Valuation of nth to Default Swaps

Transcription

1 Valuaton of nth to Default Swaps Artur Stener Kellogg College Unversty of Oxford A thess submtted n partal fulfllment of the requrements for the MSc n Mathematcal Fnance September 3, 24

2 Authentcty Statement I hereby declare that ths dssertaton s my own work and confrm ts authentcty. Name: Artur Stener Address: Zum Wetkamp Schüttorf Germany Sgned: Date: September 3, 24

3 Abstract Ths thess deals wth the valuaton of nth to default swaps n the Gaussan copula model. After a bref descrpton of the model and the numercal problems that arse n the nave Monte Carlo prcng algorthm, two more effcent valuaton methods are presented. The frst method s mportance samplng Monte Carlo, and the second method s based on a one factor copula whch reduces the dmensonalty of the prcng problem such that t can be solved wth numercal ntegraton. We extend the one factor Gaussan copula model developed by Laurent and Gregory by dervng sem-explct expressons for senstvtes wth respect to hazard rates and show how these senstvtes can be calculated n the mportance samplng Monte Carlo framework wth fnte dfferencng and (more effcently n terms of cpu tme) the lkelhood rato method. An alternatve to the Gaussan copula s the Student-t copula whch ncreases the dependency between the underlyng varables at extreme values. The use of the t-copula s dscussed for both approaches, and an addtonal modfcaton of the mportance samplng algorthm s suggested to mprove ts convergence behavour n the t-copula model. After a detaled dscusson and comparson of the models, the valuaton methods are appled n a hedgng smulaton.

4 Acknowledgements I would lke to express my grattude to d-fne GmbH for havng gven me the opportunty to partcpate n ths course and for fnancal support. Many thanks to my supervsor Dr. Dhermnder Kanth for hs valuable contrbutons, contnuous support, and patence.

5 Contents 1 Introducton Credt Dervatves Modellng Credt Rsk Outlne Modellng the Default Tme Copula Functons The Gaussan Copula The Student-t Copula Smulaton Technques Kendall s tau Posson Processes Prcng nth to Default Swaps n the Gaussan Copula Model The Gaussan Copula Model Nth to Default Credt Swaps Independent Defaults Constant Interest Rates and Constant Hazard Rates Zero Interest Rates and Homogeneous Hazard Rates Zero Interest Rates Importance Samplng Calculaton of Senstvtes to Changes n the Hazard Rates Fnte Dfferencng The Lkelhood Rato Method The One Factor Gaussan Copula Model Descrpton of the Model Prcng Formulas for nth to Default Swaps Default Leg Premum Leg Calculaton of Senstvtes to Changes n the Hazard Rates Extensons of the Models Usng the Student-t Copula Importance Samplng The One Factor Student-t Copula Model The Multfactor Gaussan Copula Model

6 6 Results Defnton of the Test Basket Comparson of the Gaussan Model and the Student-t Model Importance Samplng n the Gaussan Copula Model Calculaton of Senstvtes n the Gaussan Copula Model Importance Samplng n the Student-t Copula Model The One Factor Models Hedgng Smulatons n the Gaussan Copula Model Summary and Outlook 56 A Implementaton 58 A.1 Assumptons A.2 Approxmatons

7 Chapter 1 Introducton 1.1 Credt Dervatves A credt dervatve s a dervatve contract that has a payoff whch s condtoned on the occurrence of a credt event. The credt event s defned wth respect to payment oblgatons (e.g. bonds or loans) of one or several oblgors and s usually closely lnked to the default of an oblgor,.e. to the event that the oblgor s no longer able to honour hs payment oblgatons. If the credt event occurs, a default payment, whch often depends on the recovery rate of a specfc bond or loan, has to be made by one of the counterpartes. Besdes one or more default payments a credt dervatve can have further payments that are not default contngent. Credt dervatves emerged n the early 199s and have been the fastest growng dervatves products n the past few years. The prototype of a credt dervatve and stll the domnant product n the credt dervatves market s the credt default swap (CDS). A credt default swap s a blateral contract where one counterparty (the protecton seller) agrees to make a specfc compensaton payment to a second party (the protecton buyer) f a defned credt event occurs wth respect to a reference asset or a class of reference assets of one oblgor. In exchange for the default protecton, the protecton buyer wll make perodc payments untl ether the contract matures or the credt event occurs. A basket default swap s smlar to a CDS, except that the payments are defned wth respect to a basket of oblgors rather than to a sngle reference asset. The subject of ths thess s the ncreasngly popular nth to default swap where the compensaton payment s trggered by the nth default (or, more precsely, by the nth credt event) n the basket. Frst to default swaps, for nstance, are now among the most wdely used credt dervatves, and standardsed frst to default swaps based on the Traxx famly of credt default swap ndces have been launched recently. Although bankruptcy codes and contract laws usually force an oblgor to honour all hs payment oblgatons as long as he s able to, we cannot speak of the credt rsk of an oblgor because the payoff of a credt dervatve depends on the applcable credt events and ther exact defnton. The defnton of credt events has now been standardsed by the Internatonal Swaps and Dervatves Assocaton (ISDA) and ncludes Bankruptcy Falure to pay: f after expraton of a grace perod, the oblgor fals to make payment wth respect to prncpal or nterest on one or more of ts oblgatons. Falure to pay 1

8 sets a mnmum dollar threshold. Repudaton/moratorum (ths provson only apples to soveregn reference enttes): f the reference entty ndcates that one or more of ts oblgatons s no longer vald, or f the entty stops payment on such oblgatons. Oblgaton acceleraton: when an oblgaton has become due and payable earler than t would otherwse have been due because of a oblgor s default or smlar condton. Lke falure to pay, oblgaton acceleraton s subject to a mnmum dollar threshold payment amount. Restructurng: the ISDA defnton offers four choces, no restructurng, full restructurng, modfed restructurng and modfed modfed restructurng; the latter two lmt the maturty of delverable oblgatons to 3 and 6 months, respectvely. Reflectng the fact that the credt dervatve market s fast-expandng and open to nnovaton, the standardsaton of the documentaton has been a dynamc process wth several sgnfcant changes snce the publcaton of the frst form of confrmaton n The dsagreements arsng from the Russan default, for example, eventually led to the publcaton of the 1999 ISDA Credt Dervatves Defntons. One of the dsputes was caused by a short delay n payments due on the Cty of Moscows debt. Several sellers of protecton had not ncluded specfc provsons for grace perods to allow for techncal delays and were forced by Englsh courts to make the default payments. As a response to ths decson, the 1999 ISDA Credt Dervatves Defntons ntroduced the concept of a mnmum grace perod of three busness days. Another mportant change was the ncluson of the four choces concernng restructurng n the 23 Credt Dervatves Defntons whch address the dfferences between bankruptcy laws n the US and Europe. In general, a restructurng trggered by Chapter 11 s more accommodatng towards a troubled company than a restructurng accordng to European laws and does not always lead to losses for bond holders. Amercan market partcpants are therefore less nclned to regard restructurng as a credt event. In the remander of ths thess, we wll neglect the subtletes of the credt event defnton and use the terms credt event and default as synonyms. The underlyng assumpton s that the prcng models for nth to default swaps are calbrated to market data referrng to the same defnton of a credt event as the nth to default swap. 1.2 Modellng Credt Rsk There are two man approaches to model credt rsk. In the structural approach, the lablty structure of the frm and the evoluton of the frm s value (usually nterpreted as the total value of ts assets) determne the occurrence of defaults and the recovery rates n the event of default. Explct assumptons are made about the dynamcs of the frm s value. A frm defaults f ts value drops below a lqudaton or reorganzaton boundary dependng on the frms debts. In the frst structural models, only arrval rsk was modelled,.e. the frm s debt s gven by zero coupon bonds of a sngle maturty, and the frm can only default at the maturty of the bonds. As the frm s value s assumed to follow a geometrc Brownan moton, the probablty of default can be determned n analogy to the prcng methodology of European optons. An obvous flaw of these models s that the frm s value can decrease to almost nothng wthout trggerng default. Black and Cox corrected ths by ntroducng 2

9 safety covenants that gve bond nvestors the rght to reorganse the frm f ts value falls below a gven barrer. As a consequence, default can happen at any tme durng the bonds term. If the frm s value s drven by a geometrc Brownan moton, the default event s predctable,.e. t s not a total surprse. Ths mples that credt spreads tend to zero as tme to maturty goes to zero, whch s at odds wth market realty. Newer structural models address ths ssue by assumng that the frm s value s drven by a jump-dffuson process or by assumng that the nvestors cannot observe the value of the default barrer. In reduced form models, defaults occur wthout warnng at an exogenous default rate. The dynamcs of the default rate are specfed under the rsk-neutral (.e. market-mpled) probablty. Dependng on the complexty of the model, the default rate s assumed to be ether determnstc or gven by a stochastc process. The free parameters of the model have to be calbrated to market prces. Typcally, default probabltes, recovery rates and nterest rates are assumed to be mutually ndependent. The focus of the reduced form approach s to provde a consstent prcng methodology n a tractable model whch can be easly calbrated to market data. Unlke the structural approach, t does not lnk structural data of a frm (lke, for example, the balance sheet) wth the prces of credt-senstve nstruments. 1.3 Outlne The man purpose of ths thess s to explore the valuaton of nth to default swaps n the reduced form model developed by L [9]. In Chapter 2, the man modellng tools, copulas and Posson processes, are ntroduced. Chapter 3 presents the L model (or Gaussan copula model) and descrbes how nth to default swaps can be prced n ths model. After an llustraton of the numercal problems that arse n a nave Monte Carlo mplementaton of the prcng algorthm, the mportance samplng algorthm developed by Josh and Kanth [7] s dscussed n detal. The chapter concludes wth a presentaton of algorthms to calculate senstvtes wth respect to the hazard rates. Chapter 4 presents the one factor Gaussan copula model, a varant of the Gaussan copula model wth a restrcted correlaton structure, whch lends tself to effcent numercal evaluaton. We also show how senstvtes wth respect to the hazard rates can be calculated n the one factor framework descrbed by Laurent and Gregory [8]. Extensons of the Gaussan copula models are dscussed n Chapter 5, ncludng the replacement of the Gaussan copula by the t-copula, whch contans the Gaussan copula model as a lmt. In Chapter 6, the aforementoned models and prcng algorthms are analysed n detal. The chapter concludes wth the dscusson of a dynamc hedgng strategy. Appendx A outlnes the C++ mplementaton of the prcng algorthms. 3

10 Chapter 2 Modellng the Default Tme The most mportant problem n the prcng of basket credt dervatves s the modellng of the default tmes. In ths secton we wll ntroduce the concept of a copula functon. Copula functons allow us to separate the problem of modellng the default tmes nto two parts: 1. The specfcaton of the margnal dstrbuton functons,.e. of the dstrbuton of the default tme of each oblgor dsregardng the default tmes of the other oblgors. 2. The choce of a sutable copula whch descrbes the dependence structure between the default tmes. Ths approach sgnfcantly smplfes the calbraton of the resultng model because the nputs for the calbraton are usually separated n an analogous manner. Frst, there are the prces of bonds and sngle name default swaps from whch we can bootstrap the ndvdual term structures of default probabltes, and second there s jont defaults and correlaton nformaton. 2.1 Copula Functons The concept of copulas s not really a new one n the mathematcal world. It goes essentally back to problems posed by Hoeffdng and Fréchet more than 6 years ago concernng the maxmal and mnmal possble correlaton for bvarate dstrbutons when the margnals are fxed. The word copula was seemngly frst coned by Sklar n hs famous 1959 paper[14]. Defnton 1 A functon C : [, 1] n [, 1] s a copula f: (a) There are random varables U 1,..., U n takng values n [, 1] such that C s ther dstrbuton functon; (b) C has unform margnal dstrbutons,.e. for all {1,..., n}, u [, 1] C(1,..., 1, u, 1,..., 1) = u. Sklar s theorem shows that the decomposton of a multvarate dstrbuton n the margnal dstrbutons and a sutable copula can be acheved for any set of random varables: 4

11 Theorem 1 (Sklar) Let X 1,..., X n be random varables wth margnal dstrbuton functons F 1,..., F n and jont dstrbuton functon F. Then there exsts an n-dmensonal copula C such that for all (x 1,..., x n ) R n : F (x 1,..., x n ) = C(F 1 (x 1 ),..., F n (x n )), (2.1).e. C s the dstrbuton functon of (F 1 (x 1 ),..., F n (x n )). If F 1,..., F n are contnuous, then C s unque. Otherwse C s unquely determned on RanF 1 RanF n, where RanF denotes the range of F for = {1,..., n}. Proposton 1 The densty c assocated to a copula C s gven by c(u 1,..., u n ) = n C(u 1,..., u n ) u 1 u n. Let F be a multvarate dstrbuton such that (2.1) holds. Then the densty f of F s f(x 1,..., x n ) = n C(u 1,..., u n ) u 1 u n n f n (x n ), (2.2) where f s the densty of the margnal dstrbuton functon F (1 n). The followng proposton summarses some mportant propertes of copula functons: Proposton 2 (a) (Invarance to ncreasng transformatons.) Let (X 1,..., X n ) be a vector of random varables wth copula C. Let g : R R be a famly of n strctly ncreasng functons. Then C s agan the copula of (g 1 (X 1 ),..., g n (X n )). (b) An n-dmensonal copula C s non-decreasng n each argument,.e. =1 C(u 1,..., u n ) C(u 1,..., u 1, û, u +1,..., u n ) for all (u 1,..., u n ) [, 1] n, 1 n, u û 1. (c) (Fréchet-Hoeffdng bounds). Let C be an n-dmensonal copula. Then for every (u 1,..., u n ) [, 1] n : W n (u 1,..., u n ) C(u 1,..., u n ) M n (u 1,..., u n ), wth and W n (u 1,..., u n ) = max(u u n n + 1, ) M n (u 1,..., u n ) = mn(u 1,..., u n ). 5

12 2.1.1 The Gaussan Copula The most frequently used copula s the Gaussan copula. Important reasons for ts popularty are that t s very easy to draw random samples from t and that the dependence structure of the multvarate normal dstrbuton s well understood. The Gaussan copula s used n many models even f they do not explctly employ the concept of a copula functon. L [9], for example, has shown that the Gaussan copula s mplctly used n CredtMetrcs [5]. Defnton 2 Let X 1,..., X n be normally dstrbuted random varables wth means µ 1,..., µ n, standard devatons σ 1,..., σ n and correlaton matrx Σ. Then the dstrbuton functon C Σ (u 1,..., u n ) of the random varables ( ) X µ U := Φ, {1,..., n} σ (where Φ( ) denotes the cumulatve unvarate standard normal dstrbuton functon) s a copula and t s called the Gaussan copula wth the correlaton matrx Σ. The Gaussan copula can be wrtten as: C Σ (u 1,..., u n ) = where k Σ = Φ 1 (u 1 ) Φ 1 (u n) 1 (2π) n det(σ). k Σ exp ( 12 ( v µ)t 1 ( v µ) ) dv 1... dv n (2.3) By dfferentatng equaton (2.3) wth respect to u 1,..., u n we obtan the densty of the Gaussan copula: ( 1 ) v 1 1 (2.4) 1 c Σ (u 1,..., u n ) = exp 1 det(σ) (v 1,..., v n ) wth v = Φ 1 (u ), {1,..., n} The Student-t Copula A varant of the Gaussan copula s the t-copula, whch s of the same famly of copula functons but has heaver tals. Defnton 3 Let X 1,..., X n be normally dstrbuted random varables wth mean zero, standard devaton one and correlaton matrx Σ. Let Y be a χ 2 -dstrbuted random varable wth ν degrees of freedom whch s ndependent of (X 1,..., X n ). Then the dstrbuton functon C ν,σ (u 1,..., u n ) of the random varables U := t ν ( ν Y X ), {1,..., n} (2.5). v n 6

13 s a copula and t s called the t-copula (or Student-t copula) wth ν degrees of freedom and the correlaton matrx Σ. Here t ν s the unvarate Student-t dstrbuton functon wth ν degrees of freedom gven by t ν (x) = x Γ ( ) ν+1 ( 2 Γ ( ) ν 1 + s2 2 νπ ν ) ν+1 2 Due to the smultaneous multplcaton of all X wth ν Y, extreme co-movements of the U are more lkely n the t-copula than n the Gaussan copula, especally f the parwse correlatons of the X are low. In the models descrbed later n ths thess, oblgor defaults before a gven tme horzon T f U s lower than a specfc threshold (where the threshold depends on the oblgor and the tme horzon). If ν Y s large, then all oblgors wth X < are prone to default before the tme horzon. Hence multple defaults are more lkely n models based on the t-copula than n Gaussan copula models. The prcng mpact of the choce of the copula and the copula parameters s analysed n Secton 6.2. In the twodmensonal case, the dfference between the two copulas can be descrbed wth the concept of tal dependence: Defnton 4 If a bvarate copula C(u, v) s such that 1 + C(u, u) 2u lm = λ U > u 1 1 u then C has upper tal dependence wth parameter λ U. If C(u, u) lm = λ L > u u then C has lower tal dependence wth parameter λ L. Lower tal dependence means that as u the probablty mass of the lower left square [, u] [, u] tends to zero lke λ L u and not lke u 2, the area of the square. The t-copula has upper and lower tal dependence whereas the Gaussan copula s tal ndependent, see [3]. The t-copula can be wrtten as wth C ν,σ (u 1,..., u n ) = and ts densty s t 1 ν (u 1 ) t 1 ν (u n) Γ ( ) ν+n 2 k ν,σ = Γ ( ) ν 2 (πν) n det(σ), ds. k ν,σ ( ν ( v µ)t 1 ( v µ) ) c ν,σ (u 1,..., u n ) = det(σ) 1 Γ ( ) [ ( ν+n 2 2 Γ ν ) ] N Γ ( ) 2 ν 2 Γ ( ) ( ) ν vt v ν+1 ( ) n 2 =1 1 + v2 ν wth v = t 1 ν (u ) for {1,..., n}. 7 ν+n 2 ν+1 2 dv 1... dv n

14 2.1.3 Smulaton Technques One reason for the popularty of the Gaussan copula s that t s very easy to draw random samples from t. The algorthm makes use of the fact that for a gven postve semdefnte 1 correlaton matrx Σ R n n there exsts a matrx A R n n such that Σ = AA T. Convenent procedures to determne a pseudo-square root A are the Cholesky method or spectral decomposton, see [4] and [6]. The pseudo-square root can be used to transform uncorrelated standard normally dstrbuted random varables X 1,..., X n nto correlated standard normally dstrbuted random varables Y 1,..., Y n wth correlaton matrx Σ by settng Y = AX. The followng algorthm generates random samples from the Gaussan copula wth correlaton matrx Σ: 1. Compute a pseudo-square root decomposton Σ = AA T. 2. Draw n ndependent unformly dstrbuted numbers u nd 1,..., und n ], 1[ from a random number generator. 3. Set v := Φ 1 (u nd ) ( {1,..., N}). The vector v s a sample from n uncorrelated and standard normally dstrbuted varables. 4. Set w = A v. 5. Let u := Φ(w ), {1,..., n} where Φ( ) s the unvarate cumulatve normal dstrbuton functon. u 1,..., u n are C Σ dstrbuted. Then the The algorthm for the t-copula C ν,σ s smlar: 1. Compute a pseudo-square root decomposton Σ = AA T. 2. Draw a sample v = (v 1,..., v n ) T from n ndependent and standard normally dstrbuted random varables. 3. Set w = A v. 4. Draw a s sample from a χ 2 dstrbuton wth ν degrees of freedom. Ths can be done by generatng ν ndependent and standard normally dstrbuted random numbers v 1,..., v ν and by settng s := v v2 ν. 5. Let u := t ν ( ν s w ), {1,..., n}. Then the u 1,..., u n are C ν,σ dstrbuted. 1 Methods to salvage an nvald correlaton matrx are descrbed n [12] and [6]. 8

15 2.1.4 Kendall s tau Kendall s tau s a measure of dependence that s gven n terms of concordance and dscordance. Two pars (x, y) and (x, y ) of observatons from a vector (X, Y ) of contnuous random varables are sad to be concordant f (x x )(y y ) > and dscordant f (x x )(y y ) <. Kendall s tau s defned as the dfference between the probabltes of concordance and dscordance,.e. Defnton 5 Kendall s tau for the random varables X, Y s defned as τ(x, Y ) = P{(X X )(Y Y ) > } P{(X X )(Y Y ) < }, where (X, Y ) s an ndependent copy of (X, Y ). By defnton, Kendall s tau s nvarant under strctly ncreasng transformatons of X and Y. Ths property does not hold for the classcal lnear correlaton coeffcent - a counterexample can be found n [2]. Furthermore, for contnuous random varables X and Y, Kendall s tau only depends on the copula of X and Y and not on the margnal dstrbutons, see [2]. Kendall s tau s closely related to the correlaton matrx of a Gaussan copula or a t- copula: let Y 1,..., Y n be contnuous random varables whose copula s a Gaussan copula or a t-copula wth correlaton matrx Σ = (a j ) 1,j n. Then τ(y, Y j ) = 2 π arcsn(a j). The nonparametrc estmator of the correlaton coeffcents obtaned from an estmator of Kendall s tau and the above equaton s an effcent (low varance) estmator of Σ. Note that the lnear correlaton ρ(y, Y j ) -whch can also be estmated effcently- s n general not the correlaton parameter a j of the Gaussan or t-copula. Even f Y 1 and Y 2 are unformly dstrbuted and connected by a bvarate Gaussan copula, the correlaton of Y 1 and Y 2 s not exactly the correlaton coeffcent of the copula. More examples can be found n [6]. In the case of the t-copula, the degrees-of-freedom parameter s typcally determned wth a standard maxmum lkelhood estmaton after the correlaton matrx has been obtaned wth the estmator descrbed above. The multvarate normal dstrbuton and the multvarate Student-t dstrbuton belong to the class of so-called ellptcal dstrbutons. For ellptcal dstrbutons, the classcal lnear correlaton coeffcent s a natural measure of dependence. Kendall s tau s a good alternatve for non-ellptcal dstrbutons where the lnear correlaton coeffcent s often msleadng. 2.2 Posson Processes In ths secton, we brefly descrbe a model for the margnal dstrbutons of the default tmes. As already mentoned n the ntroducton of ths chapter, the margnal dstrbutons together wth the choce of a sutable copula are suffcent to specfy the full jont dstrbuton of the default tmes. Suppose that the cumulatve margnal dstrbuton functon F of the default tme τ s contnuously dfferentable wth densty f,.e. F (t) = t f(u)du. Gven survval untl tme t, the condtonal probablty of default n the nterval [t, t + t] s P(t τ < t + t τ t) = P(τ t) P(τ t + t) P(τ t) = F (t + t) F (t). 1 F (t) 9

16 Normalsaton to one unt of tme gves P(t τ < t + t τ t) t = 1 S(t) S(t + t) S(t), t where S(t) = 1 F (t) denotes the survval functon. normalsed to one unt of tme s P(t τ < t + t τ t) h(t) := lm = 1 t t S(t) Therefore, the default ntensty ds(t). dt The so-called ntensty-based models take the ntensty and ts dynamcs as a startng pont and derve the dstrbuton of the default tme from the ntensty. In our example, ( t ) S(t) = exp h(u)du and ( t ) F (t) = 1 exp h(u)du. The approprate mathematcal framework for ths approach are pont processes. Default tmes wth determnstc ntenstes can be modelled as the tme of the frst jump of an nhomogeneous Posson process: Defnton 6 N s called an nhomogeneous Posson process wth ntensty functon h(t), f (a) N s a non-decreasng, nteger valued process. (b) The ntal value N() =. (c) The ncrements N(t) N(s) are ndependent. (d) The ntensty h(t) s a non-negatve functon of tme only. ( ) (e) P(N(t) N(s) = n) = 1 t n n! s h(u)du e t s h(u)du for all s < t. The dstrbuton functon F (t) of the default tme s then that s F (t) := P(τ t) = P(N(t) N() 1) = 1 P(N(t) N() = ), ( F (t) = 1 exp t ) h(u)du. (2.6) The ntensty h( ) can be computed wth a bootstrappng or fttng procedure from market prces of bonds or sngle name credt default swaps. More realstc (and more complex) models than those dscussed n ths thess use pont processes wth stochastc ntenstes, such as Cox processes. Stochastc dynamcs of the ntenstes are necessary to prce credt dervatves whose payoff s drectly affected by volatlty (e.g. credt spread optons). 1

17 Chapter 3 Prcng nth to Default Swaps n the Gaussan Copula Model 3.1 The Gaussan Copula Model A popular model for prcng products whose payoffs depend on the default tmes τ 1,..., τ N of a basket of N oblgors s the Gaussan copula or L model, [9]. Its basc assumptons are 1. The default tmes τ can be descrbed wth nhomogeneous Posson processes,.e. τ s the tme of the frst jump of a Posson process. 2. The default tmes of dfferent oblgors are connected to each other by a Gaussan copula. 3. Under the prcng measure, the default-free nterest-rate dynamcs are ndependent of the default tmes. 4. For each oblgor, the recovery rate π s constant over tme 1. The ntenstes h (t) of the Posson processes are called hazard rates. Accordng to equaton (2.6), the margnal dstrbuton functons F of the default tmes are gven by ( F (t) = P(τ < t) = 1 exp t ) h (u)du. (3.1) Whle greatly smplfyng the valuaton of default-contngent products, assumpton 3 s not the most realstc representaton of econometrc observatons: central banks, for nstance, tend to lower nterest rates n tmes of recesson, and most emprcal studes fnd a negatve correlaton around 2% between default ntenstes and default-free nterest rates, see [13]. Under assumpton 3, the arbtrage-free prce of a cashflow s ts default-free prce weghted wth the probablty that the cashflow s pad. If B(s, t) denotes the prce at tme s ( s t) of a rsk-free zero-coupon bond wth prncpal $1 maturng at tme t, then the value of a ( payment of $1 due at tme t f (and only f) oblgor survves untl tme t s B(, t) exp ) t h (u)du. If a cashflow depends on the default tmes of more than one oblgor, the probablty that the cashflow s pad can be determned wth Monte Carlo 1 Ths assumpton s made to lghten notaton. The extenson of the prcng algorthms to determnstc recovery rates that are known n advance s straghtforward. 11

18 smulatons. The general prcng algorthm for default-contngent products n the Gaussan copula model s 1. Draw a sample u 1,..., u N from the Gaussan copula (see Secton 2.1.3). 2. The correspondng default tmes are gven by τ = F 1 (u ), {1,..., N}. 3. Compute the cashflows mpled by the default tmes τ 1,..., τ N. 4. Compute the present value of the cashflows wth the dscount curve. 5. Repeat steps 1-4 untl the requred number of scenaros has been smulated. 6. The average present value s then an approxmaton to the prce of the product. 3.2 N th to Default Credt Swaps In ths secton, we analyse the prcng of nth to default swaps n the L model. An nth to default swap has two legs: the premum (fee-payment) leg and the default (defaultnsurance) leg. The premum leg contans a stream of perodc payments (usually called spread payments), whch are pad by the purchaser of protecton untl ether the nth default or the maturty tme, T, whchever s earler. If n defaults do occur, then at the nth default the purchaser pays the recovery rate on the nth default and any accrued spread payment and the seller pays the notonal, see Fgure 3.1. nth Default Notonal Accrued Spread Expry Spreads Recovery Rate Spreads lost due to Default Fgure 3.1: A dagrammatc survey of the cashflows for an nth to default swap n the case that the nth default occurs before the maturty. 12

19 Let τ (n) = τ (n) (τ 1,..., τ N ) be the tme of the nth default. We denote by π (n) the recovery rate and by NA (n) the nomnal of the asset that causes the nth default. For gven τ (n) the present value of the default leg s PV Dflt (τ (n) ) = { NA (n) B(, τ (n) )(1 π (n) ) f τ (n) T f τ (n) > T (3.2) We denote by t 1,..., t P the premum payment dates and by S 1,..., S P the correspondng spread payments. The ndvdual spread payments can dffer slghtly from each other dependng on day count conventons and the swap s specfcatons but they are known n advance. The present value of the premum leg for gven τ (n) s { m PV Fee (τ (n) ) = =1 S B(, t ) + PV Accr (τ (n) ) f t m τ (n) < t m+1 P =1 S B(, t ) f τ (n) (3.3) t P If we assume lnear accural, PV Accr s gven by { PV Accr (τ (n) S ) = m B(, τ (n) ) τ (n) t m t m+1 t m f t m τ (n) < t m+1 f τ (n) (3.4) t P The value V of an nth to default swap s the rsk-neutral expectaton of the dfference between the present value of the premum leg and the present value of the default leg. Hence, from the seller of protecton s pont of vew, V = E P [PV Fee (τ (n) ) PV Dflt (τ (n) )]. We can use the formulas for PV Fee (τ (n) ) and PV Dflt (τ (n) ) to compute an estmate of ths expectaton wth the prcng algorthm descrbed n Secton Independent Defaults In some very specal cases, t s possble to derve analytcal formulas for the value of the default leg or to compute the value of the default leg wth Fourer methods. These results can be used to check the outcome of the Monte Carlo smulatons. Fourer methods wll be used agan n Chapter 4 under the slghtly more general assumpton of ndependence condtonal on the realsaton of a common factor. Throughout ths secton, we make the followng assumptons 1. The default tmes τ 1,..., τ N are ndependent. 2. All assets have the same recovery rate π,.e. π = π for {1,..., N}. 3. The (contnuously compounded) rsk-free nterest rate r s constant. 4. All assets have nomnal $1. We wll specfy addtonal assumptons at the begnnng of each subsecton. 13

20 3.3.1 Constant Interest Rates and Constant Hazard Rates Proposton 3 Suppose that assumptons 1-4 hold and that the followng condton s satsfed: 5. The hazard rates are constant over tme. Then the value of the default leg of a frst to default swap s gven by V Dflt = (1 π)(h h N ) ( ) 1 e T (r+h h N ) r + h h N and the senstvty wth respect to the hazard rate of the th oblgor s V Dflt = (1 π) T (h h N ) e T (r+h h N ) h r + h h N r ( ) + (1 π) (r + h h N ) 2 1 e T (r+h h N ). If accrued premum payments due to defaults between premum payment dates are not taken nto account, the value of the premum leg s V Fee = P S e t (r+h h N ). =1 Proof. The dstrbuton functon of the frst default s P(τ (1) < t) = 1 P(τ (1) t) = 1 P(mn(τ 1,..., τ N ) t). Because the default tmes are ndependent, we get N N P(τ (1) < t) = 1 P(τ t) = 1 e ht = 1 e (h h N )t. =1 By dfferentatng ths equaton we obtan the densty functon f (1) of the frst default =1 f (1) (t) = (h h N )e (h h N )t. Hence the value of the default leg s V Dflt = E[PV Dflt (τ (1) ] = E[(1 π)e τ (1)r 1 {τ (1) T } ] = (1 π) = (1 π) T T e tr f (1) (t)dt e tr (h h N )e (h h N )t dt = (1 π)(h h N ) r + h h N (1 e T (r+h h N ) ). The proof of the remanng statements s straghtforward. 14

21 3.3.2 Zero Interest Rates and Homogeneous Hazard Rates Suppose that assumptons 1-4 hold and 6. The rsk-free nterest rates are zero. 7. All oblgors have the same hazard rate h : R + R +. Then the dstrbuton functon of the number of defaults before the maturty s the bnomal dstrbuton B(N, q) wth q := P(τ 1 T ) = 1 e T h(u)du. Hence the probablty that n or more defaults occur s P(τ (n) T ) = 1 B(n 1; N, q) = N k=n ( ) N q k (1 q) N k. (3.5) k Because the nterest rates are zero and the payoff does not depend on the asset that causes the nth default, the value of the default leg of an nth to default swap s gven by E[PV Dflt (τ (n) )] = (1 π)e[1 {τ (n) T } ] = (1 π)p(τ (n) T ) = (1 π) Zero Interest Rates N k=n ( ) N q k (1 q) N k. k In the remander of ths secton, we allow the hazard rates to be dfferent,.e. we work under the assumptons 1-4 and 6. The rsk-free nterest rates are zero. Let N (T ) denote the ndcator functon that oblgor defaults before the maturty,.e. N (T ) = 1 {τ T }, and let q = P(τ T ). If N(T ) = N =1 N (T ) denotes the number of defaults before the maturty, then the value of an nth to default swap s V Dflt = (1 π)p(n(t ) n) = (1 π) N P(N(T ) = k). The dstrbuton of N(T ) s the convoluton of the dstrbutons of the N (T ) and can be computed wth Fourer methods: for {1,..., N} let k=n 1 q f j = a j = q f j = 1 f j > 1 be the probablty vector of the Bernoull dstrbuted random varable N (T ) and b j = m 1 k= ( ) 2π 1 a j exp m kj ts dscrete Fourer transform where m = 2 1+ log N log 2 and j {1,..., m}. 15

22 Then P(N(T ) = j) s gven by the dscrete nverse Fourer transform of the product of the fast Fourer transforms b j,.e. P(N(T ) = j) = 1 m m 1 k= [ ( exp 2π 1 m jk ) N ] b k. The dscrete Fourer transform can be computed very effcently wth an algorthm called the fast Fourer transform (FFT), see [4]. The FFT algorthm s avalable n many software packages ncludng Mcrosoft Excel. 3.4 Importance Samplng In the Monte Carlo prcng algorthm, a path wll only result n a default payoff f the nth default occurs before the maturty,.e. f τ (n) T. Fgure 3.2 s a plot of the percentages of these mportant paths for the test basket descrbed n Secton 6.1. Note that despte the relatvely hgh hazard rate of oblgor 1 less than 1% of the paths result n one or more defaults wthn the frst year. For a one year second to default swap, only.714% of the paths contrbute to the value of the default leg. Fgure 3.3 shows the dstrbuton of the dscounted payoff of the second to default swap. In our prcng algorthm we estmate the expectaton of ths dstrbuton wth Monte Carlo ntegraton. It s well known that the Monte Carlo estmator s unstable and converges slowly f the ntegrand s locally strongly peaked. Fgure 3.4 llustrates how the convergence speed decreases for hgher n. Clearly, convergence behavour also deterorates for short maturtes. =1 Percentage of Default Paths 6% 5% 4% 3% 2% 1% Percentage of Paths wth Default Payoff 1 st to default 2 nd to default 3 rd to default 4 th to default Percentage of Default Paths Percentage of Paths wth Default Payoff [Logscale] 1 st to default 2 nd to default 3 rd to default 4 th to default % Maturty [years] 1e Maturty [years] Fgure 3.2: Percentage of Monte Carlo paths that result n a default payoff for the test basket descrbed n Secton 6.1. We used 2 22 paths and Sobol numbers. The second graph shows that the estmaton of the probablty that the nth default occurs before the maturty s not stable for short maturtes; the graph does not contan values for whch no path resulted n a default payoff. M. Josh and D. Kanth [7] have adapted the concept of mportance samplng to the prcng of nth to default swaps n the Gaussan copula model. Ther procedure guarantees that every Monte Carlo path results n a default payoff and thus very sgnfcantly mproves the convergence behavour of the approxmatons. We wll frst summarse ther algorthm and then dscuss the steps n detal. 16

23 Probablty No Importance Samplng.7%.6%.5%.4%.3%.2%.1%.% Dscounted Payoff Probablty Importance Samplng 1.8% 1.6% 1.4% 1.2% 1.%.8%.6%.4%.2%.% Dscounted and weghted Payoff Fgure 3.3: Dstrbuton (.e. probablty densty functon) of the dscounted and (f mportance samplng s appled) weghted payoff of the default leg of a second to default swap on the basket descrbed n Secton 6.1. The maturty of the deal s one year. The probablty that no payoff occurs s not shown, and n the case of mportance samplng the value P( < Payoff < 11) of the frst bn has been truncated; t s 5.58%. Wthout mportance samplng, the dstrbuton s rather dscrete and the total probablty mass of non-zero payoffs s only.714%. It s clearly much easer to estmate the expectaton f mportance samplng s appled. 1.5% 1.% Value of Default Leg 1 st to default 2 nd to default 2% 1% Value of Default Leg 3 rd to default 4 th to default Relatve Error.5%.% -.5% Relatve Error % -1% -2% -3% -1.% -4% -1.5% *2 21 2*2 21 3*2 21 4*2 21 5*2 21 6*2 21 7*2 21 8*2 21 Number of Monte Carlo Scenaros -5% *2 21 2*2 21 3*2 21 4*2 21 5*2 21 6*2 21 7*2 21 8*2 21 Number of Monte Carlo Scenaros Fgure 3.4: Convergence graphs of Monte Carlo approxmatons to the values of default swaps on the test basket descrbed n Secton 6.1. All contracts mature n one year, and the Mersenne twster was used as random number generator. Notce that convergence behavour deterorates for hgher n. 17

24 In Secton 2.1.3, we have descrbed a procedure to transform N ndependent and unformly dstrbuted random numbers u nd 1,..., und N nto a sample u 1,..., u N of the Gaussan copula. The resultng default tmes n the Gaussan copula model are then gven by τ = F 1 (u ), {1,..., N}. We wll refer to the N-dmensonal transformaton (u nd 1,..., und N ) (τ 1,..., τ N ) n the followng as ξ Σ. The prncpal dea of the mportance samplng algorthm s to construct teratvely numbers u mod ], 1[ and one-dmensonal dstrbuton functons D (x) = D (x; u mod 1,..., u mod 1 ) : [, 1] [, 1] from the orgnal numbers und such that u mod = D (u nd ) and the correspondng default tmes (obtaned by applyng ξ Σ ) gve rse to a default payoff. Because the change from u nd to u mod alters the probablty of the smulated default tmes, we have to multply the payoff n the smulated scenaro wth the weghtng factor w = w(u nd 1,..., u nd N ) = N =1 dd dx + (und ). Fgure 3.3 gves an example of the probablty densty functon of the weghted payoff. Clearly, the Monte Carlo ntegraton of ths densty wll converge much faster than the ntegraton of the densty obtaned wthout mportance samplng. The remander of the secton gves a detaled descrpton of the mportance samplng algorthm: 1. Compute the Cholesky decomposton Σ = AA T where A = (a j ) 1,j N s lower trangular. The transformaton ξ Σ : (u nd 1,..., und N ) (τ 1,..., τ N ) from ndependent unforms nto default tmes whch are correlated by the Gaussan copula s then gven by [ ( ( ) )] τ = F 1 Φ a k Φ 1 u nd k, {1,..., N}. (3.6) k=1 It s mportant to note that the default tme τ only depends on the frst arguments of ψ Σ. Once u mod 1,..., u mod n 1 have been determned, the choce of umod does not alter the default tmes of the frst 1 oblgors. Ths allows us to calculate the u mod teratvely. Furthermore, we recall that the dagonal entres of A are postve. In partcular, a. 2. Draw N ndependent unformly dstrbuted numbers u nd 1,..., und N ], 1[ from a random number generator. 3. Set nr_of_defaults = and = Set and p marg p dflt ( T ) = P(τ < T ) = F (T ) = 1 exp h (s)ds = Φ ( Φ 1 (p marg ) 1 k=1 a kφ 1 (u nd a k ) ). 18

25 Accordng to equaton (3.6), the default tme τ correspondng to u mod 1,..., u mod gven by τ = F 1 [ Φ ( k=1 ( ) )] a k Φ 1 u mod k. Because u mod 1,..., u mod 1 have been already determned, τ < T s equvalent to ( ) a Φ 1 u mod 1 < Φ 1 (F (T )) k=1 ( ) a k Φ 1 u mod k. Oblgor wll therefore default before the maturty f and only f u mod 5. Set and and p mod = D (x) = u mod n nr of defaults N + 1 x pdflt p mod p dflt = D (u nd ). + 1 pdflt 1 p mod f x < p mod (x p mod ) f x p mod < p dflt. Ths transformatons ensures that gven u mod 1,..., u mod 1 the probablty that oblgor defaults s p mod. Fgure 3.5 llustrates how D alters the dstrbuton of the u mod. 6. If u nd < p mod ncrease nr_of_defaults by one. 7. If nr_of_defaults < n, ncrease by one and contnue wth step 4. s 8. Set u mod k = u nd k and D k (x) = x for k { + 1,..., N},.e. we do not alter the remanng u nd k once the requred number of defaults have been generated. 9. Set w = N =1 dd dx + (und ) wth f u nd u mod, and dd dx + (und ) = p dflt p mod 1 p dflt 1 p mod f x < p mod f x p mod dd dx + (und ) = 1 otherwse. Ths s the weghtng factor wth whch we have to scale the payoff. 19

26 1. Compute the default tmes by settng (τ 1,..., τ N ) = ξ Σ (u mod 1,..., u mod N ). 11. Sort the default tmes and determne the asset whch causes the nth default,.e. determne τ (n), π (n) and NA (n). 12. Set PV IS Dflt (τ (n) ) = wpv Dflt (τ (n) ) and ( ) PVLostFee(τ IS (n) ) = w PV Fee (T ) PV Fee (τ (n) ). The nth default tme τ (n) generated by the mportance samplng algorthm s always before the maturty T. Importance samplng s therefore only applcable to payoff functons g wth g(τ (n) ) = whenever τ (n) > T. Snce the payoff of the premum leg s non-zero f τ (n) > T, the value of the premum leg cannot be drectly estmated wth mportance samplng. The algorthm crcumvents ths problem by estmatng the lost fees and subtractng the estmate from the present value of the premum leg obtaned wth nsuffcent defaults (see next step). 13. Let ˆV Dflt (M) and ˆV LostFee (M) denote the averages of PVDflt IS and PVIS LostFee, respectvely, over M Monte Carlo scenaros. ˆVDflt (M) and ˆV Fee (M) = PV Fee (T ) ˆV LostFee (M) are then approxmatons to the values of the default leg and the premum leg. Fgure 3.6 gves an example for the dstrbuton of default tmes wth and wthout mportance samplng orgnal draw modfed draw Fgure 3.5: Illustraton of the transformaton D f p dflt =.1 and p mod =.5. 2

27 35 3 Default Tme Oblgor Default Tme Oblgor Default Tme Oblgor Default Tme Oblgor Default Tme Oblgor Default Tme Oblgor 1 Fgure 3.6: The frst graph shows the dstrbuton of the default tmes for a basket of two names wth hazard rates.2 and.3 and a Gaussan copula wth a correlaton of.2. The bottom-left graph s a magnfcaton of the frst graph, and the bottom-rght graph shows the default tmes n the same sector f mportance samplng s used to prce a frst to default swap wth a maturty of 1 year. 21

28 3.5 Calculaton of Senstvtes to Changes n the Hazard Rates Senstvtes are of central mportance n any model for prcng dervatves because they determne the tradng strategy that hedges the dervatve securty. In ths secton, we analyse methods to compute the senstvtes of the value of an nth to default swap to small changes n the hazard rates. In contrast to stock opton models where the rsk factor (the asset prce) s a sngle value, the hazard rate s a functon of tme. Hence, before calculatng senstvtes, we have to defne what knd of changes n the hazard rates we wll nvestgate. It s common market practce to assume that hazard rates are pecewse constant,.e. h (t) = h j, f t j t < t (j+1) where (t j ) j N s a strctly ncreasng sequence wth t = and lm j t j = and h j R + for all j N. If we assume that the ponts of dscontnuty t j reman fxed, all possble future hazard rate curves can be obtaned by varyng the h j. Hence we could defne senstvtes wth respect to every nterval [t j, t (j+1) ]. However, t s a convenent and plausble assumpton that (small) changes n the credt qualty of an oblgor do not alter the shape of the hazard rate curve. Therefore, we defne senstvtes wth respect to parallel shfts of the whole hazard rate curve: Defnton 7 Let g(h ; γ 1,..., γ s ) be a functon that depends on the hazard rate h : R + R + of the th oblgor and some other parameters γ 1,..., γ s. Then we call g g(h + δ; γ 1,..., γ s ) g(h, γ 1,..., γ s ) := lm h δ δ the senstvty of g wth respect to the th hazard rate. For notatonal smplcty we shall often wrte g(h ) nstead of g(h ; γ 1,..., γ s ) Fnte Dfferencng The smplest technque to estmate the senstvty wth respect to the hazard rate h of the th oblgor s to run a separate Monte Carlo smulaton wth the slghtly larger hazard rate h + δ (for a sutable chosen δ) and to set g g(h + δ) g(h ). h δ Instead of ths so-called forward dfference we can also recalculate twce and use the central dfference g g(h + δ) g(h δ). h 2δ The Lkelhood Rato Method A more advanced technque s the lkelhood rato method, [1]. It s based on the observaton that the prce of a dervatve wth dscounted payoff PV(τ 1,..., τ N ) s V = 1 1 PV(ξ Σ (u nd 1,..., u nd N ; h ))du nd 1 du nd N, 22

29 where we have wrtten ξ Σ (... ; h ) nstead of ξ Σ (...) to stress that the transformaton from ndependent unforms nto the default tmes depends on the hazard rate of the th oblgor. The dependence on the hazard rate can be moved from the dscontnuous payoff functon to the smooth densty functon by wrtng V = T T PV(τ 1,..., τ N )ψ(τ 1,..., τ N ; h )dτ 1 dτ n. where ψ s the densty functon of the default tmes. If the densty ψ s contnuously dfferentable n the hazard rates, the order of dfferentaton and ntegraton can be nterchanged,.e. V h = T T PV(τ 1,..., τ N ) ψ(τ 1,..., τ N ; h ) dτ 1 dτ n. (3.7) h The basc dea of the lkelhood rato method s that the orgnal densty ψ can be rentroduced by wrtng V h = = T T T T PV(τ 1,..., τ N ) ψ(τ 1,..., τ N ; h ) h 1 ψ(τ 1,..., τ N ; h ) ψ(τ 1,..., τ N ; h )dτ 1 dτ n PV(τ 1,..., τ N ) log ψ(τ 1,..., τ N ; h ) h ψ(τ 1,..., τ N ; h )dτ 1 dτ n. The calculaton of the senstvty s now exactly lke the orgnal prcng problem, only wth the new payoff functon PV(τ 1,..., τ N ) log ψ(τ 1,..., τ N ; h )/ h nstead of PV(τ 1,..., τ N ). To apply the lkelhood rato method n the Gaussan copula model, we have to calculate log ψ(τ 1,..., τ N ; h )/ h. The densty ψ of the default tmes can be obtaned wth Proposton 1 and dfferentaton of equaton (3.1) from the densty of the Gaussan copula (2.4): wth Takng logs gves 1 ψ(τ 1,..., τ N ) = det(σ) 1 2 log ψ(τ 1,..., τ N ) = 1 2 log det(σ) 1 2 ( exp 1 ( 2 vt Σ 1 1 ) ) N v h (τ )e τ h (s)ds =1 v = Φ 1 (F (τ )) = Φ 1 ( 1 e τ h (s)ds ). ( v T ( Σ 1 1 ) v ) + N =1 ( τ ) log h (τ ) h (s)ds. By vrtue of Defnton 7, all laws of dfferentaton hold for the operator / h. Hence v h = = ( Φ 1 1 e ) τ h (s) h 1 ( ϕ (Φ 1 1 e τ h (s)ds )) h ( 1 e τ h (s)ds ) 23

30 where ϕ(x) = 1 2π e x2 /2. Usng Defnton 7, we can evaluate the second term Smlar calculatons gve and ( 1 e ) τ h (s)ds = lm h h = e τ δ ( lm δ = τ e τ h (s)ds. log h (τ ) = 1 h h (τ ) τ h (s)ds = τ. δ+h (s)ds e τ h (s)ds δ e τδ ) 1 e τ h (s)ds δ Combnng the results we obtan the requred expresson for log ψ(τ 1,..., τ N ; h )/ h : log ψ(τ 1,..., τ N ; h ) h = N j=1 ( Σ 1 1 ) j v j v h + 1 h (τ ) τ wth v h = τ e τ h (s)ds ( ϕ (Φ 1 1 e )) τ h (s)ds and v j = Φ 1 ( 1 e τ j h j(s)ds ). 24

31 Chapter 4 The One Factor Gaussan Copula Model In the Gaussan copula model, the modellng of the default dependency was reduced to the specfcaton of a correlaton matrx. For large baskets, ths s stll a nontrval exercse. A common trck to further reduce the dmensonalty of the modellng problem s to explan the correlaton wth a small number of common factors. Laurent and Gregory [8] have appled a one factor approach n the context of the Gaussan copula model. In ther model, the prce of an nth to default swap and the senstvtes to changes n the hazard rates can be expressed n terms of two-dmensonal ntegrals. Due to ths dmensonalty reducton, the quadrature can be carred out much faster and wthout Monte Carlo smulatons. We wll descrbe the one factor Gaussan copula model n the next secton. 4.1 Descrpton of the Model In the one factor Gaussan copula model, the default tmes are gven by ( ( )) τ = F 1 Φ ρ Y + 1 ρ 2 Z, {1,..., N} where Y s the standard normally dstrbuted common factor, Z 1,..., Z N are standard normally dstrbuted random varables (the dosyncratc nose components), Y, Z 1,..., Z N are ndependent and ρ < 1. In practce, nonnegatvty of the loadng factors ρ s often mposed as a conservatve assumpton to ensure that all default tmes are postvely correlated. Furthermore, we assume that all assumptons made n Secton 3.1 hold. The random varables X := ρ Y + 1 ρ 2 Z 25

32 are normally dstrbuted wth cov(x, X j ) = { ρ ρ j f j 1 f = j Hence, accordng to defnton 2, the dependency structure of the default tmes s ndeed gven by a Gaussan copula, but the specfcaton of the full correlaton matrx (wth N(N 1)/2 degrees of freedom) has been replaced by the specfcaton of ρ 1,..., ρ N. Besdes ths often desrable reducton of dmensonalty, there s another mportant feature of the one factor model: condtonal on the realsaton of the common factor Y, the random varables X 1,..., X N are ndependent. We wll use ths observaton to derve a sem-explct formula for the prce of an nth to default swap n the next secton. 4.2 Prcng Formulas for nth to Default Swaps If the value of the common factor Y s known, t s straghtforward to compute the (rskneutral) condtonal probablty p Y t that the th oblgor defaults before tme t: p Y t := P(τ < t Y ) = P Because Z s standard normally dstrbuted, we get p Y t = Φ Z < Φ 1 (F (t)) ρ Y 1 ρ 2 Φ 1 (F (t)) ρ Y 1 ρ 2 We recall a standard result from probablty theory. Y Proposton 4 Let (Ω, F, P) be a probablty space and let X, Y : Ω R be two random varables. Then E[X] = E[E[X Y ]]. Ths relaton s called the terated expectatons theorem. We wll use the followng corollary: for any subset A Ω. P(A) = E[1 A ] = E[E[1 A Y ]] = E[P(A Y )] (4.1) By the terated expectatons theorem, the jont dstrbuton of the default tmes can be wrtten as F (t 1,..., t N ) = P(τ 1 < t 1,..., τ N < t N ) = E [P(τ 1 < t 1,..., τ N < t N Y )]. Because X 1,..., X N and thus τ 1,..., τ N are ndependent condtonal on the realsaton of the common factor Y, [ N ] [ N ] N F (t 1,..., t N ) = E P (τ < t Y ) = E p Y t = ϕ(y) p y t dy, =1 =1 =1 26