Valuation and Hedging of Credit Derivatives

Transcription

1 Valuaion and Hedging of Credi Derivaives Bielecki Tomasz R., Monique Jeanblanc, Rukowski Marek To cie his version: Bielecki Tomasz R., Monique Jeanblanc, Rukowski Marek. Valuaion and Hedging of Credi Derivaives. 3rd cycle. Marrakech Maroc, 27, pp.165. <cel-39875> HAL Id: cel hps://cel.archives-ouveres.fr/cel Submied on 24 Jun 29 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 Valuaion and Hedging of Credi Derivaives Tomasz R. Bielecki Deparmen of Applied Mahemaics Illinois Insiue of Technology Chicago, IL 6616, USA Monique Jeanblanc Déparemen de Mahémaiques Universié d Évry Val d Essonne 9125 Évry Cedex, France Marek Rukowski School of Mahemaics and Saisics Universiy of New Souh Wales Sydney, NSW 252, Ausralia Sochasic Models in Mahemaical Finance CIMPA-UNESCO-Morocco School Marrakech, Morocco, April 9-2, 27

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4 Conens 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 Hazard Funcion Approach The Toy Model Defaulable Zero-Coupon Bond wih Paymen a Mauriy Defaulable Zero-Coupon wih Paymen a Defaul Implied Defaul Probabiliies Credi Spreads Maringale Approach Key Lemma Maringales Associaed wih Defaul Time 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 Recovery a Mauriy Recovery a Defaul Generic Defaulable Claims

5 4 CONTENTS Buy-and-Hold Sraegy Spo Maringale Measure Self-Financing Trading Sraegies Maringale Properies of Prices of Defaulable Claims Hedging of Single Name Credi Derivaives Sylized Credi Defaul Swap Pricing of a CDS Marke CDS Rae Price Dynamics of a CDS Dynamic Replicaion of a Defaulable Claim Dynamic Hedging of Baske Credi Derivaives Firs-o-Defaul Inensiies Firs-o-Defaul Maringale Represenaion Theorem Price Dynamics of he i h CDS Risk-Neural Valuaion of a Firs-o-Defaul Claim Dynamic Replicaion of a Firs-o-Defaul Claim Condiional Defaul Disribuions Recursive Valuaion of a Baske Claim Recursive Replicaion of a Baske Claim Applicaions o Copula-Based Credi Risk Models Independen Defaul Times Archimedean Copulas One-Facor Gaussian Copula Hazard Process Approach General Case Key Lemma Maringales Inerpreaion of he Inensiy Reducion of he Reference Filraion Enlargemen of Filraion Hypohesis H Equivalen Formulaions Canonical Consrucion of a Defaul Time Sochasic Barrier Change of a Probabiliy Measure Represenaion Theorem Case of a Parial Informaion Informaion a Discree Times Delayed Informaion Inensiy Approach

6 CONTENTS 5 4 Hedging of Defaulable Claims 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 Toal Defaul Defaulable Asse wih Non-Zero Recovery Two Defaulable Asses wih Toal Defaul PDE Approach o Valuaion and Hedging Defaulable Asse wih Toal Defaul Defaulable Asse wih Non-Zero Recovery Two Defaulable Asses wih Toal Defaul Dependen Defauls and Credi Migraions Baske Credi Derivaives The i h -o-defaul Coningen Claims Case of Two Eniies Role of he Hypohesis H Condiionally Independen Defauls Canonical Consrucion Independen Defaul Times Signed Inensiies Valuaion of FDC and LDC Copula-Based Approaches Direc Applicaion Indirec Applicaion Jarrow and Yu Model Consrucion and Properies of he Model Exension of he Jarrow and Yu Model Kusuoka s Consrucion Inerpreaion of Inensiies Bond Valuaion Markovian Models of Credi Migraions Infiniesimal Generaor Specificaion of Credi Raings Transiion Inensiies Condiionally Independen Migraions Examples of Markov Marke Models Forward CDS Credi Defaul Swapions

7 6 CONTENTS 5.7 Baske Credi Derivaives k h -o-defaul CDS Forward k h -o-defaul CDS Model Implemenaion Sandard Credi Baske Producs Valuaion of Sandard Baske Credi Derivaives Porfolio Credi Risk

8 Inroducion The goal of hese lecure noes is o presen a survey of recen developmens in he area of mahemaical modeling of credi risk and credi derivaives. They are largely based on he following papers by T.R. Bielecki, M. Jeanblanc and M. Rukowski: Modelling and valuaion of credi risk. In: Sochasic Mehods in Finance, M. Frielli and W. Runggaldier, eds., Springer-Verlag, 24, , Hedging of defaulable claims. In: Paris-Princeon Lecures on Mahemaical Finance 23, R. Carmona e al., eds. Springer-Verlag, 24, 1 132, PDE approach o valuaion and hedging of credi derivaives. Quaniaive Finance 5 25, , Hedging of credi derivaives in models wih oally unexpeced defaul. In: Sochasic Processes and Applicaions o Mahemaical Finance, J. Akahori e al., eds., World Scienific, Singapore, 26, 35 1, Hedging of baske credi derivaives in credi defaul swap marke. Journal of Credi Risk and on some chapers from he book by T.R. Bielecki and M. Rukowski: Credi Risk: Modelling, Valuaion and Hedging, Springer-Verlag, 21. Our recen working papers by can be found on he websies: A lo of oher ineresing informaion is provided on he websies lised a he end of he bibliography of his documen. 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 defaul or 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. 7

9 8 CHAPTER. INTRODUCTION 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 is devoed o his issue. 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. These noes are organized as follows: 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 very briefly he case of a random barrier. Chaper 2 is devoed o he sudy of a simple model of credi risk wihin he hazard funcion framework. We also deal here wih he issue of replicaion of single- and muli-name credi derivaives in he sylized CDS marke. Chaper 3 deals wih he so-called reduced-form approach in which he main ool is he hazard rae process. This approach is of a purely probabilisic naure and, echnically speaking, i has a lo in common wih he reliabiliy heory. Chaper 4 sudies hedging sraegies for defaulable claims under assumpion ha some primary defaulable asses are raded. We discuss some general resuls in a semimaringale se-up and we develop he PDE approach in a Markovian se-up. Chaper 5 provides an inroducion o he area of modeling dependen defauls and, more generally, o modeling of dependen credi migraions for a porfolio of reference names. We presen some applicaions of hese models o he valuaion of real-life examples of credi derivaives, such as: CDSs and credi defaul swapions, firs-o-defaul CDSs, CDS indices and CDOs. Le us menion ha he proofs of mos resuls can be found in Bielecki and Rukowski [12], Bielecki e al. [5, 6, 9] and Jeanblanc and Rukowski [59]. We quoe some of he seminal papers; he reader can also refer o books by Bruyère [25], Bluhm e al. [18], Bielecki and Rukowski [12], Cossin and Piroe [33], Duffie and Singleon [43], Frey, McNeil and Embrechs [49], Lando [65], or Schönbucher [83] for more informaion. A he end of he bibliography, we also provide some web addresses where aricles can be downloaded. Finally, i should be acknowledged ha several resuls especially wihin he reduced-form approach were obained independenly by various auhors, who worked under differen se of assumpions and/or wihin differen se-ups. For his reason, we decided o omi deailed credenials in mos cases. We hope ha our colleagues will accep our apologies for his deficiency, and we sress ha his by no means signifies ha any resul given in wha follows ha is no explicily aribued is ours. Begin a he beginning, and go on ill you come o he end: hen sop. Lewis Carroll, Alice s Advenures in Wonderland

10 Chaper 1 Srucural Approach In his chaper, we presen he so-called srucural approach o modeling credi risk, which is also known as he value-of-he-firm approach. This mehodology refers direcly 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. I is worh noing ha his was hisorically he firs approach used in his area i goes back o he fundamenal papers by Black and Scholes [17] and Meron [76]. 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 A, 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, A 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, A is assumed o be a process of finie variaion, wih A =. We assume wihou menioning ha all random objecs inroduced above saisfy suiable inegrabiliy condiions. 9

11 1 CHAPTER 1. STRUCTURAL APPROACH Probabiliies P and Q. The probabiliy P is assumed o represen he real-world or saisical probabiliy, as opposed o a maringale measure also known as a risk-neural probabiliy. Any maringale measure will be denoed by Q in wha follows. Defaul ime. 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 ] R + is some deerminisic funcion, ermed he barrier. Predicabiliy of defaul ime. 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, A, 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 = X1 {τ>t } + X1 {τ T }. Definiion The dividend process D of a defaulable coningen claim X, A, X, Z, τ, which seles a ime T, equals D = X d T 1 { T } + 1 H u da u + Z u dh u. I is apparen ha D is a process of finie variaion, and 1 H u da u = 1 {τ>u} da u = A τ 1 {τ } + A 1 {τ>}. ],] ],] ],] Noe ha if defaul occurs a some dae, he promised dividend A A, 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 }. ],] ],]

12 1.1. BASIC ASSUMPTIONS 11 Remark In principle, he promised payoff X could be incorporaed ino he promised dividends process A. However, his would be inconvenien, since in pracice he recovery rules concerning he promised dividends A 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, A, 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 A, 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 L1 {τ>t } + X1 {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 [76] posulaes ha he recovery payoff upon defaul ha is, 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.

13 12 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 [76]. 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 A =, Z =, and X d T = V T 1 {VT <L} + L1 {VT L}

14 1.2. CLASSIC STRUCTURAL MODELS 13 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 [76] derived a closed-form expression for is arbirage price. Le N denoe he sandard Gaussian cumulaive disribuion funcion: Nx = 1 x e u2 /2 du, x R. 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 φ 2 = D, T φ1 V B, T = LN d V, T. Credi spreads. 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 + Nd 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

15 14 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 Nd 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 [16] exend Meron s [76] research in several direcions, by aking ino accoun such specific feaures of real-life deb conracs as: safey covenans,

16 1.2. CLASSIC STRUCTURAL MODELS 15 deb subordinaion, and resricions on he sale of asses. Following Meron [76], 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 Brownian moion 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 [16] corresponds o β 1 = β 2 = 1. To summarize, we consider he following model: X = L, A, 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, R + R + : < < T, v > Ke γt }, uke γt γt, = β 2 Ke

17 16 CHAPTER 1. STRUCTURAL APPROACH and he erminal condiion uv, T = min β 1 v, L. 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 [16], 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. 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

18 1.2. CLASSIC STRUCTURAL MODELS 17 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 [21, 22] an analysis of converible bonds, Nielsen e al. [78] a random barrier and random ineres raes, Leland [69], Leland and Tof [7] a sudy of an opimal capial srucure, bankrupcy coss and ax benefis, Longsaff and Schwarz [72] a consan barrier and random ineres raes, Brigo [23]. Oher sopping imes. In general, one can sudy he bond valuaion problem for he defaul ime given as τ = inf { R + : V L}, where L is a deerminisic funcion and V is a geomeric Brownian moion. However, here exiss few explici resuls. Moraux s model. Moraux [77] propose o model he defaul ime as 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, ha is, g b V = sup { s : V s = b}.

19 18 CHAPTER 1. STRUCTURAL APPROACH The Parisian sopping ime is he firs ime a which he process V is below he level b for a ime period of lengh greaer han D, ha is, G,b D V = inf { R + : g b V 1 {V <b} D}. Clearly, his ime is a sopping ime. Le τ = G,b D V. In he case of Black-Scholes dynamics, i is possible o find he join law of τ, V τ Anoher defaul ime is he firs ime where he process V has spend more han D ime below a level, ha is, τ = inf{ R + : A V > D} where A V = 1 {V s >b} ds. The law of his ime is relaed o cumulaive opions. Campi and Sbuelz model. Campi and Sbuelz [26] assume ha he defaul ime is given by a firs hiing ime of by a CEV process, and hey sudy he difficul problem of pricing an equiy defaul swap. More precisely, hey assume ha he dynamics of he firm are ds = S r κ d + σs β dw dm where W is a Brownian moion and M he compensaed maringale of a Poisson process i.e., M = N λ, and hey define τ = inf { R + : S }. In oher erms, Campi and Sbuelz [26] se τ = τ β τ N, where τ N is he firs jump of he Poisson process and τ β = inf { R + : X } where in urn dx = X r κ + λ d + σx β dw. Using ha he CEV process can be expressed in erms of a ime-changed Bessel process, 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 [85] sudies he case where he dynamics of he firm is dv = V µ λν d + σ dw + dx where W is a Brownian moion, X a compound Poisson process, ha is, X = N 1 ey i 1 where law ln Y i = Na, b 2 wih ν = expa + b 2 /2 1. Noe ha for his choice of parameers he process V e µ is a maringale. Zhou firs sudies Meron s problem in ha seing. Nex, he gives an approximaion for he firs passage problem when he defaul ime is τ = inf { R + : 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 [16] 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

20 1.2. CLASSIC STRUCTURAL MODELS 19 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 { R + : 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 or, equivalenly, u V = c r + v α+1 c r vα V α u V = c r α α v v 1 + v. 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 D V = c r 1 q + v q, v q = V α = 1 V α c r + σ 2 /2 Furher developmens. We end his secion by menioning ha oher imporan developmens in he area of opimal capial srucure were presened in he papers by Leland [69], Leland and Tof [7], Chrisensen e al. [31]. Chen and Kou [29], Dao [34], Hilberink and Rogers [53], LeCourois and Quiard-Pinon [68] sudy he same problem, bu hey model he firm s 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.

21 2 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 ] R, σ : [, T ] R d and b, T : [, T ] R 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 df V, T = κf V, T d + F V, T σ b, T dw T where he process W T = W bu, T du, [, T ], is a d-dimensional Brownian moion under P T. For any [, T ], we se FV κ, T = F V, T e R T κu du. Then 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 R 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 R 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,

22 1.4. RANDOM BARRIER 21 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 [24]. 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, Longsaff and Schwarz [72], Nielsen e al. [78], or Saá-Requejo and Sana-Clara [81]. 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 [41], 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 [2]. As we shall see, he mehod we use here is close o he general mehod presened in Chaper 3. We suppose here ha he defaul barrier is a random variable η defined on he underlying probabiliy space Ω, P. The defaul occurs a ime τ where τ = inf{ : V η}, where V is he value of he firm and, for simpliciy, V = 1. Noe ha {τ > } = {inf u V u > η}. 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 > η}. Noe ha m V is a decreasing process Independen Barrier In a firs sep we assume ha, under he risk-neural probabiliy Q, a random variable η modelling is independen of he value of he firm. We denoe by F η he cumulaive disribuion funcion of η, i.e., F η z = Qη z. We assume ha F η is differeniable and we denoe by f η is derivaive.

23 22 CHAPTER 1. STRUCTURAL APPROACH Lemma Le F = Qτ F and Γ = ln1 F. Then Proof. If η is independen of F, hen f η m V u Γ = F η m V u dmv u. F = Qτ F = Qm V η F = 1 F η m V. The process m V is decreasing. I follows ha Γ = ln F η m V, hence dγ = f ηm V F η m V dmv and f η m V u Γ = F η m V u dmv u as expeced. Example Assume ha η is uniformly disribued on he inerval [, 1]. Then, Γ = ln m V. The compuaion of he expeced value E Q e Γ T fv T requires he knowledge of he join 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 { R + : 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 posulae 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 [5] or Babbs and Bielecki [2]. 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 R S, T = 1 {τ>} T ln E T fη Ψ u Q e Fη Ψ u dψ u F. Laer on, we will 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 η Ψ u F η 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.

24 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 informaion 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 also sudy hedging sraegies based on credi defaul swaps and/or defaulable zerocoupon bonds. Finally, we also presen a credi risk model wih several defaul imes. 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 a ime of a risk-free zero-coupon bond wih mauriy T equals T B, T = exp rs ds. 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, Q. We denoe by F he cumulaive funcion of he random varible τ defined as F = Qτ = fs ds and we assume ha F < 1 for any >. Oherwise, here would exiss a dae for which F = 1, so ha he defaul would occurs before or a wih probabiliy 1. We emphasize ha he random payoff of he form 1 {T <τ} canno be perfecly hedged wih deerminisic zero-coupon bonds, which are he only radeable primary asses in our model. To hedge he risk, we shall laer posulae ha some defaulable asse is raded, e.g., a defaulable zero-coupon bond or a credi defaul swap. 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 Bond wih Paymen a Mauriy A defaulable zero-coupon bond DZC in shor, or a corporae zero-coupon bond, wih mauriy T and he rebae recovery δ paid a mauriy, consiss of: The paymen of one moneary uni a ime T if defaul has no occurred before ime T, i.e., if τ > T, A paymen of δ moneary unis, made a mauriy, if τ T, where < δ < 1. 23

25 24 CHAPTER 2. HAZARD FUNCTION APPROACH Value of he Defaulable Zero-Coupon Bond The fair value of he defaulable zero-coupon bond is defined as he expecaion of discouned payoffs D δ, T = B, T E Q 1{T <τ} + δ1 {τ T } = B, T E Q 1 1 δ1{τ T } = B, T 1 1 δf T. 2.1 In fac, his quaniy is a ne presen value and is equal o he value of he defaul free zero-coupon bond minus he expeced loss, compued under he hisorical probabiliy. Obviously, his value is no a hedging price. The ime- value depends wheher or no defaul has happened before his ime. If defaul has occurred before ime, he paymen of δ will be made a ime T, and he price of he DZC is δb, T. If he defaul has no ye occurred, he holder does no know when i will occur. The value D δ, T of he DZC is he condiional expecaion of he discouned payoff B, T 1 {T <τ} + δ1 {τ T } given he informaion available a ime. We obain where he pre-defaul value D δ is defined as D δ, T = 1 {τ } B, T δ + 1 {<τ} Dδ, T D δ, T = E Q B, T 1{T <τ} + δ1 {τ T } < τ = B, T 1 1 δqτ T < τ = B, T 1 1 δ Q < τ T Q < τ = B, T 1 1 δ F T F 1 F. 2.2 Noe ha he value of he DZC is disconinuous a ime τ, unless F T = 1 or δ = 1. In he case F T = 1, he defaul appears wih probabiliy one before mauriy and he DZC is equivalen o a paymen of δ a mauriy. If δ = 1, he DZC is simply a defaul-free zero coupon bond. Formula 2.2 can be rewrien as follows D δ, T = B, T EDLGD DP where he expeced discouned loss given defaul EDLGD is defined as B, T 1 δ and he condiional defaul probabiliy DP is defined as follows DP = Q < τ T Q < τ = Qτ T < τ. In case he paymen is a funcion of he defaul ime, say δτ, he value of his defaulable zerocoupon is D δ, T = E Q B, T 1{T <τ} + B, T δτ1 {τ T } T = B, T QT < τ + δsfs ds.

26 2.1. THE TOY MODEL 25 If he defaul has no occurred before, he pre-defaul ime- value D δ, T saisfies To summarize, we have D δ, T = B, T E Q 1 {T <τ} + δτ1 {τ T } < τ QT < τ = B, T Q < τ + 1 T δsfs ds. Q < τ D δ, T = 1 {<τ} Dδ, T + 1 {τ } δτ B, T. Hazard Funcion Le us recall he sanding assumpion ha F < 1 for any R +. funcion Γ by seing Γ = ln1 F We inroduce he hazard for any R +. Since we assumed ha F is differeniable, he derivaive Γ = γ = where f = F. This means ha 1 F = e Γ = exp γs ds = Qτ >. f 1 F, The quaniy γ is he hazard rae. The inerpreaion of he hazard rae is he probabiliy ha he defaul occurs in a small inerval d given ha he defaul did no occur before ime Noe ha Γ is increasing. Then, formula 2.2 reads where we denoe 1 γ = lim Qτ + h τ >. h h D δ, T = B, T = R,d T R,d T = exp 1 F T 1 F + δ B, T R,d T T F T F + δ 1 F, rs + γs ds. In paricular, for δ =, we obain D, T = R,d T. Hence he spo rae has simply o be adjused by means of he credi spread equal o γ in order o evaluae DZCs wih zero recovery. The dynamics of D δ can be easily wrien in erms of he funcion γ as d D δ, T = r + γ D δ, T d B, T γδ d. The dynamics of D δ, T will be derived in he nex secion. If γ and δ are consan, he credi spread equals 1 B, T ln T D δ, T = γ 1 T ln 1 + δe γt 1 and i converges o γ1 δ when goes o T. For any < T, he quaniy γ, T = f,t 1 F,T where F, T = Qτ T τ >

27 26 CHAPTER 2. HAZARD FUNCTION APPROACH and f, T dt = Qτ dt τ > is called he condiional hazard rae. I is easily seen ha T F, T = 1 exp γs, T ds. Noe, however, ha in he presen seing, we have ha and hus γs, T = γs. 1 F, T = Qτ > T T Qτ > = exp γs ds Remark In case τ is he firs jump of an inhomogeneous Poisson process wih deerminisic inensiy λ, f = Qτ d d = λ exp λs ds = λe Λ where Λ = λs ds and Qτ = F = 1 e Λ. Hence he hazard funcion is equal o he compensaor of he Poisson process, i.e., Γ = Λ. Conversely, if τ is a random ime wih densiy f, seing Λ = ln1 F allows us o inerpre τ as he firs jump ime of an inhomogeneous Poisson process wih he inensiy equal o he derivaive of Λ Defaulable Zero-Coupon wih Paymen a Defaul Here, a defaulable zero-coupon bond wih mauriy T consiss of: The paymen of one moneary uni a ime T if defaul has no ye occurred, The paymen of δτ moneary unis, where δ is a deerminisic funcion, made a ime τ if τ T. Value of he Defaulable Zero-Coupon The value of his defaulable zero-coupon bond is D δ, T = E Q B, T 1{T <τ} + B, τδτ1 {τ T } = QT < τb, T + T B, sδs df s T = GT B, T B, sδs dgs, 2.3 where G = 1 F = Q < τ is he survival probabiliy. Obviously, if he defaul has occurred before ime, he value of he DZC is null his was no he case for he recovery paymen made a bond s mauriy, and D δ, T = 1 {<τ} Dδ, T where D δ, T is a deerminisic funcion he predefaul price. The pre-defaul ime- value D δ, T saisfies Hence B, D δ, T = E Q B, T 1{T <τ} + B, τδτ1 {τ T } < τ = QT < τ Q < τ B, T + 1 Q < τ T B, sδs df s. T RG D δ, T = GT B, T B, sδs dgs.