CREDIT RISK MODELING

Transcription

1 CREDIT RISK MODELING 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 Cener for he Sudy of Finance and Insurance Osaka Universiy, Osaka, Japan

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3 Conens 1 Srucural Approach Noaion and Definiions Defaulable Claims Risk-Neural Valuaion Formula Defaulable Zero-Coupon Bond Meron s Model Firs Passage Times Disribuion of he Firs Passage Time Join Disribuion of Y and τ Black and Cox Model Bond Valuaion Black and Cox Formula Corporae Coupon Bond Opimal Capial Srucure Exensions of he Black and Cox Model Sochasic Ineres Raes Random Barrier Independen Barrier Hazard Funcion Approach Elemenary Marke Model Hazard Funcion and Hazard Rae Defaulable Bond wih Recovery a Mauriy Defaulable Bond wih Recovery a Defaul Maringale Approach Condiional Expecaions Maringales Associaed wih Defaul Time Predicable Represenaion Theorem Girsanov s Theorem Range of Arbirage Prices Implied Risk-Neural Defaul Inensiy Price Dynamics of Simple Defaulable Claims

4 4 CONTENTS 2.3 Pricing of General Defaulable Claims Buy-and-Hold Sraegy Spo Maringale Measure Self-Financing Trading Sraegies Maringale Properies of Arbirage Prices Single Name Credi Derivaives Sylized Credi Defaul Swap Marke CDS Spread Price Dynamics of a CDS Replicaion of a Defaulable Claim Baske Credi Derivaives Firs-o-Defaul Inensiies Firs-o-Defaul Represenaion Theorem Price Dynamics of Credi Defaul Swaps Valuaion of a Firs-o-Defaul Claim 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 Models Independen Defaul Times Archimedean Copulae Hazard Process Approach Hazard Process and is Applicaions Condiional Expecaions Hazard Rae Valuaion of Defaulable Claims Defaulable Bonds Maringales Associaed wih Defaul Time F-Inensiy of Defaul Time Reducion of he Reference Filraion Enlargemen of Filraion Hypohesis H Equivalen Forms of Hypohesis H Canonical Consrucion of a Defaul Time Sochasic Barrier Predicable Represenaion Theorem Girsanov s Theorem Invariance of Hypohesis H Case of he Brownian Filraion

5 CONTENTS Exension o Orhogonal Maringales G-Inensiy of Defaul Time Single Name CDS Marke Sanding Assumpions Valuaion of a Defaulable Claim Replicaion of a Defaulable Claim Muli-Name CDS Marke Valuaion of a Firs-o-Defaul Claim Replicaion of a Firs-o-Defaul Claim Hedging of Defaulable Claims Semimaringale Marke Model Dynamics of Asse Prices Trading Sraegies Unconsrained Sraegies Consrained Sraegies Maringale Approach Defaulable Asse wih Zero Recovery Hedging wih a Defaulable Bond Defaulable Asse wih Non-Zero Recovery Two Defaulable Asses wih Zero Recovery PDE Approach Defaulable Asse wih Zero Recovery Defaulable Asse wih Non-Zero Recovery Two Defaulable Asses wih Zero Recovery Modeling Dependen Defauls Baske Credi Derivaives The kh-o-defaul Coningen Claims Case of Two Credi Names Condiionally Independen Defauls Canonical Consrucion Hypohesis H Independen Defaul Times Signed Inensiies Valuaion of FTDC and LTDC Copula-Based Approaches Direc Approach Indirec Approach One-facor Gaussian Copula Model Jarrow and Yu Model

6 6 CONTENTS Consrucion of Defaul Times Case of Two Credi Names Kusuoka s Model Model Specificaion Bonds wih Zero Recovery Baske Credi Derivaives Credi Defaul Index Swaps Collaeralized Deb Obligaions Firs-o-Defaul Swaps Sep-up Corporae Bonds Valuaion of Baske Credi Derivaives Modeling of Credi Raings Infiniesimal Generaor Transiion Inensiies for Credi Raings Condiionally Independen Credi Migraions Examples of Markovian Models Forward Credi Defaul Swap Credi Defaul Swapions Spo kh-o-defaul Credi Swap Forward kh-o-defaul Credi Swap Model Implemenaion A Poisson Processes 23 A.1 Sandard Poisson Process A.2 Inhomogeneous Poisson Process A.3 Condiional Poisson Process A.4 The Doléans Exponenial A.4.1 Exponenial of a Process of Finie Variaion A.4.2 Exponenial of a Special Semimaringale

7 Inroducion The goal of his ex is o give a survey of echniques used in mahemaical modeling of credi risk and o presen some recen developmens in his area, wih he special emphasis on hedging of defaulable claims. I is 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, 24, , Hedging of defaulable claims. In: Paris-Princeon Lecures on Mahemaical Finance 23, R. Carmona e al., eds. Springer, 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, 26, 35 1, Hedging of baske credi derivaives in credi defaul swap marke. Journal of Credi Risk 3 27, Pricing and rading credi defaul swaps in a hazard process model. Forhcoming in Annals of Applied Probabiliy. 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 ha is, he collaeral asses or he reference credi defaul swaps. 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. 7

8 8 CHAPTER. INTRODUCTION Our main goal is o presen a comprehensive inroducion o he mos imporan mahemaical ools ha are used in arbirage valuaion of defaulable claims, which are also known under he name of credi derivaives. We also examine in some deail he imporan issue of hedging hese claims. This ex is 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. In paricular, we presen he classic srucural models, pu forward by Meron [124] and Black and Cox [25], and we menion some varians and exensions of hese models. We also sudy very succincly he case of a srucural model wih a random defaul riggering barrier. Chaper 2 is devoed o he sudy of an elemenary model of credi risk wihin he hazard funcion framework. We focus here on he derivaion of pricing formulae for defaulable claims and he dynamics of heir prices. We also deal here wih he issue of replicaion of singleand muli-name credi derivaives in he sylized credi defaul swap marke. Resuls of his chaper should be seen as a firs sep oward more pracical approaches ha are presened in he foregoing chapers. Chaper 3 deals wih he alernaive reduced-form approach in which he main modeling ool is he hazard process. We examine he pricing formulae for defaulable claims in he reduced-form seup wih sochasic hazard rae and we examine he behavior of he sochasic inensiy when he reference filraion is reduced. Special emphasis is pu on he so-called hypohesis H and is invariance wih respec o an equivalen change of a probabiliy measure. As an applicaion of mahemaical resuls, we presen here an exension of hedging resuls esablished in Chaper 2 for he case of deerminisic pre-defaul inensiies o he case of sochasic defaul inensiies. Chaper 4 is devoed o a sudy of hedging sraegies for defaulable claims under he assumpion ha some primary defaulable asses are raded. We firs presen some heoreical resuls on replicaion of defaulable claims in an absrac semimaringale marke model. Subsequenly, we develop he PDE approach o he valuaion and hedging of defaulable claims in a Markovian framework. For he sake of simpliciy of presenaion, we focus in he presen ex on he case of a marke model wih hree raded primary asses and we deal wih a single defaul ime only. However, an exension of he PDE mehod o he case of any finie number of raded asses and several defaul imes is readily available. Chaper 5 provides an inroducion o he area of modeling dependen defauls and, more generally, o modeling of dependen credi raing migraions for a porfolio of reference credi names. We presen here some applicaions of hese models o he valuaion of real-life examples of acively raded credi derivaives, such as: credi defaul swaps and swapions, firs-odefaul swaps, credi defaul index swaps and ranches of collaeralized deb obligaions. For he reader s convenience, we presen in he appendix some well known resuls regarding he Poisson process and is generalizaions. We also recall here he definiion and basic properies of he Doléans exponenial of a semimaringale. The deailed proofs of mos resuls can be found in papers by Bielecki and Rukowski [2], Bielecki e al. [12, 13, 16] and Jeanblanc and Rukowski [99]. We also quoe some of he seminal papers, bu, unforunaely, we were no able o provide here a survey of an exensive research in he area of credi risk modeling. For more informaion, he ineresed reader is hus referred o original papers by oher auhors as well as o monographs by Ammann [2], Bluhm, Overbeck and Wagner [28], Bielecki and Rukowski [2], Cossin and Piroe [55], Duffie and Singleon [68], McNeil, Frey and Embrechs [123], Lando [19], or Schönbucher [138]

9 Chaper 1 Srucural Approach We sar by presening a raher brief overview of he srucural approach o credi risk modeling. Since i is based on he modeling of he behavior of he oal value of he firm s asses, i is also known as he value-of-he-firm approach. In order o model credi evens he defaul even, in paricular, his mehodology refers direcly o economic fundamenals, such as he capial srucure of a company. 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. Hisorically, his was he firs approach used in his area i can be raced back o he fundamenal papers by Black and Scholes [26] and Meron [124]. The presen exposiion is largely based on Chapers 2 and 3 in Bielecki and Rukowski [2]; he ineresed reader may hus consul [2] for more deails. 1.1 Noaion and Definiions We fix a finie horizon dae T >. The underlying probabiliy space Ω, F, P is endowed wih some reference filraion F = F T, and is sufficienly rich o suppor he following random quaniies: he shor-erm ineres rae process r and hus also a defaul-free erm srucure model, he value of he firm process V, which is inerpreed as a sochasic model for he oal value of he firm s asses, he barrier process v, which is used o specify he defaul ime τ, he promised coningen claim X represening he liabiliies o be redeemed o he holder of a defaulable claim a mauriy dae T T, he process A, which models he promised dividends, ha is, he liabiliies ha are redeemed coninuously or discreely over ime o he holder of a defaulable claim, he recovery claim X represening he recovery payoff received a ime T if defaul occurs prior o or a he claim s mauriy dae T, he recovery process Z, which specifies he recovery payoff a ime of defaul if i occurs prior o or a he mauriy dae T. The probabiliy measure P is aimed 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 Defaulable Claims 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 9

10 1 CHAPTER 1. STRUCTURAL APPROACH o be a process of finie variaion wih A =. We assume wihou menioning ha all random objecs inroduced above saisfy suiable inegrabiliy condiions. Wihin he srucural approach, he defaul ime τ is 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 +. Typically, he se T is he inerval [, T ] or [, in he case of perpeual claims. In classic firs-passage-ime srucural models, he defaul ime τ is given by he formula τ = inf { > : [, T ] and V v}, where v : [, T ] R + is some deerminisic funcion, ermed he barrier. Remark In mos srucural models, he underlying filraion F is generaed by a sandard Brownian moion. In ha case, he defaul ime τ will be an F-predicable sopping ime as any sopping ime wih respec o a Brownian filraion, meaning ha here exiss a sricly increasing sequence of F-sopping imes announcing he defaul ime. Provided ha defaul has no occurred before or a ime T, he promised claim X is received in full a he claim s mauriy dae T. Oherwise, depending on he marke convenion regarding a paricular conrac, eiher he amoun X is received a mauriy T, or he amoun Z τ is received a ime τ. If defaul occurs a mauriy of he claim, ha is, on he even {τ = T }, we adop he convenion ha only he recovery paymen X is received. I is someimes convenien o consider simulaneously boh kinds of recovery payoff. Therefore, in his chaper, a generic defaulable claim is formally defined as a quinuple X, A, X, Z, τ. In oher chapers, we se X = and we consider a quadruple X, A, Z, τ, formally idenified wih a claim X, A,, Z, τ. In some cases, we will also se A = so ha a defaulable claim will reduce o a riple X, Z, τ, o be idenified wih 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 defaul process H = 1 { τ} and we denoe by D he process modeling all cash flows received by he owner of a defaulable claim. Le us wrie X d T = X1 {τ>t } + X1 {τ T }. Definiion The dividend process D of a defaulable coningen claim X, A, X, Z, τ wih mauriy dae T equals, for every R +, D = XT d 1 [T, [ + 1 H u da u + Z u dh u. I is apparen ha he process D is of finie variaion, and 1 H u da u = 1 {u<τ} da u = A τ 1 { τ} + A 1 {<τ}. ],] ],] ],] ],]

11 1.1. NOTATION AND DEFINITIONS 11 Noe ha if defaul occurs a some dae, he promised dividend paymen A A, which is due o occur 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}. ],] Remark In principle, he promised payoff X could be easily incorporaed ino he promised dividends process A. This would no be convenien, however, since in pracice he recovery rules concerning he promised dividends A and he promised claim X are no he same, in general. For insance, in he case of a defaulable coupon bond, i is frequenly posulaed ha if defaul occurs hen he fuure coupons are los, whereas a sricly posiive fracion of he face value is received by he bondholder. We are in a 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 a defaulable claim X, A, X, Z, τ is given as S = B E Q B 1 u dd u F. 1.1 ],T ] Noe ha he discouned ex-dividend price S = S B 1, [, T ], saisfies S = E Q Bu 1 dd u F ],T ] ],] B 1 u dd u. Hence i is a supermaringale submaringale, respecively under Q if and only if he dividend process D is increasing decreasing, respecively. The process S c, which is given by he formula S c = B E Q Bu 1 dd u F = S + B Bu 1 dd u, ],T ] ],] is called he cumulaive price of a defaulable claim X, A, X, Z, τ 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 referred o as he corporae discoun bond in he sequel 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 recovery rae upon defaul. For a corporae bond, i is naural o assume ha X L, so ha for random variable δt we obain he following bounds δt 1.

12 12 CHAPTER 1. STRUCTURAL APPROACH where 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, 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 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 According o Meron s [124] model, he recovery payoff upon defaul ha is, on he even {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 is equal here o δt = V T /L and he wriedown rae upon defaul equals wt = 1 V T /L. For simpliciy, we assume ha he savings accoun B is non-sochasic ha is, he shorerm ineres 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 Qτ T F p, where p = Qτ 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 w, 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 ha 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 will reurn o various convenions regarding he recovery values of corporae bonds laer on in his ex see, in paricular, Secion Meron s Model 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 following sochasic differenial equaion SDE dv = V r κ d + σv dw

13 1.2. MERTON S MODEL 13 wih V >, where κ is he consan payou raio dividend yield and he process W is a sandard Brownian moion under he maringale measure Q. The posiive consan σ V represens he volailiy. We firs presen he classic model pu forward by Meron [124], who proposed o base he valuaion of a corporae bond on he following posulaes: a firm has a single liabiliy wih he promised erminal payoff L, inerpreed as a zero-coupon bond wih mauriy T and face value L >, he 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 defaul ime τ in Meron s model equals τ = T 1 {VT <L} + 1 {VT L}. Using he presen noaion, a corporae bond is described by A =, Z =, and X d T = V T 1 {VT <L} + L1 {VT L} so ha X = V T. In oher words, he bond s payoff a mauriy dae T equals DT, T = min V T, L = L max L V T, = L L V T +. The las 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 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 admis he following represenaion DV = D, T = LB, T P, where P is he price of a pu opion wih srike L and expiraion dae T. Hence he value EV of he firm s equiy a ime equals EV = V e κt DV = V e κt 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 can be seen as holders of he call opion wih srike L and expiry T on he oal value of he firm s asses. Using he opion-like feaures of a corporae bond, Meron [124] derived a closed-form expression for is arbirage price. Le N denoe he sandard Gaussian cumulaive disribuion funcion Nx = 1 2π x e u2 /2 du, x R. 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.

14 14 CHAPTER 1. STRUCTURAL APPROACH The unique replicaing sraegy for a corporae 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. Le us now examine credi spreads in Meron s model. 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 = d + V, T = ln Γ σ2 V T. 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 D, T < LB, T is valid. This condiion is in urn equivalen o he sric posiiviy of he corresponding credi spread, as defined by formula 1.2 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 ineres rae r. Indeed, we have B, T = e r,t T = e rt. Le us denoe by r d, T he coninuously compounded yield a ime < T on he corporae bond, so ha D, T = Le rd,t T. From he las equaliy, i follows ha r d, T = ln D, T ln L. T The credi spread S, T is defined as he excess reurn on a defaulable bond, ha is, for any < T, S, T = r d, T r, T = 1 LB, T ln T D, T. 1.2 In Meron s model, he credi spread S, T is given by he following expression S, T = ln Nd σ V T + Γ N d T The propery S, T > is consisen wih he real-life feaure ha corporae bonds have an expeced reurn in excess of he risk-free ineres rae. Indeed, he observed yields on corporae bonds are sysemaically higher han yields on Treasury bonds wih maching noional amouns and mauriies. Noe, however, ha when ime converges o mauriy dae T hen 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 credi spread a ime T as ST, T := lim T S, T >.

15 1.3. FIRST PASSAGE TIMES 15 hen, by sraighforward compuaions, we obain ha {, on he even {VT > L}, ST, T =, on he even {V T < L}. I is frequenly argued in he financial lieraure ha, for realisic values of model s parameers, he credi spreads produced by Meron s model for bonds wih shor mauriies are far below he spreads observed in he marke. 1.3 Firs Passage Times Before we presen an exension of Meron s model, pu forward by Black and Cox [25], le us presen some well-known mahemaical resuls regarding firs passage imes, which will prove useful in wha follows. Le W be a sandard one-dimensional Brownian moion under Q wih respec o is naural filraion F. Le us define an auxiliary process Y by seing, for every R +, Y = y + ν + σw, 1.3 for some consans ν R and σ >. Le us noice ha Y inheris from W he srong Markov propery wih respec o he filraion F Disribuion of he Firs Passage Time Le τ sand for he firs passage ime o zero by he process Y, ha is, τ = inf { R + : Y = }. 1.4 I is known ha in an arbirarily small inerval [, ] he sample pah of he Brownian moion sared a passes hrough origin infiniely many imes. Using Girsanov s heorem and he srong Markov propery of he Brownian moion, i is hus easy o deduce ha he firs passage ime by Y o zero coincides wih he firs crossing ime by Y of he level, ha is, wih probabiliy 1, τ = inf { R + : Y < } = inf { R + : Y }. In wha follows, we will wrie X = ν + σw for every R +. Lemma Le σ > and ν R. Then for every x > we have Q sup X u x x νs x νs = N u s σ e 2νσ 2x N s σ s 1.5 and for every x < Q inf X u x = N u s x + νs σ s e 2νσ 2x N x + νs σ. 1.6 s Proof. To derive he firs equaliy, we will use Girsanov s heorem and he reflecion principle for a Brownian moion. Assume firs ha σ = 1. Le P be he probabiliy measure on Ω, F s given by so ha he process W dp dq = e νw s ν 2 2 s, Q-a.s., := X = W + ν, [, s], is a sandard Brownian moion under P. Also dq dp = eνw s ν2 2 s, P-a.s.

16 16 CHAPTER 1. STRUCTURAL APPROACH Moreover, for x >, Q sup X u > x, X s x = E P e νw s ν2 2 s 1 { sup u s W. u u s >x, W s x} We se τ x = inf { : W = x} and we define an auxiliary process W, [, s] by seing W = W 1 {τx } + 2x W 1 {τx <}. By virue of he reflecion principle, W is a sandard Brownian moion under P. Moreover, we have { sup W u > x, W s x} = {Ws x} {τ x s}. u s Le J := Q sup W u + νu x. u s Then we obain J = QX s x Q sup X u > x, X s x u s = QX s x E P e νw s ν2 2 s 1 { sup u s Wu >x, W s x} = QX s x E P e νf W s ν2 2 s 1 { sup f u s Wu >x, W f s x} = QX s x E P e ν2x W ν2 s 2 s 1 {W s x} = QX s x e 2νx E P e νw s ν2 2 s 1 {W s x} = QW s + νs x e 2νx QW s + νs x x νs x νs = N e 2νx N. s s This ends he proof of he firs equaliy for σ = 1. For any σ >, we have Q sup σw u + νu x = Q sup W u + νσ 1 u xσ 1, u s u s and his implies 1.5. Since W is a sandard Brownian moion under Q, we also have ha, for any x <, Q inf σw u + νu x = Q sup σw u νu x, u s u s and hus 1.6 easily follows from 1.5. Proposiion The firs passage ime τ given by 1.4 has he inverse Gaussian probabiliy disribuion under Q. Specifically, for any < s <, Qτ s = Qτ < s = Nh 1 s + e 2νσ 2 y Nh 2 s, 1.7 where N is he sandard Gaussian cumulaive disribuion funcion and Proof. Noice firs ha where X u = νu + σw u. h 1 s = y νs σ s, h 2 s = y + νs σ. s Qτ s = Q inf Y u = Q inf X u y, 1.8 u s u s

17 1.3. FIRST PASSAGE TIMES 17 From Lemma 1.3.1, we have ha, for every x <, Q inf X u x x + νs = N u s σ s e 2νσ 2x N x + νs σ, s and his yields 1.7, when combined wih 1.8. The following corollary is a consequence of Proposiion and he srong Markov propery of he process Y wih respec o he filraion F. Corollary For any < s we have, on he even { < τ}, Y νs Qτ s F = N σ + e 2νσ 2 Y Y + νs N s σ. s We are in a posiion o apply he foregoing resuls o specific examples of defaul imes. We firs examine he case of a consan lower hreshold. Example Suppose ha he shor-erm ineres rae is consan, ha is, r = r for every R +. Le he value of he firm process V obey he SDE dv = V r κ d + σv dw wih consan coefficiens κ R and σ V >. Le us also assume ha he barrier process v is consan and equal o v, where he consan v saisfies v < V, so ha he defaul ime is given as τ = inf { R + : V v} = inf { R + : V < v}. We now se Y = lnv / v. Then i is easy o check ha ν = r κ 1 2 σ2 V and σ = σ V in formula 1.3. By applying Corollary 1.3.1, we obain, for every s > on he even { < τ}, ln v V Qτ s F = N νs v 2aN ln v V + + νs, σ V s V σ V s where we denoe a = ν σ 2 V This resul was used in Leland and Tof [117]. = r κ 1 2 σ2 V σ 2 V Example Le he value process V and he shor-erm ineres rae r be as in Example For a sricly posiive consan K and an arbirary γ R +, le he barrier funcion be defined as v = Ke γt for R +, so ha he funcion v saisfies d v = γ v d, v = Ke γt. We now se Y = lnv / v and hus he coefficiens in 1.3 are ν = r κ γ 1 2 σ2 V and σ = σ V. We define he defaul ime τ by seing τ = inf { : V v}. From Corollary 1.3.1, we obain, for every < s on he even { < τ}, ln v V Qτ s F = N νs 2ea v ln v V + N + νs, σ V s σ V s where ã = This formula was employed by Black and Cox [25]. ν σ 2 V V = r κ γ 1 2 σ2 V σ 2 V..

18 18 CHAPTER 1. STRUCTURAL APPROACH Join Disribuion of Y and τ We will now find he join probabiliy disribuion, for every y and s >, I := QY s y, τ s F = QY s y, τ > s F, where τ is given by 1.4. Le us denoe by M W and m W he running maximum and minimum of a one-dimensional sandard Brownian moion W, respecively. More explicily, Ms W = sup u s W u and m W s = inf u s W u. I is well known ha for every s > we have QM W s > = 1, Qm W s < = 1. The following classic resul commonly referred o as he reflecion principle is a sraighforward consequence of he srong Markov propery of he Brownian moion. Lemma We have ha, for every s >, y and x y, QW s x, M W s y = QW s 2y x = QW s x 2y. 1.9 We need o examine he Brownian moion wih non-zero drif. Consider he process X ha equals X = ν + σw. We wrie M X s = sup u s X u and m X s = inf u s X u. By virue of Girsanov s heorem, he process X is a Brownian moion, up o an appropriae re-scaling, under an equivalen probabiliy measure and hus we have, for any s >, QM X s > = 1, Qm X s < = 1. Lemma For every s >, he join disribuion of X s, M X s is given by he expression QX s x, M X s for every x, y R such ha y and x y. y = e 2νyσ 2 QX s 2y x + 2νs Proof. Since I := Q X s x, Ms X y = Q Xs σ xσ 1, Ms Xσ yσ 1, where X σ = W + νσ 1, i is clear ha we may assume, wihou loss of generaliy, ha σ = 1. We will use an equivalen change of probabiliy measure. From Girsanov s heorem, i follows ha X is a sandard Brownian moion under he probabiliy measure P, which is given on Ω, F s by he Radon-Nikodým densiy recall ha σ = 1 Noe also ha where he process W easily seen ha dp dq = e νw s ν 2 2 s, Q-a.s. dq dp = eνw s ν2 2 s, P-a.s., = X = W + ν, [, s] is a sandard Brownian moion under P. I is I = E P e νw s ν2 2 s 1 {Xs x, M X s y} = E P e νw s ν2 2 s 1 {W s x, M W s y} Since W is a sandard Brownian moion under P, an applicaion of he reflecion principle 1.9 gives I = E P e ν2y W ν2 s since clearly 2y x y. 2 s 1 {2y W s x, M W s = E P e ν2y W ν2 s 2 s 1 {W s 2y x} = e 2νy E P e νw s ν2 2 s 1 {W s 2y x}, y}.

19 1.3. FIRST PASSAGE TIMES 19 Le us define one more equivalen probabiliy measure, P say, by seing Is is clear ha d P dp = e νw s ν2 2 s, P-a.s. I = e 2νy E P e νw s ν2 2 s 1 {W s 2y x} = e 2νy PW s 2y x. Furhermore, he process W = W + ν, [, s] is a sandard Brownian moion under P and we have ha I = e 2νy P Ws + νs 2y x + 2νs. The las equaliy easily yields he assered formula. I is worhwhile o observe ha a similar remark applies o all formulae below QX s x, M X s y = QX s < x, M X s > y. The following resul is a sraighforward consequence of Lemma Proposiion For any x, y R saisfying y and x y, we have ha Q X s x, Ms X y x 2y νs = e 2νyσ 2 N σ. s Hence Q X s x, M X s y = N for every x, y R such ha x y and y. x νs σ e 2νyσ 2 N s x 2y νs σ s Proof. For he firs equaliy, noe ha x 2y νs QX s 2y x + 2νs = Q σw s x 2y νs = N σ, s since σw has Gaussian law wih zero mean and variance σ 2. For he second formula, i is enough o observe ha QX s x, M X s y + QX s x, M X s y = QX s x and o apply he firs equaliy. I is clear ha for every y, and hus QM X s y = QX s y + QX s y, M X s y QM X s y = QX s y + e 2νyσ 2 QX s y + 2νs. Consequenly, QM X s y = 1 QM X s y = QX s y e 2νyσ 2 QX s y + 2νs. This leads o he following corollary. Corollary The following equaliy is valid, for every s > and y, y νs y νs QMs X y = N σ e 2νyσ 2 N s σ. s

20 2 CHAPTER 1. STRUCTURAL APPROACH We will now focus on he disribuion of he minimal value of X. Observe ha we have, for any y, Q sup σw u νu y = Q inf X u y, u s u s where we have used he symmery of he Brownian moion. Consequenly, for every y we have Qm X s y = QM e X s y, where he process X equals X = σw ν. I is hus no difficul o esablish he following resul. Proposiion The join probabiliy disribuion of X s, m X s saisfies, for every s >, x + νs 2y x + νs Q X s x, m X s y = N σ e 2νyσ 2 N s σ s for every x, y R such ha y and y x. Corollary The following equaliy is valid, for every s > and y, y + νs y + νs Qm X s y = N σ e 2νyσ 2 N s σ. s Recall ha we denoe Y = y + X, where X = ν + σw. We wrie m X s = inf u s X u, m Y s = inf u s Y u. Corollary We have ha, for any s > and y, y + y + νs QY s y, τ s = N σ e 2νσ 2 y y y + νs N s σ. s Proof. Since QY s y, τ s = Q Y s y, m Y s = Q X s y y, m X s y, he assered formula is raher obvious. More generally, he Markov propery of Y jusifies he following resul. Lemma We have ha, for any < s and y, on he even { < τ}, y + Y + νs QY s y, τ s F = N σ s e 2νσ 2 Y y Y + νs N σ. s Example Assume ha he dynamics of he value of he firm process V are dv = V r κ d + σv dw 1.1 and se τ = inf { : V v}, where he consan v saisfies v < V. By applying Lemma o Y = lnv / v and y = lnx/ v, we obain he following equaliy, which holds for x v on he even { < τ}, lnv /x + νs QV s x, τ s F = N where ν = r κ 1 2 σ2 V 2a v N V and a = νσ 2 V. σ s ln v 2 lnxv + νs σ s,

21 1.4. BLACK AND COX MODEL 21 Example We consider he seup of Example 1.3.2, so ha he value process V saisfies 1.1 and he barrier funcion equals v = Ke γt for some consans K > and γ R. Making use again of Lemma 1.3.4, bu his ime wih Y = lnv / v and y = lnx/ vs, we find ha, for every < s T and an arbirary x vs, he following equaliy holds on he even { < τ} lnv / v lnx/ vs + νs QV s x, τ s F = N σ V s 2ea v lnv / v lnx/ vs + νs N σ V s V where ν = r κ γ 1 2 σ2 V and ã = νσ 2 V. Upon simplificaion, his yields lnv /x + νs QV s x, τ s F = N σ V s 2ea v ln v 2 lnxv + νs N, σ V s where ν = r κ 1 2 σ2 V. V Remark Noe ha if we ake x = vs = Ke γt s hen clearly 1 QV s vs, τ s F = Qτ < s F = Qτ s F. Bu we also have ha lnv / vs + νs ln v/v νs 1 N = N σ V s σ V s and ln v 2 ln vsv + νs ln v/v + νs N = N. σ V s σ V s By seing x = vs, we rediscover he formula esablished in Example , 1.4 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 have pu forward various srucural models for valuaion of a corporae deb in which his resricive and unrealisic feaure was relaxed. In mos of hese models, he ime of defaul was defined as he firs passage ime of he value process V o eiher deerminisic or 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 firm value process V. Black and Cox [25] exend Meron s [124] research in several direcions by aking ino accoun such specific feaures of real-life deb conracs as: safey covenans, deb subordinaion, and resricions on he sale of asses. Following Meron [124], hey assume ha he firm s sockholders

22 22 CHAPTER 1. STRUCTURAL APPROACH 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. The so-called safey covenans provide he bondholders wih he righ o force he firm o bankrupcy or reorganizaion if he firm is doing poorly according o some gauge. 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 oal value of firm s asses his his lower hreshold, he bondholders ake over he firm. Oherwise, defaul eiher occurs a mauriy dae T or no, depending on wheher he inequaliy V T < L holds or no. 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. Formally, he defaul ime equals by convenion inf = + τ = inf { [, T ] : V v }. The recovery process Z and he recovery payoff X are proporional o he value process, specifically, Z = β 2 V and X = β 1 V T for some consans β 1, β 2 [, 1]. Noe ha he case examined by Black and Cox [25] corresponds o β 1 = β 2 = 1, bu, of course, he exension o he case of arbirary β 1 and β 2 is immediae. To summarize, we consider he following defaulable claim X = L, A =, X = β1 V T, Z = β 2 V, τ = τ τ, where he early defaul ime τ equals τ = inf { [, T [ : V v} and τ sands for Meron s defaul ime, ha is, 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 for every [, T ] or, more explicily, Ke γt Le rt, so ha, in paricular, K L. This addiional condiion is imposed in order o guaranee ha he payoff o he bondholder a he defaul ime τ will never exceed he face value of he deb, discouned a a risk-free rae. Since he dynamics for he value process V are 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 corporae bond saisfies he following PDE u v, + r κvu v v, σ2 V v 2 u vv v, ruv, = on he domain {v, R + R + : < < T, v > Ke γt }

23 1.4. BLACK AND COX MODEL 23 wih he boundary condiion uke γt γt, = β 2 Ke and he erminal condiion uv, T = min β 1 v, L. Alernaively, he price D, T = uv, of a defaulable bond has he following probabilisic represenaion, on he even { < τ} = { < τ}, D, T = E Q Le rt 1 { τ T, VT L} F + β 1 E Q V T e rt 1 { τ T, VT <L} F + β 2 K E Q e γt τ e r τ 1 {< τ<t } F. Afer defaul ha is, on he even { τ} = { τ}, we clearly have D, T = β 2 vτb 1 τ, T B, T = β 2 Ke γt τ e r τ. We wish find explici expressions for he condiional expecaions arising in he probabilisic represenaion of he price D, T. To his end, we observe ha: he firs wo condiional expecaions can be compued by using he formula for he condiional probabiliy QV s x, τ s F, o evaluae he hird condiional expecaion, i suffices o 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 V. 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 will rely on he knowledge of he probabiliy law of he firs passage ime of he geomeric ha is, exponenial Brownian moion o an exponenial barrier. All relevan resuls regarding his issue were already esablished in Secion 1.3 see, in paricular, Examples and Proposiion Assume ha m 2 +2σ 2 r γ >. Prior o defaul, ha is, on he even { < τ}, he price process D, T = uv, of a defaulable bond equals D, T = LB, T N h 1 V, T R 2b 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 R 2b+2 N h5 V, T N h 6 V, T + β 2 V R θ+ζ N h 7 V, T + R θ ζ N h 8 V, T, where R = 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

24 24 CHAPTER 1. STRUCTURAL APPROACH 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 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 Before proceeding o he proof of Proposiion 1.4.1, we will esablish an elemenary lemma. Lemma For any a R and b > we have, for every y >, y ln x + a x dn = e 1 ln y + a b 2 b2 a 2 N b b and y 1.11 ln x + a x dn = e 1 ln y + a + b 2 b2 +a 2 N b b Le a, b, c R saisfy b < and c 2 > 2a. Then we have, for every y >, y b cx e ax dn = d + c x 2d gy + d c hy, d where d = c 2 2a and where we denoe gy = e bc d N b dy, hy = e bc+d N y b + dy. y Proof. The proof of is sandard. For 1.13, observe ha y b cx y b cx fy := e ax dn = e ax n b x x 2x 3/2 c 2 dx, x where n is he probabiliy densiy funcion of he sandard Gaussian law. Noe also ha g x = e bc c 2 2a b c n 2 2ax b x 2x c2 2a 3/2 2 x b cx = e ax n b x 2x d 3/2 2 x and h x = e bc+ c 2 2a n b cx = e ax n x b + c 2 2ax x b 2x + 3/2 d 2. x b 2x + c2 2a 3/2 2 x

25 1.4. BLACK AND COX MODEL 25 Consequenly, and Hence f can be represened as follows fy = 1 2 g x + h x = e ax b b cx x n 3/2 x g x h x = e ax d b cx x n. 1/2 x y g x + h x + c d g x h x dx. Since lim y + gy = lim y + hy =, we conclude ha we have, for every y >, This ends he proof of he lemma. fy = 1 c gy + hy + gy hy. 2 2d Proof of Proposiion To esablish he assered formula, i suffices o evaluae he following condiional expecaions: D 1, T = LB, T QV T L, τ T F, D 2, T = β 1 B, T E Q VT 1 {VT <L, τ T } F, D 3, T = Kβ 2 B e γt E Q e γ r τ 1 {< τ<t } F. For he sake of noaional convenience, we will focus on he case = of course, he general resul will follow easily. Le us firs evaluae D 1, T, ha is, he par of he bond value corresponding o no-defaul even. From Example 1.3.4, we know ha if L vt = K hen QV T L, τ T = N wih R = v/v. I is hus clear ha ln V L + νt ln v 2 σ R 2ea LV N + νt T σ T D 1, T = LB, T N h 1 V, T R 2ea N h 2 V, T. Le us now examine D 2, T ha is, he par of he bond s value associaed wih defaul a ime T. We noe ha D 2, T β 1 B, T = E L Q VT 1 {VT <L, τ T } = K x dqv T < x, τ T. Using again Example and he fac ha he probabiliy Q τ T does no depend on x, we obain, for every x K, dqv T < x, τ T = dn ln x V νt ln v 2 σ + R 2ea xv dn + νt T σ. T Le us denoe and K 2 = K 1 = L K L K ln x ln V νt x dn σ T 2 ln v ln x ln V + νt x dn σ T.

26 26 CHAPTER 1. STRUCTURAL APPROACH Using , we obain K 1 = V e r κt N where ν = ν + σ 2 = r κ σ2. Similarly, Since ln v 2 K 2 = V Re 2 r κt LV N + νt σ T we conclude ha D 2, T is equal o ln L V νt ln K σ V N νt T σ, T N D 2, T = β 1 B, T K 1 + R ea K 2, ln v 2 KV + νt σ T D 2, T = β 1 V e κt N h 3 V, T N h 4 V, T + β 1 V e κt R 2ea+2 N h5 V, T N h 6 V, T. I remains o evaluae D 3, T, ha is, he par of he bond value associaed wih he possibiliy of he forced bankrupcy before he mauriy dae T. To his end, i suffices o calculae he following expeced value v E Q e γ r τ T 1 { τ<t } = v e γ rs dq τ s, where see Example V. 2ea ln v/v νs v Q τ s = N σ ln v/v + νs + N s σ. s Noe ha v < V and hus ln v/v <. Using 1.13, we obain T v e γ rs dn = V ã + ζ 2ζ ln v/v νs σ s R θ ζ N h 8 V, T V ã ζ R θ+ζ N h 7 V, T 2ζ and v 2ea+1 T V 2ea = V ã + ζ 2ζ e γ rs ln v/v + νs dn σ s R θ+ζ N h 7 V, T V ã ζ R θ ζ N h 8 V, T. 2ζ Consequenly, D 3, T = β 2 V R θ+ζ N h 7 V, T + R θ ζ N h 8 V, T. Upon summaion, his complees he proof for =. Le us consider some special cases of he Black and Cox pricing formula. 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 he pricing formula reduces o D, T = D 1, T + D 3, T, where D 1, T = LB, T N h 1 V, T R 2â N h 2 V, T, D 3, T = V R θ+ζ N h 7 V, T + R θ ζ N h 8 V, T.

27 1.4. BLACK AND COX MODEL 27 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 R θ+ζ = LB, T, V R θ ζ = V R 2â+1 = LB, T R 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. Noe ha his 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 Corporae Coupon Bond We now posulae ha he shor-erm rae r > and ha a defaulable bond, of fixed mauriy T and face value L, pays coninuously coupons a a consan rae c, so ha A = c for every R +. The coupon paymens are disconinued as soon as he defaul even occurs. Formally, we consider here a defaulable claim specified as follows X = L, A = c, X = β1 V T, Z = β 2 V, τ = inf { [, T ] : V < v } wih he Black and Cox barrier v. Le us denoe by D c, T he value of such a claim a ime < T. I is clear ha D c, T = D, T + A, T, where A, T sands for he discouned value of fuure coupon paymens. The value of A, T can be compued as follows T T A, T = E Q ce rs 1 { τ>s} ds F = ce r e rs Q τ > s F ds. Seing =, we hus obain D c, T = D, T + c T e rs Q τ > s ds = D, T + A, T, where recall ha we wrie σ insead of σ V 2ea lnv / v + νs v Q τ > s = N σ ln v/v + νs N s σ. s An inegraion by pars formula yields T e rs Q τ > s ds = 1 r V T 1 e rt Q τ > T + e rs dq τ > s. We assume, as usual, ha V > v, so ha ln v/v <. Arguing in a similar way as in he las par of he proof of Proposiion specifically, using formula 1.13, we obain T ea+ ζ e v e rs ln v/v + dq τ > s = N ζσ 2 T V σ T ea ζ e v ln v/v N ζσ 2 T V σ T,