A general first-passage-time model for multivariate credit spreads and a note on barrier option pricing. Inaugural-Dissertation

Transcription

1 A general firs-passage-ime model for mulivariae credi spreads and a noe on barrier opion pricing Inaugural-Disseraion zur Erlangung des Dokorgrades an den Naurwissenschaflichen Fachbereichen Mahemaik der Jusus-Liebig-Universiä Giessen vorgeleg von Sefanie Kammer 12. Sepember 27

2 ii Dekan: Prof. Dr. Bernd Baumann Guacher: Prof. Dr. Ludger Overbeck Jusus-Liebig-Universiä Giessen Prof. Dr. Winfried Sue Jusus-Liebig-Universiä Giessen exerner Bereuer: Prof. Dr. Wolfgang Schmid Frankfur School of Finance & Managemen Dispuaion: November 27

3 Für meine Omas und Opas

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5 Preface Afer finishing my diploma in mahemaics in November 22 I was no oally sure abou doing a PhD, so I sared working wih KPMG as a quaniaive advisor where my main ask was o price any financial produc. There I was more and more convinced ha I needed deeper heory for my undersanding. I wen back o he universiy of Giessen and go a research posiion wihin a projec ha is financed by BMBF Bundesminiserium für Bildung und Forschung which I graefully acknowledge. Wih his disseraion I never wen back o pure mahemaics, bu a leas I found my place somewhere in beween deep heory and pure applicaion. I have o hank many people who helped me o come his far and made my ime so enjoyable wih workshops, winerschools, coffee breaks and cockails: I wan o hank my supervisors Prof. Dr. Ludger Overbeck, Prof. Dr. Wolfgang Schmid, and Prof. Dr. Winfried Sue for many discussions and advices. Prof. Sue, hanks for accompanying me hrough my whole universiy life! Three years ago, I had o promise my grandma ha she can come o my dispuaion in order o mee you! I furher wan o hank all my colleagues and friends a he universiy of Giessen, he Frankfur School of Finance & Managemen, and he universiy of Ulm; in paricular Rolf Klaas, Swanje Becker, Chrisina Niehammer, Naalie Packham, and las bu no leas Rüdiger Kiesel. I wan o hank my grea friends from he good old Coba ime: Andreas, Anonis and Tino. I s always a pleasure o cach up wih you! Andreas, a very big hank-you goes o you for having spen such an enjoyable ime wih me here in Frankfur and such an amoun of ime wih my hesis. I am also very graeful o Nick Bingham for his big effor o inser housands of commas and hyphens ino my wriing: I shall remember he nice sory of Cinderella! Once more, hank you o all of my friends! I am mos graeful o my bes and oldes friends, Meli and Olli, for being here always when necessary - for almos all my life! Finally, I wan o hank my family: Mama, Papa, Oma und Opa und nochmal Oma und Opa, danke, dass ihr mich so lange unersüz hab! iii

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7 Conens 1 Firs-passage imes Firs-passage-ime framework The aim of his chaper and our applicaion Survey: firs-passage-ime models Brownian moion wih drif Join survival probabiliy - Brownian moion Join survival probabiliy - Brownian moion wih drif Deerminisically ime-changed Brownian moion Join survival probabiliy - deerminisically ime-changed Brownian moion Classical jump-diffusion approach Firs-passage jump-diffusion approach Subordinaed Lévy processes Sable processes a firs passage Time-changed Lévy processes Summary and conclusion Sochasic ime-change model Wha does a coninuous sochasic ime change look like? Firs-passage-ime disribuion Mulivariae firs-passage-ime disribuion under Brownian independence Join survival probabiliy under Brownian correlaion Join survival probabiliy under Brownian correlaion and separae ime changes Numerical ime-change densiies Explici ime-change densiies Analyzing a simple ime-change model Calibraion o a defaul-probabiliy curve Mulivariae defaul probabiliies Join survival probabiliy under Brownian correlaion Even Correlaion v

8 vi CONTENTS Even correlaion agains ime Credi spread Credi-risk framework Credi-defaul-swap spread Annual credi-spread paymens Coninuous credi-spread paymens Coninuous credi-spread paymens in discree ime Credi-spread formula Credi-spread curve and is dynamics Esimaing credi-spread volailiy Volailiy esimae of a log-normal spread process Volailiy esimae of a normal spread process Conclusion Meron model Model framework Speed-of-defaul probabiliy Calibraion - hreshold level Credi spread Insananeous spread Simulaions Credi-spread dynamics Spread dynamics for fixed mauriy T Spread dynamics for fixed ime o mauriy M Simulaion: spread dynamics and spread pahs Simulaion: spread volailiy Survey: exensions of he Meron model CrediGrades model Longsaff and Schwarz: CIR ineres raes Collin-Dufresne and Goldsein: mean-revering ineres raes and barrier Jacobs and Li: wo-facor model Duffie and Lando: incomplee accouning informaion Giesecke and Goldberg: incomplee defaul-barrier informaion Zhou: jump-diffusion model Comparing srucural and reduced-form models Survey: mulivariae exensions Brownian correlaion in bivariae models Cariboni and Schouens: variance-gamma model Conclusion: Meron-ype models

9 CONTENTS vii 5 Overbeck & Schmid model Model framework Calibraion - hreshold level Credi Spread Insananeous spread Simulaions Credi-spread dynamics Simulaion: spread dynamics and spread pah Simulaion: spread volailiy Conclusion Sochasic ime-change model Model framework Credi spread Simulaions Credi-spread dynamics Firs-o-defaul swap Firs-o-defaul spread on wo credis Firs-o-defaul spread on n credis Explici condiional ime-change densiies The simple ime change The CIR-ype ime change Conclusion Applicaions o opion pricing Heson model Revisied: original Heson call price Revisied: analyical Heson call price Sochasic ime-change model European call Barrier opions The Dufresne ime change Idea: adding a leverage effec A Technical deails 141 A.1 General derivaive for a ime-dependen inegral A.2 Gamma, Bessel and modified Bessel funcion A.3 Proof: European call of Theorem

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11 Lis of Figures 1 Classical srucural approach xiv 1.1 Srucural firs-passage-ime approach Asse-value pahs for CIR-ype ime changes Oher asse-value pahs for CIR-ype ime changes CIR-ype ime-change disribuions Mulivariae asse-value process Densiies of he simple ime change Calibraion of he simple ime change Calibraion of he simple ime change Calibraion of he simple ime change Cumulaive defaul-rae curves by S&P Calibraion of he simple ime change o a CCC-curve Defaul-probabiliy curve and join defaul-probabiliy curves for wo resp. hree uncorrelaed asses Join defaul-probabiliy curves for wo resp. hree asses Join survival-probabiliy curves for several Brownian correlaion parameers Join survival-probabiliy curves for several Brownian correlaions ρ and ˆσ = 1 dashed resp. ˆσ = 5 solid Even correlaion versus ime, for fixed Brownian correlaion years credi-spread hisories of IBM and GM Credi-spread vol esimaion under he log-normal model Credi-spread vol esimaion under he normal model Defaul speed Survival-probabiliy erm srucures Survival-probabiliy pahs and corresponding credi-spread pahs Brownian pah and corresponding credi-spread erm srucures for K = 1, 2, 3, 4, 5 a =, 2.5, 5, 7.5, Brownian pah and resuling spread dynamics n = Brownian pah and resuling spread dynamics n = ix

12 x 4.7 Spread vol for = 1, M = 1 and Q, + M = 99% Spread vol for = 1, M = 5 and Q, + M = 99% CrediGrades model Credi-spread erm srucure under complee dashed curve and incomplee solid curve informaion from Duffie & Lando Defaul speed σ and analyical ime change T Survival-probabiliy erm srucure for K = 5.3, 4.2, 3.6, 3.1, 2.7 and T = Time-ransformed Brownian pah firs plo and corresponding credi-spread erm srucures ill defaul for he hreshold K = Time-ransformed Brownian pah and corresponding credispread erm srucures for he hreshold K = Defauling ime-ransformed Brownian moion, survival probabiliy, spread dynamics and resuling credi-spread pahs Time-ransformed Brownian moion, survival probabiliy, spread dynamics and resuling credi-spread pahs Spread vol for = 1, M = Spread vol for = 5, M = Defaul-probabiliy curves and credi-spread curves under G = B2 s ds Defaul-probabiliy curves and credi-spread curves under G = g + B2 s ds for various saring values g

13 Lis of Tables 1.1 Gamma subordinaor and inverse-gaussian subordinaor Time-change examples ha yield a Laplace ransform and hus a numerical ime-change disribuion Examples for ime changes wih analyical densiy Parameer ses calibraed o IPτ.5 =.6 and IPτ < 5 =.73 g = and Y = Model parameers µ and K calibraed o F = 1 e λ in wo poins for λ = 1%, λ = 7% resp. λ = 1% for σ = 1, ˆσ = Average S&P defaul raes [in %] Calibraion o an average CCC defaul-rae curve a wo poins, 1 and 2 [in years] for σ = 1 and ˆσ = Varying ime-change parameer ˆσ, resuling JSPs and corresponding Brownian correlaion parameers Even correlaion and corresponding Brownian correlaion for five resp. en years JDPs under marginal defaul-raes of.7 and xi

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15 Inroducion A firs-passage ime FPT is defined as he firs ime poin a sochasic process crosses some hreshold level, defaul boundary or criical barrier ha is usually a consan, bu can be a random variable or even a sochasic process iself. When we aim a general resuls for firs-passage imes we will call his sochasic process he firs-passage process or underlying process. This model approach a firs-passage process falling below some hreshold is called hreshold model/approach. A firs-passage ime will be a sopping ime wih respec o a filraion ha holds he necessary informaion o answer he quesion wheher a firs passage happened. In his disseraion we will inroduce a new class of firs-passage processes: Brownian moion ime-changed wih a coninuous sochasic non-decreasing process. Our main applicaion will be in credi risk, where he firs-passage process will be inerpreed as firm-value process, asse-value process or abiliyo-pay process - depending on one s paricular ineres. The hreshold level will be a funcion of he firm s liabiliies. The hreshold approach hus connecs equiy and deb of a firm. In his conex he hreshold model is also called srucural model, see Definiion 1.8, and he firs-passage ime is called defaul ime because i indicaes a company s defaul even. A defaul even may no imply a oal defaul liquidaion of he firm bu can also indicae a raing downgrade 1. As lieraure on credi risk and credi derivaives we sugges Bluhm & Overbeck 26, Schönbucher 23 and Marin, Reiz & Wehn 26. There is anoher applicaion in barrier opion pricing. Here he underlying process is he opion s underlying. The hreshold level is he barrier value and he firs-passage ime indicaes a cerain knock-in or knock-ou even. Our focus is on models ha yield a firs-passage-ime disribuion which 1 Raing sysems measure he crediworhiness of borrowing companies. The borrowers are ranked in raings. Exernal raings are assigned by raing agencies - he mos famous ones are Sandard & Poor s S&P, Moody s and Fich - and inernal raings are esablished by he credi insiue iself. An improving crediworhiness urns ino a raing upgrade and a worsening crediworhiness ino a raing downgrade. The ransiion probabiliies are given by a migraion marix. Compare Bluhm e al. 23 and Bluhm & Overbeck 26. xiii

16 xiv INTRODUCTION can be represened by an inegral and/or series. We will call his an analyical disribuion. In he menioned applicaions, credi risk and barrier opion pricing, his allows for a simpler calibraion of he model o given marke daa, eiher a defaul-probabiliy curve or barrier opion prices. In he following we summarize he advances of he srucural model: The classical srucural approach, inroduced by Meron 1974, considers a geomeric Brownian moion as asse-value process and assumes ha he firm s deb consis of only one issued zero-coupon bond wih some face value. This classical approach allows for a defaul only a one dae, he mauriy T of he bond, and defaul happens when he asse value a mauriy is less han he face value K. This is illusraed in Figure 1. In order o Figure 1: Classical srucural approach: defaul happens a a fixed ime T when he firm value a ha ime, Y T, lies below he pre-specified defaul hreshold K, ha is when Y T < K. deermine he defaul disribuion Meron applied he well-known opion pricing resul by Black & Scholes We name a few exensions of Meron s classical approach: Geske & Johnson 1984 considered couponbearing bonds. Leland 1994 and Leland & Tof 1996 applied a generalized geomeric Brownian moion and examined an opimal capial srucure, i.e. mauriy and amoun, of corporae deb. Their model is able o predic various credi-spread erm srucures. 2 Shimko, Tejima & van Devener 1993 inroduced sochasic ineres raes following a Vasicek process, see Vasicek Regarding Lévy processes 3 a defaulime disribuion under he classical approach can be deermined when a closed-form probabiliy densiy is available. This is for example he case for 2 We inroduce credi spreads in Chaper 3 and analyze hem in Chaper 4, 5 and 6. 3 A Lévy process is defined by independen and saionary incremens and sochasic coninuiy, see Def. 1.3 and cf. Sao 1999, Kyprianou 26b or Applebaum 25.

17 INTRODUCTION xv he jump-diffusion model, see Zhou 1997a and Subsecion 1.2.6, and for some subordinaed Lévy processes, see Subsecions and Black & Cox 1976 exended he classical approach o he firspassage approach: They considered a geomeric Brownian moion as abiliyo-pay process coninuously in ime, an exponenially ime-dependen defaul boundary and allowed for defaul a any ime, whenever he hreshold has been hi. Their approach yields an analyical firs-passage-ime disribuion; compare Harrison 1985 and see Secion There are several exensions of his original firs-passage approach. A firs survey on firspassage imes was given by Abrahams An overview of more recen firs-passage models can be found in Bielecki & Rukowski 22 and Elizalde 25 - among hese are he following: Kim, Ramaswarny & Sundaresan 1993 inroduced sochasic ineres raes following a Cox- Ingersoll-Ross CIR process; see Cox, Ingersoll & Ross There is a posiive probabiliy of defaul a mauriy. Longsaff & Schwarz 1995b used Vasiceck ineres raes. Nielsen, Saà-Requejo & Sana- Clara 1993 applied sochasic ineres raes modeled by a Hull & Whie process, and inroduced a sochasic defaul barrier. Boh papers, however, show ha inroducing a sochasic process for he risk-free ineres rae has only a small effec on credi spreads. Briys & de Varenne 1997 analyzed correlaed asse-value and ineres-rae processes. Regarding Lévy processes an analyical FPT formula is no available. Bu, a leas Kyprianou 26a and Alili & Kyprianou 25 were able o deermine overshoo and undershoo densiies for some specific, ime-changed Lévy processes a firs-passage, see Subsecions and For a jump-diffusion model wih double-exponenially disribued jumps, Kou & Wang 23 derived he Laplace ransform of he FPT disribuion, see Subsecion 1.2.7, in our erms i is no an analyical FPT disribuion. The srucural models menioned above assume ha he asse-value process is adaped o he marke filraion. In realiy his is no rue; see Buffe 22. For his Duffie & Lando 21 and Giesecke 24, 26 analyzed he role of informaion in srucural models and inroduced models wih incomplee accouning informaion abou he firm asses, as well as abou he liabiliy-dependen hreshold barrier. These models belong o he class of hybrid models because hey combine he advanages of reduced-form models and srucural models, ha is, racabiliy o marke daa and economic inuiion. For compleeness, one more word abou he class of reduced-form models. They are also called inensiy-based models or hazard-rae models because defaul probabiliies are modeled hrough a possibly random inensiy or hazard rae process. Defaul ime is an exogenous random variable and he cause of defaul is no furher specified hrough a firm value or asse value. These wo main model classes, srucural and reduced-form models, are compared in Jarrow & Proer 24. Oher hybrid models where defaul inensiy explicily depends on firm value were

18 xvi INTRODUCTION considered by Madan & Unal 1998 or Ammann Ammann herein analyzed counerpary risk in a Meron-ype framework. We also wan o name some lieraure of he more recen class of credibarrier models, where he focus lies on an influencing facor oher han he firm value. Hull & Whie 21 and Avellaneda & Zhu 21 modeled he disance-o-defaul in a srucural framework wih ime-dependen hreshold boundary. They assumed ha he disance-o-defaul is no observable; insead i is used o build a risk-neural measure ha leads o realisic spread curves. Gordy & Heifield 21 modeled he disance-o-defaul hrough a one-facor model and focus on he changes in disance-o-defaul over ime, in order o model raing ransiions. They found ha he process of raing ransiions is no closely ied o a defaul-indicaing process such as he disance-o-defaul process. Therefore hey suggesed o incorporae hrough-he-cycle raings. The credi-barrier model by Albanese e al. 23 and Albanese & Chen 27 considered he credi raing as driving process which was calibraed o migraion raes 4 and credi spreads. I capures all imporan firm informaion. Models wih jumps or sochasic volailiy are necessary in order o fi he whole marix of migraion raes, ha is, no only he probabiliies of reaining he same raing level, bu also he ransiion probabiliies of raings changes. Wih he jus given overview of one-dimensional srucural models, including he classical approach, he firs-passage approach, hybrid models wih incomplee informaion and credi-barrier models, we have seen ha here are srucural models, even wih jumps, where a classical defaul-ime disribuion can be obained. When regarding firs-passage-ime models hose based on Brownian moion can lead o analyical FPT disribuions. Bu when including jump processes, as far as we know, analyical FPT disribuions are no available. Bu, a leas, here are some examples where a numerical approximaion for he FPT disribuion can be yielded, when a Laplace ransform or Fourier ransform of he FPT disribuion is available. These will be said o have a numerical FPT disribuion, in order o differeniae from he analyical FPT disribuion. We also wan o focus on mulivariae srucural models and desire analyical join firs-passage ime disribuions ha yield join defaul probabiliies. Naurally, we also resric ourselves o processes based on Brownian moion. For wo correlaed Brownian moions and an exponenially imedependen barrier, Zhou 1997b derived a join FPT disribuion by applying resuls of Cox & Miller, Harrison 1985 and Rebholz In his conex Rebholz and Fischer 23 also considered he wo-dimensional Brownian moion wih drif. Overbeck & Schmid 25 exended he resul of Zhou by applying a deerminisic ime change 5 on each Brow- 4 See foonoe 1. 5 A deerminisic ime change is a posiive non-decreasing funcion; see Definiion 1.1.

19 INTRODUCTION xvii nian moion and deermined he join survival probabiliy for wo correlaed processes. In addiion, heir model perfecly fis he marginal defaulprobabiliy erm srucures. Sill, an abiliy-o-pay process given by a Brownian moion or a deerminisically ime-changed Brownian moion does no have enough degrees of freedom o adap marginal defaul probabiliies, join defaul probabiliies, and credi-spread dynamics he evoluion of he credi-spread erm srucure in ime 6. Furhermore he dependence beween wo asse-value processes relies on only one correlaion parameer. We analyze he Meron model and he Overbeck & Schmid model in he Chapers 4 and 5. In his disseraion we will consider firs-passage-ime models ha have a coninuous sochasic ime-change. For his we here give a shor hisory of he ime change: Bochner 1949 firs inroduced a ime-changed Brownian moion. Feller 1966 firs presened subordinaors as a ime change o Markov processes. Clark 1973 inroduced Brownian moion wih an independen ime change as a price process in finance. Monroe 1978 showed ha a very general semi-maringale can be embedded in Brownian moion via a ime change. Ikeda & Waanabe 1981 sudied ime-change models for solving SDEs. Øksendal 199 sudied when a sochasic inegral can be represened as a ime change of a diffusion. Geman, Madan & Yor 2 and Carr & Wu 23 inroduced subordinaed Lévy process. Schouens 23, 24 and Cariboni & Schouens 27 used hese in derivaive pricing. As he second par of he inroducion we here give he srucure of his docoral hesis: Chaper 1 sars wih an inroducion o he framework for modeling firspassage imes and gives definiions ha are used in all he chapers. For he firs ime we inroduce a general coninuous sochasic ime-change model on Brownian moion in a FPT-seing and derive analyical formulas for he FPT disribuion in one and several dimensions. The mulivariae model inroduces a dependence srucure via he ime change. Our wo-dimensional model allows for an addiional dependency of correlaed Brownian moions, he so-called Brownian correlaion, and also yields an analyical FPT disribuion. We give ime-change examples ha are close a hand and ha yield numerical ime-change disribuions in erms of a Laplace ransform and, as a consequence, numerical FPT disribuions. Furhermore we give less obvious ime-change examples for which we derive an analyical ime-change disribuion. Whenever a condiional ime-change disribuion is necessary, which is he case when we deermine credi-spread dynamics, numerical approximaions are no pracical. Our general ime-change model can also 6 See Conclusion 3.5.

20 xviii INTRODUCTION be used across asses in order o model dependence srucures on he one hand and o model dynamics on he oher hand. Up o now, mulivariae processes for dependency modeling include only eiher correlaion beween he driving processes or a ime change. Our bivariae model conains boh; see Secion Chaper 2 analyzes one specific ime-change model where he ime change is given by he inegral over an independen squared Brownian moion. This is he simples model in our general model class. We refer o i as he simple ime-change model. We calibrae he model o defaul-probabiliy curves and yield a good fi for non-invesmen-grade companies. Using hese calibraed model parameers we simulae and plo join and mulivariae defaul probabiliies or survival probabiliies, respecively. We analyze he influence of he ime change on he join survival probabiliy and furhermore he relaionship beween defaul correlaion and even correlaion. Chaper 3 inroduces he basic credi produc, he credi defaul swap CDS, which is a conrac beween a proecion seller and a proecion buyer. We deermine he CDS spread under annual and under coninuous proecion paymens following Hull & Whie 2 and Schmid 24b, respecively. Then we inroduce he credi-spread dynamics. In order o show ha i is reasonable and moreover necessary o consider credi-spread dynamics when modeling credi-spread curves, we empirically sudy credispread volailiy of five years markes CDS spreads. In Chaper 4 we deermine he credi spread and he credi-spread dynamics under he FPT approach of he Meron model. The model has no degrees of freedom o influence hese credi-spread dynamics. A he end of he chaper we review some exensions of he Meron model. The exensions add randomness o he defaul barrier CrediGrades model, he ineres raes, or he business clock. Oher exensions assume incomplee accouning informaion or include jumps ino he firs-passage process. The main advanages of hese models are ha hey can yield a posiive insananeous credi spread and also oher credi-spread shapes han he hump-shaped erm srucure under he Meron model. Chaper 5 sudies he deerminisic ime-ransformaion model by Overbeck & Schmid. The model perfecly fis he defaul-probabiliy curve and hus he survival-probabiliy erm srucure. We make an addiional assumpion of an available defaul-probabiliy densiy derivaive of he given defaulprobabiliy curve which implies an analyical formula for he credi-spread dynamics. Again credi-spread dynamics are a funcion of asse value and hreshold and canno be influenced.

21 INTRODUCTION xix In Chaper 6 he general sochasic coninuous ime-change model is applied o credi-spread modeling. Wih he ime change arbirarily many degrees of freedom can be included. Furhermore, when choosing a ime change wih sar above zero he model is of incomplee informaion and yields non-zero insananeous credi spreads. The ime change has several inerpreaions e.g. as economic ime or amoun of informaion flow. The exension of he sochasic ime-change model o a mulivariae model is sraighforward and insers a dependency via a join ime change. The model is applicable for mulivariae producs, especially muli-credi producs. A ime change should no always be chosen all he same for he underlying credis. Insead, he credi-spread dynamics of each underlying have o be sudied and he ime change should be chosen accordingly. We derive an analyical formula for he coninuously-paid credi spread. Assuming an absoluely coninuous business clock we can also deermine an analyical formula for he credi-spread dynamics. Credi-spread dynamics should in paricular be considered when credi conracs have a long ime o mauriy and when credi producs are credi-spread sensiive. We name a few examples: consan-o-mauriy CMS swaps, credi-spread opions, credi baskes, k-h-o-defaul swaps, collaeralized-deb obligaions CDOs, credi-spread variance swaps and leveraged credi producs; cf. for example Schönbucher 23 or Hun & Kennedy 24. We ake he firs-odefaul swap as an example o show ha our mulivariae model can be used o yield closed formulas for more complex credi producs. This is due o he analyical formula for he join defaul probabiliy. For all hese formulas he condiional ime-change densiy is needed. For his we give wo explici examples, he simple ime change of Chaper 2 and a CIR-ype ime change. Chaper 7 shows how our sochasic ime-change model can be applied o opion pricing, especially pricing of barrier opions. The ime-change model is a sochasic volailiy model, and we give he ime-change model ha is equivalen o he Heson model which is well-known for opion pricing. Under sochasic volailiy models he risk-neural measure is no unique and we choose o price under he minimal maringale measure. Assuming he general ime-change model, allowing for correlaion beween spo and ime change, we derive a closed pricing-formula for he European call. Under no correlaion and zero ineres raes we show how o derive pricing formulas for barrier opions applying our FPT resuls of Chaper 1. The degrees of freedom of he ime change can be used o produce desired volailiy feaures.

22 xx INTRODUCTION

23 Chaper 1 Firs-passage imes under a general coninuous sochasic ime-change model This chaper inroduces he general sochasic framework which will be he basis hroughou his hesis. For definiions and noaions from sochasic calculus we follow Proer 24 when considering sochasic processes in general, Karazas & Shreve 1991, Seele 21 and Klebaner 25 for processes based on Brownian moion and Schouens 23, Con & Tankov 24 and Kyprianou 26b for Lévy processes. In a srucural seing wih underlying process Y and a pre-specified hreshold level K, he firs-passage ime FPT is given by he ime poin he process Y firs ouches or crosses he hreshold. In credi risk he underlying process is he asse-value process. The firs-passage ime will be a sopping ime wih respec o a filraion ha holds he necessary informaion only hen we can observe wheher a firs-passage even happened or no. In general he hreshold boundary can be a sochasic process iself, see Figure 1.1, bu in ha case he asse-value process Y can be adjused so ha he new hreshold K is a consan. Noe ha hen he adjused model has a differen inerpreaion. We inroduce a general coninuous ime-change model, a process ha lives in a ransformed ime, under anoher clock, see Definiion 1.1. Under his ime-change model we derive analyical formulas for firs-passage-ime disribuions. We sar wih he firs-passage model in one dimension, hen exend he model o wo and higher dimensions. Join firs-passage-ime disribuions are derived under he following asse dependencies: independen Brownian processes bu idenical ime change, correlaed Brownian processes and idenical ime change, 1

24 2 CHAPTER 1. FIRST-PASSAGE TIMES Figure 1.1: Srucural FPT approach: defaul happens a he firs-passage ime τ, where he firm value process firs crosses he defaul hreshold K. correlaed Brownian processes under separae ime changes. A he end of his chaper we give some ime-change examples. Table 1.1 liss Laplace ransforms of disribuions of ime changes close a hand. These lead o numerical FPT disribuions. For one hing i is no clear wheher here are indeed ime-change processes wih an analyical disribuiion such ha an analyical FPT disribuion can be obained. We give examples. These are lised in Table 1.3. In Chaper 6 we analyze he evoluion of credi speads under he sochasic ime-change model. In order o obain analyical condiional survival probabiliies, analyical firs-passage-ime disribuions and herewih also analyical ime-change disribuions become necessary. 1.1 Firs-passage-ime framework We assume a probabiliy space Ω, F, IP, where Ω represens he saes of he world, F is he σ-algebra conaining all possible evens of ineres and IP : F [, 1] is he probabiliy measure. The probabiliy space is assumed o be equipped wih a filraion IF = F. A filraion is a nondecreasing family of sub-σ-algebras of F, ha is, F s F F for all s. In his hesis any sochasic process X will live on Ω [,, someimes only on Ω [, T], T <, and will be adaped o IF, i.e. for all, X will be F -measurable. The underlying process will hroughou be denoed by Y and generaes he filraion IF Y = F Y, F Y := {, Ω} and F Y := σy s : s, ha is he smalles filraion holding all he informaion abou Y, and is called naural filraion of Y. Brownian moions will be denoed by W and B and i is assumed ha hey sar in zero.

25 1.1. FIRST-PASSAGE-TIME FRAMEWORK 3 Definiion 1.1 Time change A deerminisic ime change is a posiive, non-decreasing funcion and a sochasic ime change a posiive, non-decreasing sochasic process. The non-decreasing propery of he ime ransformaion can be undersood as saying ha informaion ha has been obained once, will never be los. Thus a sochasic ime change can conain neiher a pure Brownian moion nor negaive jumps. A sochasic ime change adds sochasic volailiy o a process. The original clock will someimes be called normal clock and he new clock will be called business clock. The ime change may be inerpreed as experienced ime, ha runs faser when he informaion flow is bigger or speeds up. In oher words, experienced ime is a measure of he amoun of informaion arrival. 1 Definiion 1.2 Absolue coninuiy A sochasic process X is called absoluely coninuous if here exiss a process h such ha X = h + h s ds. If X is a posiive, increasing process we can subsiue he posiive process h by a process g wih g 2 = h. Definiion 1.3 Lévy process A real-valued càdlàg 2 IF-adaped sochasic process L wih independen, saionary incremens ha is sochasic coninuous, i.e., ǫ > : lim s IP L L s > ǫ =, is said o be a Lévy process. Every Lévy process L can be associaed wih an infiniely divisible 3 random variable, hrough L 1. Any infiniely divisible disribuion is specified 1 Geman, Madan & Yor 2 show ha a ime change of a Lévy process represens a measure of aciviy in he economy and herefore is a speed of he economy. 2 Càdlàg is he abbreviaion for coninu à droie e limies à gauche righ coninuous pahs wih lef limis. 3 The disribuion of a random variable X is said o be infiniely divisible if for all n IN here exis iid random variables X n 1,..., X n n such ha X L = X n X n n.

26 4 CHAPTER 1. FIRST-PASSAGE TIMES by is characerisic riple b, A, ν wih b IR, A IR + and a Borel measure ν saisfying ν{} =, IR 1 x2 ν dx <, where he so-called characerisic exponen ψ 1 u = iub u2 A 2 + yields he characerisic funcion IR e iux 1 iuxii { x <1} ν dx IE [ e iul 1 ] = e ψ 1u. Thus also L is idenified hrough b, A, ν called Lévy riple and he socalled Lévy exponen ψ u ψ 1 u describes he characerisic funcion IE [ e iul] = e ψu. ν is said o be he Lévy measure, and is densiy, when his exiss, he Lévy densiy. A linear drif is he simples Lévy process. Brownian moion wih drif is he only non-rivial Lévy process wih coninuous pahs. A Lévy process may have small < 1 infinie-variaion jumps and large 1 finie-variaion 4 jumps. A Lévy process can be spli up in he jus-menioned descripive componens: L = b + AW + xj L ν ds, dx = b + x 1 xν dx } {{ } b x 1 IR + AW } {{ } coninuous maringale xj L ds, dx + } {{ } Compound poisson process x <1 xj L ds, dx x <1 xν dx } {{ } pure jump maringale where J L is he jump measure or Poisson measure, a random measure couning he jumps of magniude 1 resp. < 1. The Lévy measure ν is is compensaor, in ha he inegral over he small jumps under he compensaed jump measure J L ν is a maringale. This decomposiion is called Lévy-Iô decomposiion. 4 A process X is said o be of finie variaion if almos all pahs are of finie variaion on each compac inerval, i.e. for any decreasing pariion Π m of his inerval ha is ending o zero for m, w.r.. he maximum-norm i is rue ha lim m Π m X k X k 1 <. Oherwise he process is said o be of infinie variaion.

27 1.1. FIRST-PASSAGE-TIME FRAMEWORK 5 As lieraure on Lévy processes we sugges Papapanoleon 25, Sao 1999, Kyprianou 26b, Con & Tankov 24 and Applebaum 25. Definiion 1.4 Subordinaor A subordinaor is an increasing Lévy process, ha is i has a nonnegaive drif, no diffusion and only posiive jumps ha are of finie variaion. Subordinaors wihou drif are fully characerized hrough heir jump measure ν. Since subordinaors are posiive non-decreasing processes hey can be used as ime-change processes. Examples are he gamma process or he inverse Gaussian process, whose corresponding Lévy densiies are given in Table 1.1. Definiion 1.5 Sopping ime On a measurable, filered space Ω, F, IF a random variable T is called sopping ime of he filraion IF = F, if {T } F for all. Definiion 1.6 Laplace ransform The Laplace ransform of a posiive random variable T is given by l T u := IE [ e ut] = e u IPT d, u. Inegraion by pars yields he following equivalence for he Laplace ransform: l T u = u e u IPT d. 1.1 Definiion 1.7 Defaul ime under he classical srucural approach Given he underlying process Y, a hreshold level K, and a fixed ime T [, he classical defaul ime is defined by { T if Y T < K τ := oherwise, a discree random variable on IR { } whose disribuion is specified by IPτ = T. Definiion 1.8 Firs passage ime FPT under he srucural approach Given he underlying process Y and a hreshold level K, he firspassage ime or defaul ime is defined by τ := inf{s : Y s < K}. The firs-passage-ime disribuion, also called defaul probabiliy, is denoed by IPτ and for u he condiional defaul probabiliy densiy given he informaion F Y is given by IPτ du F Y.

28 6 CHAPTER 1. FIRST-PASSAGE TIMES Since {τ } = {inf s Y s < K} F Y for all, Definiion 1.5 ells us ha he firs-passage ime τ is a IF Y -sopping ime. Considering a FPT is only ineresing, when he firs-passage even has no already occurred a ime zero; so we will always assume K Y. Definiion 1.9 Mulivariae firs-passage-ime disribuion Given underlying processes Y i, hreshold levels K i and defaul imes τ i = inf{s : Y s < K}, i = 1,...,n, he join firs-passage-ime disribuion or join defaul probabiliy JDP is given by Furhermore IPτ 1,...,τ n. IPτ 1 >,...,τ n > is called join survival probabiliy JSP. The wo-dimensional JDP, JSP and marginal probabiliies are in he following relaion: IPτ 1, τ 2 = IPτ 1 >, τ 2 > + IPτ 1 + IPτ The following definiions characerize he dependencies in a mulivariae model. Definiion 1.1 Asse correlaion The correlaion beween wo asse value processes Y 1 and Y 2, is called asse correlaion. ρ A := Corr Y 1, Y 2 CovY 1 =, Y 2 VarY 1 VarY 2, Asse correlaion does no display he whole dependence beween wo sochasic processes, as i does no oally specify he join disribuion. We will say ha he join disribuion is described by he asse dependence. Concerning our processes, asse dependence will be due o correlaed driving Wiener processes on he one hand and a join or dependen ime change on he oher. Therefore he nex definiion is o disinguish he cause of dependence. Assume he Brownian moions W 1 and W 2 are correlaed wih consan correlaion parameer ρ. Then here exiss a Brownian moion W, independen of W 1, such ha W 2 can be wrien in erms of W 1 and W : W 2 = ρw ρ 2 W.

29 1.1. FIRST-PASSAGE-TIME FRAMEWORK 7 Definiion 1.11 Brownian correlaion A consan correlaion beween Brownian moions will be called Brownian correlaion. If here is zero correlaion we say here is Brownian independence. Definiion 1.12 Even correlaion / defaul correlaion The correlaion beween he defaul evens {τ 1 } and {τ 2 }, ρ E := Corr II {τ1 }, II {τ2 } = Cov II }, II = {τ1 {τ2 } Var II {τ1 } Var II{τ2 } IPτ 1, τ 2 IPτ 1 IPτ 2 IPτ1 1 IPτ 1 IPτ 2 1 IPτ 2. is called even correlaion or defaul correlaion. We define m := minipτ 1, IPτ 2 and M := maxipτ 1, IPτ 2. Then IPτ 1, τ 2 m, and some simple algebraic ransformaions lead o he following naural bounds for he even correlaion: IPτ 1 IPτ 2 IPτ1 1 IPτ 1 IPτ 2 1 IPτ 2 ρ E m1 M M 1 m. 1.3 The nex lemma connecs he sochasic inegral w.r.. Brownian moion o a ime-changed Brownian moion. We will need his relaionship in order o deermine dynamics under our sochasic ime-change model in Secion 6.3. The remark following he lemma considers he special case where he ime change is deerminisic. This will be applied o derive he dynamics under he Overbeck & Schmid model in Secion 5.4. Firs of all we inroduces he quadraic variaion. Definiion 1.13 Quadraic variaion Le X be a coninuous maringale. The quadraic variaion process X is defined by X := X 2 2 X s dx s. If X is in addiion square-inegrable, i.e. IE[X 2 ] <, he quadraic variaion process X is he unique adaped, increasing process for which X = X 2 = and X2 X is a maringale. Then also, if Π m = {,..., m } are pariions of [, ] ending o zero for m in erms of he maximumnorm Π m = max 1 k m k k 1, we have ha he so-called realized

30 8 CHAPTER 1. FIRST-PASSAGE TIMES quadraic variaion or sample quadraic variaion m 2 k=1 Xk X k 1 ends o he quadraic variaion 5, i.e. 2 Π m Xk X k 1 X in probabiliy. Π m Furhermore, for wo sochasic processes X and Y he realized covariaion is given by m Xk X k 1 Yk Y k 1. k=1 Lemma Time-changed Brownian moion Le g be a sochasic process on Ω, F, IP ha is adaped o he filraion IF = F, has càglàd 7 pahs and saisfies IE[ g2 s ds] < for every. Furhermore W, F be a Brownian moion. Then he sochasic inegral g s dw s is a coninuous local maringale wih quadraic variaion g s dw s = gs 2 ds =: G, mean zero and variance [ ] [ ] Var g 2 s dw s = IE g s dw s Furhermore G 1 = inf { u : = IE [ g s dw s ] = IE[G ]. g s dw s is a IF-sopping ime. Then here exiss a Brownian moion B, F G 1 such ha, a.s., g s dw s = B G, <. Remark 1.15 Deerminisically ime-changed Brownian moion In paricular he las lemma holds for a deerminisic ime change G = g2 s ds. In fac, hen g s dw s and W G are boh Gauss-processes having he same covariance srucure i.e. This will be used in Chaper 5. g s dw s L = WG. 5 Cf. Karazas & Shreve 1991, Secion Cf. Karazas & Shreve 1991, Secion 3.4, B. 7 Càglàd is he abbreviaion for coninu à gauche e limies à droie lef coninuous pahs wih righ limis. u > }

31 1.1. FIRST-PASSAGE-TIME FRAMEWORK 9 Remark 1.16 Quadraic variaion of ime-change processes Geman, Madan & Yor 22 considered he composie process of a Brownian moion W ime changed wih a righ-coninuous process A, Y = W A, and analyzed wheher he ime change can be observed under he informaion of he underlying process IF Y. This is imporan because if he ime change is known o all marke paricipans coninuously in ime, hen i is useable for hedging and maringale models are available for pricing. Oherwise pricing under he filraion IF Y is criical. If he ime change is coninuous and square inegrable, i.e. IE[A 2 ] <, i is given by he quadraic variaion of he ime-changed process cf. Geman, Madan & Yor 2: A = Y. Geman, Madan & Yor showed ha a disconinuous ime change canno be recovered by he observed composie process i.e. by is realized quadraic variaion. For a general disconinuous ime change hey obained IE [ [Y, Y ] A 2] = IE[2[A, A] ] >, so he ime change is no deermined by he quadraic variaion. 8 They considered a variey of disconinous ime changes, analyzed wheher he realized quadraic variaion is a leas a sufficien saisic for he ime change and found ha his is only he case for he Gamma ime change The aim of his chaper and our applicaion We aim for a srucural firs-passage-ime model τ ha leads o an analyical firs-passage-ime disribuion, in one and more dimensions. We will assume a consan hreshold level K and for he underlying process Y we search a model class ha, in he mulivariae model, inherens some dependency srucure. In our applicaion o modeling credi-spread curves, he srucural approach has a realisic inerpreaion of he defaul even and he available analyical FPT disribuions simplify calibraions. Moreover, dependencies in our mulivariae model, i.e. business ime and asse correlaion, can be explained. Las bu no leas our model yields credi-spread dynamics and allows for inpu on he credi-spread volailiy. This is especially necessary and imporan for long-erm and leveraged credi producs. When i comes o calibraing a specific model see Chaper 2 and 5 we someimes assume ha he marke provides us wih a FPT disribuion F T. Usually, for simpliciy, we will assume i is given by he 8 [Y, Y ] denoes he quadraic variaion for a semimaringale, in general. Our Def is well-defined because in case Y is coninuous we have [Y, Y ] = Y.

32 1 CHAPTER 1. FIRST-PASSAGE TIMES exponenial disribuion F = 1 e λ, λ >. 1.4 Then on a discree grid { =,..., m = T } we wan o fi he model exacly a leas a wo poins, F i! = IPτ i, e.g. he liquid credi spread poins 1 = 5 years and 2 = 7 years. 1.2 Survey: defaul-ime and firs-passage-ime models In his chaper we wan o keep he resuls as general as possible and herefore do no focus on he inerpreaion in credi risk and he link o equiy and deb. This we pospone o he Chapers 3, 4, 5 and 6, where we give an inroducion o credi risk, especially o he credi spread, and analyze he Meron model, he Overbeck & Schmid model and our sochasic ime change model in deail Brownian moion wih drif Meron 1974 inroduced he classical hreshold model o finance, where he considered underlying process is given by geomeric Brownian moion and he hreshold level is supposed o be consan. A a fixed poin in ime T, one is ineresed in wheher a ha ime he underlying process crossed he hreshold or no. This model is called he Meron model. I is equivalen o considering Brownian moion wih volailiy and drif parameer, Y = σw + µ, and a consan hreshold barrier K given by he naural logarihm of he original hreshold value. Please noe ha hroughou his hesis W denoes a Brownian moion saring a zero, µ a consan drif, σ a consan volailiy, K a consan hreshold level, Φ he sandard normal cumulaive disribuion funcion and φ is densiy. The defaul-ime disribuion for he classical approach see Definiion 1.7 is specified by K µt IP τ = T = IPσW T + µt < K = Φ σ. 1.5 T Black & Cox 1976 sared o consider he Meron model in a firspassage ime FPT approach. So again geomeric Brownian moion was chosen o be he underlying process, and he hreshold was assumed o be exponenially ime dependen. Equivalenly, we consider he firs-passageime problem for Brownian moion wih drif and a consan hreshold barrier. The FPT disribuion can be derived via he reflecion principle 9 since 9 See Harrison 1985 or Karazas & Shreve 1991, Secion 2.6.

33 1.2. SURVEY: FIRST-PASSAGE-TIME MODELS 11 K W, and yields IPτ = IP min [σw s + µs] < K s K µ = Φ σ + e 2 µk K + µ σ 2 Φ σ Join survival probabiliy - Brownian moion Zhou 21 deermined he JSP of he wo-dimensional Brownian moion σ 1 W 1, σ 2 W 2, wih correlaion parameer ρ, no crossing he upper bound K 1, K 2 by considering he equivalen problem of solving a parial differenial equaion PDE. Le FK 1, K 2, = IP max σ 1Ws 1 < K 1, max σ 2Ws 2 < K 2 s s 1.7 be he join survival probabiliy and fx 1, x 2, he corresponding ransiion probabiliy densiy saisfying FK 1, K 2, = K1 K2 fx 1, x 2, dx 1 dx 2. Tha is, f is he probabiliy densiy ha a paricle being a x 1, x 2 a ime zero and no having ye crossed he boundary K 1, K 2 will no reach ha boundary in he ime inerval [, ]. The ransiion probabiliy saisfies he Kolmogorov forward equaion 1, σ f 2 f x 2 + ρσ 1 σ 2 1 subjec o specific boundary condiions: + σ2 2 2 f x 1 x 2 2 x 2 2 x 1 < K 1, x 2 < K 2, = f fx 1, K 2, = fk 1, x 2, =, >, K1 K2 fx 1, x 2, dx 1 dx 2 1, >, f, x 2, = fx 1,, = fx 1, x 2, = δx 1 δx 2, where δ is he Dirac dela funcion. The firs hree condiions are due o he fac ha f is a densiy and he las one displays he saring value of he process. Zhou derived he soluion by various ransformaions of he PDE, 1 See for example Cox & Miller 1972, Karazas & Shreve 1991.

34 12 CHAPTER 1. FIRST-PASSAGE TIMES in paricular by ransforming he x 1, x 2 coordinaes ino polar coordinaes r, θ, yielding he PDE 1 2 f r 2 θ f r r + 2 f r 2 = 2 f. As a soluion he obained he ransiion probabiliy densiy: 2 fr, θ, = r 2 +r 2 σ 1 σ 2 1 ρ 2 α e 2 n=1 sin nπθ α sin nπθ α I nπ α rr, 1.8 where he abbreviaions r, θ and α are given in Theorem 1.17 and I ν denoes he modified Bessel funcion of he firs kind wih order ν 11. The JSP F of equaion 1.7 is yielded by inegraing f. We will always consider he conrary problem of no crossing a lower barrier. Thus before applying Zhou s soluions for he JSP F and ransiion probabiliy densiy f we have o reflec our firs-passage processes a he x- axes. Therefore he nex heorem saes he JSP resul in our erms where surviving means no crossing a lower barrier. We will apply Zhou s resuls various imes. Theorem 1.17 Zhou; JSP of a wo-dimensional Brownian moion no crossing a lower barrier Le Y 1 = σ 1 W 1 and Y 2 = σ 2 W 2 be correlaed Wiener processes wih correlaion parameer ρ and K 1 <, K 2 < he hreshold levels. Then Y 1 and Y 2 have he following join survival probabiliy: IP τ 1 >, τ 2 > IP min σ 1Ws 1 > K 1, min σ 2Ws 2 > K 2 s s = 2r e r2 [ 1 4 2π n sin nπθ α where θ = n=1,3,5,... an 1 π + an 1 r = Y 2 K 2 σ 2 sinθ, an 1 ρ α = π + an 1 σ 1 K 2 1 ρ 2 σ 2 K 1 ρσ 1 K 2 σ 1 K 2 1 ρ 2 σ 2 K 1 ρσ 1 K 2 1 ρ 2 1 ρ 2 ρ I1 2 nπ α +1 r 2 + I1 4 2 nπ α 1 if σ 1K 2 1 ρ 2 σ 2 K 1 ρσ 1 K 2 > oherwise, if ρ < oherwise. ] r The Bessel funcion is given in Def. A.3, Borodin & Salminen 22 and sudied in more deail in Revuz & Yor 25.,

35 1.2. SURVEY: FIRST-PASSAGE-TIME MODELS Join survival probabiliy - Brownian moion wih drif We proceed much as in he previous subsecion for Brownian moion wihou drif. Denoe he JSP of he wo-dimensional Brownian moion wih drif and correlaion parameer ρ by F µ K 1, K 2, = IP max σ1 Ws 1 + µ 1 s < K 1, max σ2 Ws 2 + µ 2 s < K 2. s s 1.9 The corresponding ransiion probabiliy densiy for a paricle being a x 1, x 2 in ime zero and no reaching he boundary K 1, K 2 in he ime inerval [, ] will be denoed by f µ x 1, x 2,. The soluion for he ransiion probabiliy densiy in polar coordinaes can be derived by changing he measure so ha he considered Brownian moion wih drif becomes a Brownian moion wihou drif under he new measure, and hen using he resul for f in equaion 1.8. Compare e.g. Fischer 23. The soluion is he following: f µ r, θ, = 2 σ 1 σ 2 1 ρ 2 α e A3 nπθ sin α n=1 2 +A 1r cos θ+a 2 r sin θ r2 +r 2 2 sin nπθ α I nπ α rr. 1.1 The JSP F µ is yielded when inegraing he ransiion probabiliy densiy in equaion 1.1 and is saed in he 21 paper by Zhou - an updae of Zhou 1997b. The derivaion can be found in Fischer 23 resp. Rebholz Again we give he JSP resul in he way ha is suiable for us, i.e. for he conrary problem of no crossing a lower barrier. Thus before applying he JSP F µ and ransiion probabiliy densiy f µ we have o reflec our firs-passage processes a he x-axes. The resul, he JSP for no crossing a lower barrier, is given by he nex heorem. Theorem 1.18 JSP of a wo-dimensional Brownian moion wih drif no crossing a lower barrier Le Y 1 = σ 1 W 1 and Y 2 = σ 2 W 2 be correlaed Wiener processes wih correlaion parameer ρ and K 1 <, K 2 < he hreshold levels. Then Y 1 and