Contagion models with interacting default intensity processes

Transcription

1 ICCM 2007 Vol. II 1 4 Conagion models wih ineracing defaul inensiy processes Kwai Sun Leung Yue Kuen Kwok Absrac Credi risk is quanified by he loss disribuion due o unexpeced changes in he credi qualiy of he counerpary in a financial conrac. Defaul correlaion risk refers o he risk ha a bundle of risky obligors may defaul ogeher. To undersand he clusering phenomena in correlaed defauls, we consider credi conagion models which describe he propagaion of financial disress from one risky obligor o anoher. We presen he conagion model of porfolio credi risk of muliple obligors wih ineracing defaul inensiy processes where he defaul of one firm may rigger he increase of defaul inensiy of oher relaed firms. As an applicaion, we consider how correlaed defaul risks beween he proecion seller and he underlying eniy may affec he credi defaul premium in a credi defaul swap Mahemaics Subjec Classificaion: 60G55, 62M05. Keywords and Phrases: credi risk, defaul correlaion, reduced form models, credi conagion, ineracing inensiies, credi defaul swap 1. Inroducion Credi risk is he likelihood ha a conracual pary may no mee is obligaions, like paymen of coupons or principal in a bond conrac, hus causing a financial loss of he counerpary. Broadly speaking, financial loss due o a credi even is quanified by he loss disribuion due o expeced changes in he credi qualiy (downgrade or defaul) of a conracual pary. The hree basic aribues for quanifying defaul loss is he probabiliy of defaul, loss given defaul and exposure a defaul. Risk Managemen Insiue, Naional Universiy of Singapore, rmilks@nus.edu.sg Deparmen of Mahemaics, Hong Kong Universiy of Science and Technology, maykwok@us.hk

2 2 Kwai Sun Leung and Yue Kuen Kwok There are wo major approaches ha aemp o describe he defaul processes of risky obligors, commonly known as he srucural models and reduced form (inensiy) models. The srucural models use he coningen claims approach in opion pricing heory where he value of firm s asses is used as he underlying sae variable. A firm defauls when he firm asses are insufficien o honor conracual paymens. This approach aemps o provide a srucural inerpreaion of defaul. The reduced form approach assumes ha defaul occurs unpredicably a an exogenous inensiy or hazard rae. There is no srucural inerpreaion of defaul, and he inensiy is calibraed from marke prices. The dynamics of defaul are prescribed under a pricing measure in he framework of poin processes. Le τ denoe he random defaul ime of a risky obligor. The defaul process is defined by { 1 if τ H = 1 {τ } = 0 oherwise. (1.1) Noe ha H is a poin process wih one jump of size one upon defaul. One major concern in he pricing and managemen of credi risk in an invesmen porfolio is he occurrence of muliple defauls of differen obligors wihin he porfolio. This correlaion risk is direcly linked o he iner-dependence beween defaul evens. The developmen of quaniaive models for analyzing correlaed defaul risk has recenly become a focus of aenion for academics, regulaors, and praciioners. Iner-dependence beween defauls sems from a leas wo sources. Firs, he financial healh of a firm varies wih general macroeconomic facors. Since differen firms are affeced by common macroeconomic facors, we have iner-dependence beween heir defauls hrough hese facors. Anoher iner-dependen defaul srucures are caused by direc links beween firms such as business relaions, like borrower-lender relaionship. The likelihood of defaul of a commercial bank is likely o increase if some of is major borrowers or counerparies defaul. For example, he Souh Korean banking crisis is commonly aribued o non-performing of a primary firm so ha he likelihood of defaul of he secondary firm depends on he credi even of he primary firm. To inroduce defaul correlaion under he reduced form framework, one may se he defaul inensiy dynamics be driven by a common se of macroeconomic facors. Condiioned o he realizaion of he macroeconomic sae variables, he defaul imes are condiionally independen. The conagion models ake one sep furher by inroducing addiional dependence o accoun for defaul clusering, an empirical fac ha defaul imes end o concenrae in cerain periods of ime. To model he phenomenon ha he defaul of one firm may increase he likelihood of defaul of oher relaed firms, Jarrow and Yu (2001) and Yu (2007) creae he defaul conagion effec by inroducing a posiive jump

3 Conagion models wih ineracing defaul inensiy processes 3 in he defaul inensiy whenever here is an occurrence of defaul of a counerpary. For example, considering a porfolio of 3 obligors, he iner-dependen defaul inensiies of he 3 obligors under he conagion model wih ineracing defaul inensiy processes may be formulaed as λ A = a 10 + a 12 1 {τb } + a 13 1 {τc } + a 14 1 {τb,τ C } λ B = a 20 + a 21 1 {τa } + a 23 1 {τc } + a 24 1 {τa,τ C } λ C = a 30 + a 31 1 {τa } + a 32 1 {τb } + a 34 1 {τa,τ B }, (1.2) where λ A is he defaul inensiy of obligor A, ec. Informaion Srucure We characerize he credi risk model by inroducing a collecion of Cox processes (also known as doubly sochasic Poisson processes). Le he informaion srucure in he economy wih a rading period [0, T] be described by he filered probabiliy space ( Ω, F, {F } T =0, P ), where F = F T and P is he risk neural (equivalen maringale) probabiliy measure. Le I be he number of firms in he economy, and F X = σ(x s ; 0 s ) denoes he marke informaion generaed by he macroeconomic facors X. Also, F i = σ(ni s ; 0 s ) denoes he defaul informaion generaed by he defaul process N i of firm i I. Therefore, he complee informaion on he macroeconomic facors and he defaul processes of all firms up o ime is F = F X F I, where F I = F 1 F I and F i F j represens he smalles σ-algebra conaining F i and F j. Furhermore, he filraion generaed by F i = F 1 F i 1 F i+1 F I represens he complee defaul informaion of all firms oher han ha of he i h firm, up o ime T. Hence, G i = F i F X T F i T conains he complee informaion on he marke bu excludes he defaul informaion of firm i up o ime. Defaul Time Following he sandard reduced form approach o model defaul risk, we characerize he sopping ime (defaul imes) τ i of he i h firm in he Cox process framework. Specifically, we define τ i by { } τ i = inf : λ i s ds E i (1.3) 0

4 4 Kwai Sun Leung and Yue Kuen Kwok and {E i } i I is a se of independen uni exponenial random variables. The probabiliy space is hen enlarged o accommodae {E i } i I, which are independen of FT X and Fi T for each i. Each τi is characerized by he non-negaive and F-measurable process λ i such ha for each > 0 and he process 0 λ i s ds <, P-a.s. M i = H τ i 0 λ i s ds is a P-maringale wih respec o F. Here, λ i is called he defaul inensiy of τ i. This provides an inuiion behind a formal definiion of he Cox process. Suppose ha a curren ime, firm i has no ye defauled so ha τ i >. Wih respec o he above characerizaion, he condiional and uncondiional survival probabiliies of firm i are given by ( P(τ i > T G i ) = 1 ) T {τ i >} exp λ i s ds, P(τ i > T F X ) = 1 {τ i >}E ( exp ( T λ i s ds ) F ). (1.4) In he nex secion, we presen he Markov chain framework of he conagion model of correlaed defauls, exending a similar formulaion presened by Frey and Backhaus (2004). Markovian chain approach has also been applied by Avellaneda and Wu (2001) o model he defaul saus of a porfolio of risky obligors. The compuaion procedure ha calculaes he join disribuion of defaul imes is exemplified. In Secion 3, we apply he conagion model of ineracing inensiies o analyze he correlaed defaul risks of he proecion seller and he underlying reference eniy in a credi defaul swap. The procedure of calibraing he parameer funcions in he model formulaion is also explained. The paper is ended wih conclusive summaries and remarks in he las secion. 2. Markov chain framework Considering a porfolio of N firms, we associae a random defaul ime τ i wih firm i in he porfolio. The defaul saus of he porfolio is given by he defaul process H = (H 1, H2,, HN ) {0, 1}N = S, (2.1)

5 Conagion models wih ineracing defaul inensiy processes 5 where H i = 1 {τ i } for i = 1, 2,, N. Here, H is visualized as a finie sae Markov chain and S is he sae space of H. The macroeconomic variables are described by he d-dimensional sochasic process Ψ = (Ψ ) [0,T] wih sae space D R d. Le y S, where y is a vecor of defaul indicaors of he risky obligors in he porfolio. For noaional convenience, we define he flipped sae y i S by y i (i) = 1 y(i) and y i (j) = y(j), j {1, 2,, N} {i}. (2.2) In oher words, o obain y i from y, only he i h componen of y is flipped from 1 o 0 or 0 o 1 while all oher componens remain he same value. Le D([0, ), E) denoe he space of righ coninuous funcions wih lef limi from [0, ) ino he Polish space E. We define a measurable space (Ω, F) in he following manner: and Ω = Ω 1 Ω 2 where Ω 1 = D([0, ), D) and Ω 2 = D([0, ), S) F = F 1 F 2 where F i is he Borel σ-field of Ω i, i = 1, 2. For each ω Ω, we wrie ω = (ω 1, ω 2 ) where ω i Ω i, i = 1, 2. We model Γ on (Ω, F) as follows: wih Γ : [0, ) Ω D S Γ (ω) = (Ψ (ω 1 ), H (ω 2 )) = (ω 1 (), ω 2 ()). Suppose he informaion available o he invesor in he marke a ime include he hisory of macroeconomic variables and defaul saus of he porfolio up o ime. Mahemaically, he filraion (F ) 0 on (Ω, F) is given by F = F Ψ F 1 F 2 F N where F Ψ = σ(ψ s : 0 s ) F i = σ(h i s : 0 s ), i = 1, 2,, N. For each γ = (ψ, y) D S, we define a family of probabiliy measure Pγ on (Ω, F, (F ) 0 ) as Pγ = µ ψ κy(ω 1, dω 2 ).

6 6 Kwai Sun Leung and Yue Kuen Kwok Here, µ ψ is a probabiliy measure on Ω 1 which gives he law of Ψ, κy is a ransiion kernel from (Ω 1, F 1 ) o (Ω 2, F 2 ), which models he condiional disribuion of H for a given rajecory of Ψ. Le S denoe he number of saes in S. For y i, y j S, he infiniesimal generaor Λ [Ψ] () = (Λ ij ( ω 1 )) S S for H given he pah of Ψ is defined as follows. (a) For i j { [1 yi (k)]λ Λ ij ( ω 1 ) = k (Ψ (ω 1 ), y i ), if y j = y k i for some k. 0 else (2.3a) The ransiion rae Λ ij equals λ k (Ψ (ω 1 ), y i ) when y j can be obained from y i by flipping is k h elemen from 0 o 1, indicaing defaul of he k h obligor in he porfolio. The facor 1 y i (k) is included since y i (k) = 1 is an absorbing sae. (b) For i = j Λ ii ( ω 1 ) = j i Λ ij ( ω 1 ) = N [1 y i (k)]λ k (Ψ (ω 1 ), y i ). k=1 (2.3b) Here, λ i (Ψ, H ) is a sricly posiive F-progressively measurable process. Precisely, λ i (Ψ, H ) is he maringale defaul inensiy of firm i, τi ha is, H i λ i (Ψ s, H s )ds is a {F }-maringale. 0 By convenion, we order he defaul indicaor vecors according o he ordering of he obligors inside he porfolio. The firs sae y 1 corresponds o no defaul of any obligor, he second sae corresponds o defaul of he firs obligor only, he hird sae corresponds o defaul of he second obligor only, ec., he las sae y S corresponds o defaul of all obligors. Condiional ransiion probabiliies Noe ha H can be visualized as a condiional ime-inhomogeneous Markov chain. For 0 < s <, we denoe he ransiion densiy marix condiional on he pah of Ψ by P(, s ω 1 ) = (p ij (, s ω 1 )) S S. (2.4) The ransiion densiy marix P(, s ω 1 ) can be obained by solving he corresponding Kolmogorov equaions. The backward Kolmogorov equaion akes he form dp(, s ω 1 ) d = Λ [Ψ] ()P(, s ω 1), P(s, s ω 1 ) = I. (2.5)

7 Conagion models wih ineracing defaul inensiy processes 7 The individual ransiion probabiliy p ij (, s ω 1 ) saisfies he following sysem of ODE: S dp ij (, s ω 1 ) = Λ ik ( ω 1 )p kj (, s ω 1 ) d k=1{. (2.6) 1 if i = j p ij (s, s, y i, y j ω 1 ) = 0 if i j Alernaively, he forward Kolmogorov equaion akes he form dp(, s ω 1 ) ds = P(, s ω 1 )Λ [Ψ] (s), P(, ω 1) = I. (2.7) The soluion of P(, s ω 1 ) can be obained by solving eiher Eq. (2.5) or Eq. (2.7), and P(, s ω 1 ) is deerminisic for a given pah of (Ψ ) 0 [ha is, condiional on Ψ = ω 1 ]. Marginal disribuion of he defaul ime Once he condiional ransiion densiy marix P(, s ω 1 ) has been found, i can be used o derive he marginal disribuion of τ i, i = 1, 2,, N. The marginal disribuion funcion of he defaul ime τ i of Obligor i is defined by F i ( i ) = P r [τ i i ], i = 1, 2,, N. (2.8) Le µ ψ (ω 1 ) be he probabiliy measure which gives he law of Ψ. To obain F i ( i ), we sum over all saes j wih defaul of he i h obligor [observing he requiremen ha y j (i) = 1] of all ransiion probabiliies moving from sae 1 (none of he obligors defauls) o sae j, and subsequenly inegrae over he disribuion of µ ψ (ω 1 ). This gives F i ( i ) = p 1j (0, i ω 1 )dµ ψ (ω 1 ). (2.9) y j (i)=1 Join disribuion of he defaul imes The join disribuion of he defaul imes is defined as F( 1, 2,, N ) = P r [τ 1 1,, τ N N ]. (2.10) To express F( 1, 2,, N ) in erms of p ij ( k, k k+1 ω 1 ), we consider he decomposiion of he even {τ 1 1,, τ N N } ino he union of he following muually exclusive sub-evens. Wihou loss of generaliy, we assume 1 2 N. The firs sub-even is he defaul of all obligors wihin [0, 1 ], whose probabiliy is given by p 1M (0, 1 ω 1 ). The second sub-even corresponds o he defaul of all obligors wihin (0, 2 ], while Obligor 1 bu no all obligors have defauled by 1. Similarly, in

8 8 Kwai Sun Leung and Yue Kuen Kwok he hird sub-even, all obligors have defauled by 3. However, Obligor 1 mus defaul wihin (0, 1 ], Obligor 2 mus defaul wihin (0, 2 ] while no all obligors have defauled by 2. In he las sub-even, Obligor k mus defaul wihin (0, k ], k = 1, 2,, N 1, while no all obligors have defauled by N 1. In addiion o he above requiremens, we also require ha once an obligor has defauled, he defaul sae is an absorbing sae. For noaional convenience, we define S(n) = {y S : y(i) = 1 for 1 i n, y(k) = 0 a leas for some k > n}, n = 1, 2,, N 1. (2.11) The defaul indicaor vecor y jn a ime n mus be chosen from S(n) since he firs n obligors have defauled wihin (0, n ] bu no all of he obligors have defauled by n. Assuming 1 2 N, he join disribuion funcion can be expressed as F( 1, 2,, N ) = p 1M (0, 1 ω 1 ) + Ω 1 y j1 S(1) y j2 S(2) y j1 S(1). y jn 1 S(N 1) y j1 S(1) p 1j1 (0, 1 ω 1 )p j1m( 1, 2 ω 1 ) + p 1j1 (0, 1 ω 1 )p j1j 2 ( 1, 2 ω 1 )p j2m( 2, 3 ω 1 ) + + p 1j1 (0, 1 ω 1 )p j1j 2 ( 1, 2 ω 1 ) p jn 1M( N 1, N ω 1 ) dµ ψ (ω 1 ), (2.12) where y jn observes he propery: y jn (l) y jn 1 (l) for l = 1, 2,, N, n = 2,, N 1, This condiion is dicaed by defaul is an absorbing sae. Tha is, once y jn 1 (l) becomes one hen y jn (l) canno be zero. 3. Counerpary risk of credi defaul swaps In a vanilla credi defaul swap (CDS), he proecion buyer pays periodic premium o he proecion seller. In reurn, he buyer is eniled o receive compensaion from he seller on finanical loss upon defaul of he underlying reference eniy. There have been several papers (Kim

9 Conagion models wih ineracing defaul inensiy processes 9 and Kim, 2003; Leung and Kwok, 2005; Walker, 2005) which discuss specifically on he counerpary risk in credi defaul swaps. In his secion, we would like o propose a simple defaul conagion model ha examines how correlaed defaul risks beween he proecion seller and he underlying reference eniy may affec he credi defaul swap premium. In paricular, our model assumes ha he defaul inensiy of he proecion seller and reference eniy are subjec o a posiive jump in value upon he occurrence of an exernal shock even. To pu ino real life perspecive of our model, we may consider a credi defaul swap on a risky Korean bond whose proecion seller is a Korean financial insiuion. Though he Korean financial insiuion may offer proecion on he Korean bond a a lower credi defaul swap premium, we may query wheher he reducion in swap premium would be sufficien o compensae for he higher counerpary risk. This is because he Korean proecion seller may share higher level of correlaed risk wih he Korean reference eniy upon he arrival of a counry wide shock (like he 1997 economic meldown in Korea). Model formulaion Le τ C and τ R denoe he random defaul ime of he counerpary and reference asse, respecively, and τ S be he random ime of arrival of he exernal shock S. The arrival of he shock is modeled as a Poisson even wih consan mean inensiy λ S. Prior o he arrival of he shock, he defaul inensiies λ C and λ R are assumed o be a C () and a R (), where a C () and a R () are deerminisic funcions of. Upon arrival of S, λ C jumps from a C () o α R a C (), and similarly, λ R jumps from a R () o α R a R (). Here, he proporional facors α C and α R are assumed o be posiive consans, wih α C > 1 and α R > 1. In summary, he defaul inensiies of he hree evens are given by λ R = a R ()[(α R 1)1 {τs } + 1] λ C = a C ()[(α C 1)1 {τs } + 1] λ S = λ S. (3.1) Our assumed model falls wihin he framework of a conagion model wih ineracing inensiies. The probabiliies of ransiion beween various saes of even occurrences can be solved using he Markov chain formulaion. Accordingly, we le H = (H R H C H S ), (3.2) { where H R 1 if τr = 0 if τ R >, and similar definiion for HC and H S. There are eigh possible saes of he defaul process H. The infiniesimal

10 10 Kwai Sun Leung and Yue Kuen Kwok generaor Λ() can be readily found o be Λ 11 = [a R () + a C () + λ S ], Λ 12 () = a R (), Λ 13 () = a C (), Λ 14 () = λ S, Λ 22 = [a C () + λ S ], Λ 25 = a C (), Λ 26 = λ S, Λ 33 = [a R () + λ S ], Λ 35 = a R (), Λ 37 = λ S, Λ 44 = [α R a R () + α C a C ()], Λ 46 = α R a R (), Λ 47 = α C a C (), Λ 55 = λ S, Λ 58 = λ S, Λ 66 = α C a C (), Λ 68 = α C a C (), Λ 77 = α R a R (), Λ 78 = α R a R (), while all oher enries are zero. The ransiion probabiliy marix P is governed by he forward Kolmogorov equaion dp(, u) du = P(, u)λ(u), 0 u, (3.3) wih P(, ) = I. Since Λ(u) is upper riangular, individual ransiion probabiliy p ij (, u) can be solved successively in a sequenial manner. Some of hese probabiliy values are found o be p 11 (, T) = e R T [ar(u)+ac(u)+λs] du p 13 (, T) = e R T [ar(u)+λs] du [1 e R T ac(u) du ] p 14 (, T) T = λ S e R T [ar(u)+ac(u)] du e R T s [(αr 1)aR(u)+(αC 1)aC(u)] du λss ds. The marginal disribuion for τ R is given by P r [τ R > T F ] = p 11 (, T) + p 13 (, T) + p 14 (, T) + p 17 (, T) = e R T + λ S T [ ar(u) du λs(t ) e e R T s (αr 1)aR(u) du λs(s ) ds ]. (3.4) Credi swap premium Le T be he mauriy dae of he CDS and assume uni value for he par of he underlying reference asse. We assume ha he swap premium paymens are made coninuously a a consan swap rae C(T). We assume ρ o be he deerminisic recovery rae of he reference asse upon defaul. The coningen compensaion paymen of 1 ρ is made by he proecion seller during (, + d] provided ha here has been no defaul during (0, ) and defaul of he reference asse occurs during he infiniesimal ime inerval (, + d]. The expeced

11 Conagion models wih ineracing defaul inensiy processes 11 presen value of coningen compensaion paymen over (, + d] is (1 ρ)e r [p 11 (0, )a R () + p 14 (0, )α R a R ()] d. The probabiliy of no defaul up o ime is given by p 11 (0, ) + p 14 (0, ) and he expeced presen value of he swap premium paymen over (, + d] is C(T)e r [p 11 (0, )+p 14 (0, )] d. By equaing he expeced presen value of he swap premium paymen and coningen compensaion paymen upon defaul over he whole period [0, T], we obain C(T) = (1 ρ) T 0 e r [p 11 (0, )a R () + p 14 (0, )α R a R ()] d T. (3.5) 0 e r [p 11 (0, ) + p 14 (0, )] d Subsiuing he known soluions of p 11 () and p 14 (), we obain an analyic expression for C(T). When here is no defaul risk of he counerpary, we hen have a C = 0. In his case, he credi defaul swap rae wihou counerpary risky is given by C(T) = (1 ρ) R T 0» a R ()e R 0 r+a R (u) du e λ S + α R R 0 λ S e R s (α R 1)a R (u) du λ S s ds «R T0 e R r+a 0 R (u) du e λ S + R λ 0 S e R «[(αr 1)a R (u) du λ S s ds d Calibraion of he parameer funcions The parameer funcion α R () in he inensiy α R can be calibraed using he erm srucure of prices of defaulable bonds issued by he reference eniy. Le B R (, T) denoe he ime- price of he defaulable bond wih uni par and zero recovery upon defaul. Under he risk neural measure P and consan riskfree ineres rae r, he defaulable bond price B R (, T) is given by B R (, T) = e r(t ) E P [1 {τr>t } F ] = e r(t ) P r [τ R > T F ]. (3.7) We can also esablish he following relaion beween he parameer funcion α R () and he erm srucure of B R (, T): B R T (, T) = rb R(, T) + a R (T)B R (, T) + a R (T)(α R 1)B R (, T) 4. Conclusion d. (3.6) a R (T)(α R 1)e R T ar(u) du e λs(t ) r(t ). (3.8) A robus and versaile defaul correlaion models should reflec he following wo empirical facs: (i) defaul of one firm may rigger an increase of he defaul inensiies of oher relaed firms, (ii) defaul imes end o concenrae in cerain periods of ime (clusers of defaul). In his paper, we presen he Markov chain framework of modeling defaul conagion via he ineracing inensiies approach, and apply he Markov

12 12 Kwai Sun Leung and Yue Kuen Kwok chain echniques in calculaing he join defaul disribuion of he random defaul imes of muliple risky obligors wihin a porfolio. We develop he hree-firm conagion model o analyze he counerpary risk of he proecion seller of a credi defaul swap. To model he correlaed risk of defauls, he proecion seller and reference eniy are subjec o a posiive jump in defaul inensiy upon he arrival of an exernal shock. We obain he credi defaul swap premium wih and wihou defaul risk of he proecion seller. We also manage o calibrae he parameer funcions in he conagion model using marke prices of raded bonds. Acknowledgemen This research was suppored by he Research Grans Council of Hong Kong, HKUST6425/05H. References [1] M. Avellaneda & L. Wu, Credi conagion: pricing cross counry risk in he Brady deb markes, Inernaional Journal of Theoreical and Applied Finance, 4(6) (2001), [2] R. Frey & J. Backhaus, Credi derivaives in models wih ineracing defaul inensiies: a Markovian approach. Working paper of Universiy of Leipzig (2006). [3] R. Jarrow & F. Yu, Counerpary risk and he pricing of defaulable securiies, Journal of Finance, 56(5) (2001), [4] M.A. Kim & T.S. Kim, Credi defaul swap valuaion wih counerpary risk, Journal of Risk, 6(2)(2001), [5] S.Y. Leung & Y.K. Kwok, Credi defaul swap valuaion wih counerpary risk, Kyoo Economics Review, 74(1) (2005), [6] M.B. Walker, Credi defaul swaps wih counerpary risk: A calibraed Markov model. Working paper of Universiy of Torono (2005). [7] F. Yu, Correlaed defauls in inensiy-based models, Mahemaical Finance, 17(2) (2007),