for he value of a corporae deb is available, and hus a numerical approach is required see, for insance, Kim e al. [134], Longsaff and Schwarz [152], Nielsen e al. [161], or Saá-Requejo and Sana-Clara [167]. 1.4 Random Barrier In he case of full informaion and Brownian filraion, he firs hiing ime of a deerminisic barrier is predicable. This is no longer he case when we deal wih incomplee informaion as in Duffie and Lando [71], see also Chaper 2, Secion 2.2.7, or when an addiional source of randomness is presen. We presen here a formula for credi spreads arising in a special case of a oally inaccessible ime of defaul. For a more deailed sudy we refer o Babbs and Bielecki [7]. As we shall see, he mehod we use here is close o he general mehod presened in Chaper 4. We suppose here ha he defaul barrier is a random variable D defined on he underlying probabiliy space Ω, P. The defaul occurs a ime τ where τ = inf{ : V D}, where V is he value of he firm and, for simpliciy, V = 1. Noe ha {τ > } = {inf u V u > D}. We shall denoe by m V he running minimum of V, i.e. m V = inf u V u. Wih his noaion, {τ > } = {m V > D}. Noe ha m V is a decreasing process Independen barrier In a firs sep we assume ha, under he risk-neural probabiliy Q, he barrier D is independen of he value of he firm. We denoe by F D he cumulaive disribuion funcion of he r.v. D, i.e. F D z = QD z. We assume ha F D is differeniable and we denoe f D is derivaive.

24 16 CHAPTER 1. STRUCTURAL APPROACH Lemma Le F = Qτ F and Γ = ln1 F. Then Proof: If D is independen of F, f D m V u Γ = F D m V u dmv u. F = Qτ F = Qm V D F = 1 F D m V. The process m V is decreasing. I follows ha Γ = ln F D m V, hence dγ = f Dm V F D m V dmv and f D m V u Γ = F D m V u dmv u. Example Assume ha D is uniformly disribued on he inerval [, 1]. Then, Γ = ln m V. The compuaion of quaniies as Ee Γ T fv T requires he knowledge of he joined law of he pair V T, m V T. We posulae now ha he value process V is a geomeric Brownian moion wih a drif, ha is, we se V = e Ψ, where Ψ = µ + σw. I is clear ha τ = inf { : Ψ ψ}, where Ψ is he running minimum of he process Ψ: Ψ = inf {Ψ s : s }. We choose he Brownian filraion as he reference filraion, i.e., we se F = F W. Le us denoe by Gz he cumulaive disribuion funcion under Q of he barrier ψ. We assume ha Gz > for z < and ha G admis he densiy g wih respec o he Lebesgue measure noe ha gz = for z >. This means ha we assume ha he value process V hence also he process Ψ is perfecly observed. In addiion, we suppose ha he bond invesor can observe he occurrence of he defaul ime. Thus, he can observe he process H = 1 {τ } = 1 {Ψ ψ}. We denoe by H he naural filraion of he process H. The informaion available o he invesor is represened by he enlarged filraion G = F H. We assume ha he defaul ime τ and ineres raes are independen under Q. Then, i is possible o esablish he following resul see Giesecke [92] or Babbs and Bielecki [7]. Noe ha he process Ψ is decreasing, so ha he inegral wih respec o his process is a pahwise Sieljes inegral. Proposiion Under he assumpions saed above, and addiionally assuming L = 1, Z and X =, we have ha for every < T 1 S, T = 1 {τ>} T ln E T f D Ψ u P e F D Ψ u dψ u F. In he nex chaper, we shall inroduce he noion of a hazard process of a random ime. For he defaul ime τ defined above, he F-hazard process Γ exiss and is given by he formula Γ = f D Ψ u F D Ψ u dψ u. This process is coninuous, and hus he defaul ime τ is a oally inaccessible sopping ime wih respec o he filraion G. To be compleed

25 1.5. COMMENTS ON STRUCTURAL MODELS Commens on Srucural Models We end his chaper by commening on meris and drawbacks of he srucural approach o credi risk. Advanages An approach based on he volailiy of he oal value of a firm. The credi risk is hus measured in a sandard way. The random ime of defaul is defined in an inuiive way. The defaul even is linked o he noion of he firm s insolvency. Valuaion and hedging of defaulable claims relies on similar echniques as he valuaion and hedging of exoic opions in he sandard defaul-free Black-Scholes seup. The concep of he disance o defaul, which measures he obligor s leverage relaive o he volailiy of is asses value, may serve o reflec credi raings. Dependen defauls are easy o handle hrough correlaion of processes corresponding o differen names. Disadvanages A sringen assumpion ha he oal value of he firm s asses can be easily observed. In pracice, coninuous-ime observaions of he value process V are no available. This issue was recenly addressed by Crouhy e al.[54], Duffie and Lando [71], Jeanblanc and Valchev [122], who showed ha a srucural model wih incomplee accouning daa can be deal wih using he inensiy-based mehodology. The paper of Guo [97] presens a case wih delayed informaion. See also Secion An unrealisic posulae ha he oal value of he firm s asses is a radeable securiy. This approach is known o generae low credi spreads for corporae bonds close o mauriy. I requires a judicious specificaion of he defaul barrier in order o ge a good fi o he observed spread curves. Oher issues A major problem wih applying srucural models is he difficuly wih esimaion of he volailiy of asses value. For he classical Meron s model, here exiss a simple formula ha relaes his volailiy o he volailiy of he firm s equiy, which in principle can be easily esimaed. However, no such simple expression exiss in case of firs-passage-ime models. Cerain marke-oriened echnologies, such as CrediGrades, aemp o produce such a formula. Srucural models discussed above were a mos one-facor models, wih he only facor being he shor-erm ineres rae. Two- and hree-facor srucural models have been also developed and closed-form valuaion formulae were derived in some special cases.

26 18 CHAPTER 1. STRUCTURAL APPROACH

27 Chaper 2 Hazard Funcion Approach We provide in his chaper a deailed analysis of he relaively simple case of he reduced form mehodology, when he flow of informaions available o an agen reduces o he observaions of he random ime which models he defaul even. The focus is on he evaluaion of condiional expecaions wih respec o he filraion generaed by a defaul ime wih he use of he hazard funcion. We sudy hedging sraegies based on CDS and/or wih DZC. We also presen a model wih wo defaul imes. In he following chapers, we shall sudy he case when an addiional informaion flow - formally represened by some filraion F - is presen, wih he use of he hazard process. 2.1 The Toy Model We begin wih he simple case where a riskless asse, wih deerminisic ineres rae rs; s is he only asse available in he defaul-free marke. The price of a risk-free zero-coupon bond wih mauriy T is B, T = exp T, rsds whereas is ime price B, T is B, T def = exp T rsds Defaul occurs a ime τ where τ is assumed o be a posiive random variable wih densiy f, consruced on a probabiliy space Ω, G, P. We denoe by F he cumulaive funcion of he r.v. τ defined as F = Pτ = fsds and we assume ha F < 1 for any < T, where T is he mauriy dae Oherwise here exiss < T such ha F = 1, and defaul occurs a.s. before. We emphasize ha he risk is no hedgeable. Indeed, a random payoff of he form 1 {T <τ} canno be perfecly hedged wih deerminisic zero-coupon bonds which are he only radeable asses in our model. To hedge he risk, we shall assume laer on ha some defaulable asse is raded, e.g., a defaulable zero-coupon bond or a CDS Credi Defaul Swap. Remark I is no difficul o generalize he sudy presened in wha follows o he case where τ does no admi a densiy by dealing wih he righ-coninuous version of he cumulaive funcion. The case where τ is bounded can also be sudied along he same mehod. We leave he deails o he reader Defaulable Zero-coupon wih Paymen a Mauriy A defaulable zero-coupon bond DZC in shor- or a corporae bond- wih mauriy T and rebae R paid a mauriy, consiss of 19