Contagion models with interacting default intensity processes Yue Kuen KWOK Hong Kong University of Science and Technology This is a joint work with Kwai Sun Leung. 1
Empirical facts Default of one firm may trigger the default of other related firms. Default times tend to concentrate in certain periods of time (clusters of default). Sources of dependence between defaults common macro-economic factors and sectors correlation of the individual credit quality process to the economic cycle and sector (industrial/geographic) indices default contagion (i) direct economic links between firms (ii) information effect e.g. accounting scandal of WorldCom 2
Default correlation occurrence of multiple defaults in a portfolio of risky assets. Generate the distribution of default losses. Optimize the regulatory capital in credit portfolios, under BASEL II (banks build their own internal monitoring systems). Pricing of correlation-dependent products, like the tranches in collateralized debt obligations (exposure to prioritized losses in a portfolio of loans/bonds), requires modeling not only the credit process for each individual obligor but also the correlation of the credit processes. 3
Modeling of correlation of defaults Copula approach A copula function transforms marginal probabilities into joint probabilities. Let τ i denote the default time of Firm i and let F i (t) = P(τ i t). The joint default probability is given by F(t 1,, t n ) = P(τ 1 t 1,, τ n t n ) where C d is the copula function. = C d (F 1 (t 1 ),, F n (t n )), CreditMetrics (industrial software): Assume that each credit index is Gaussian, they use the Gaussian copula function to define the joint correlation of credit processes: C(u 1,, u n ) = N n (N 1 (u 1 ),, N 1 (u n );Ω). 4
Conditionally independent defaults This approach introduces credit risk dependence through the dependence of the obligors default intensities on a common set of state variables. The default rates are independent once we fix the realization of state variables. A random default time vector (τ 1,, τ n ) T has a d-dimensional conditional independent structure with conditioning macro-economic state variables Ψ = (ψ 1 ψ d ) T, with d < n, such that conditional on Ψ, the random default times τ 1,, τ n are independent. 5
Infectious defaults (Davis and Lo, 2001) Once a firm defaults, it may bring down other firms with it. Let H i be the default indicator of Firm i. Let Y ij be an infection variable, which equals 1 when the default of Firm i triggers the default of Firm j. The default of Firm i arises either due to firm specific causes or infectious effects. Let X i denote the default indicator of Firm i due to firm specific causes. H i = X i + (1 X i ) 1 (1 X j Y ji ) direct default of firm i j i default of firm j triggers default of firm i We then randomize X i and Y ji to generate the dependence structure of defaults.. 6
Information structure The macroeconomic variables are described by a d-dimensional stochastic process Ψ = (ψ t ) t [0,T]. The overall state of the system is Γ t = (ψ t, H t ). The information available to the investor in the market at time t include the path history of macroeconomic variables and default status of the portfolio up to time t. The filtrations are generated collectively by the information contained in the state variables and the default processes so that F Ψ t = σ(ψ s : 0 s t) F i t = σ(h i s : 0 s t), i = 1,2,, N. F t = F Ψ t F 1 t F2 t FN t. 7
Martingale default intensity and default time (reduced form approach) The default time τ i of Firm i is defined by { t } τ i = inf t : 0 λi s ds E i and {E i } i I is a set of independent unit exponential random variables. Here, λ i t (ψ t, H t) is the martingale default intensity of Firm i. This definition ensures that is a F t -martingale. H i t =1 {t τ i } H i t t τi 0 λ i (ψ s, H s ) ds is the single-jump process associated with the default time τ i. The default times τ i are independent given the whole history of Γ t and default correlation arises due to correlation of the intensities induced by their common dependence on Γ t. 8
Default contagion with interacting intensities The default status of the assets in the portfolio is given by the default indicator process H t = (H 1 t H 2 t HN t ) {0,1}N = E, where H i t =1 {τ i t} and E is the state space of the default status. Characterize the default intensity of firm i by λ t,i (H t ) = a i + j i b i,j 1 {τj t}, t τ i, (A) and λ t,i = 0 for t > τ i. Also, a i 0 and b i,j are constants such that λ t,i is non-negative. The jumps are independent of the order in which the defaults have occurred. 9
The default intensity for Firm 5 when the first default time T 1 = τ 7, the second default time T 2 = τ 3 and the third default time T 3 = τ 1. The successive defaults of Firm 7, Firm 3 and Firm 1 put Firm 5 at a higher risk. 10
Matrix-analytic method The contagion model is seen to be a Markov jump process. Conditional on the given trajectory of Ψ, the process H is a timeinhomogeneous Markov chain with initial value y S, where S is the state space of H. With respect to a given state y S, we may flip its i th component from 0 to 1, resulting in the flipped state ỹ i. For example, take y = (1 0 0) S = {0,1} 3, then ỹ 2 = (1 1 0) and ỹ 3 = (1 0 1). For y i, y j S, y j = ỹ k i for some k means y j is obtained by flipping the k th component in y i. 11
Infinitesimal generator Λ of the process H t For y i, y j S, i j, the transition rate Λ ij equals λ k (ψ, y j ) when y j can be obtained from y i by flipping its k th element from 0 to 1. The infinitesimal generator Λ [ψ] (t) = (Λ ij (t ψ)) S S for H at time t given Ψ t = ψ is defined by (a) for i j Λ ij (t ψ) = { [1 yi (k)]λ k (ψ, y j ), if y j = ỹ k i for some k 0 else. (1a) Here, y i (k) denotes the k th component of y i. Once the k th firm has defaulted (signified by y i (k) = 1), it stays at the default state. (b) for i = j Λ ii (t ψ) = j iλ ij (t ψ) = N k=1 [1 y i (k)]λ k (ψ, y i ). (1b) 12
Illustration of the construction for m = 3. Arrows indicate possible transitions, and the transition intensities are given on top of the arrows. {0} no default; {i} default of Firm i; {i, j} default of Firms i and j; {1,2,3} default of all 3 firms. 13
Define the conditional transition density matrix on ψ s = ψ as P(t, s ψ ) = (p ij (t,s ψ )) S S = (p(t, s, y i, y j ψ )) S S. Kolmogorov backward equation dp(t, s ψ ) dt = Λ [ψ] (t)p(t,s ψ ), P(s, s ψ ) = I. The (i, j) th entry p ij (t, s ψ ) satisfies the following system of ODE: S dp ij (t, s, y i, y j ψ ) = Λ ik (t ψ)p kj (t, s, y dt k, y j ψ ). (2a) k=1 p ij (s, s, y i, y j ψ ) =1 {yj } (y i ) Since default state is absorbing, Λ is upper triangular. 14
Using the results in eqs (1a,b), eq. (2a) can be expressed as dp ij (t, s, y i, y j ψ ) + N k=1 dt [1 y i (k)]λ k (ψ, y i )[p ij (t, s,ỹ k i, y j ψ ) p ij (t, s,y i, y j ψ )] = 0 with auxiliary condition: p ij (s,s, y i, y j ψ ) =1 {yj } (y i ). (2b) Marginal distribution of default times F i (t i ) = P r [τ i t i ] = y j (i)=1 p ij (0, t i ψ) dµ ψ (ω), where we sum over all states j with default of the i th obligor [y j (i) = 1] and subsequently integrate over the distribution µ ψ (ω). Here, µ ψ (ω) is the probability measure which gives the law of Ψ. 15
Three-Firm Model The inter-dependent default intensities of the 3 firms are defined as λ A t λ B t = a 1 + b 12 1 {τb t} + b 131 {τc t} + b 141 {τb t,τ C t} = a 2 + b 21 1 {τa t} + b 231 {τc t} + b 241 {τa t,τ C t} λ C t = a 3 + b 31 1 {τa t} + b 321 {τb t} + b 341 {τa t,τ B t}. We assume an extra jump in default intensity if the other two firms have defaulted, allowing the interaction between the default events on the intensity of surviving firms. The state space S of H = (H A t, HB t, HC t ) is given by S = {(0,0,0),(1,0,0),(0,1,0),(0,0,1),(1,1,0),(1,0,1),(0,1,1),(1,1,1)}. State 1 (0, 0, 0) State 2 (1, 0, 0) State 3 (0, 1, 0) State 4 (0, 0, 1) State 5 (1, 1, 0) State 6 (1, 0, 1) State 7 (0, 1, 1) State 8 (1, 1, 1) 16
Λ = The infinitesimal generator Λ of the process H is given by (a 1 + a 2 + b 3 ) a 0 a 2 a 3 0 0 0 0 0 (a 2 + b 21 0 0 a 2 + b 21 a 3 + b 31 0 0 +a 3 + b 31 ) 0 0 (a 0 + b 12 0 a 1 + b 12 0 a 3 + b 32 0 +a 3 + b 32 ) 0 0 0 (a 1 + b 13 a 0 + b 13 a 2 + b 23 0 +a 2 + b 21 ) 0 0 0 0 (a 3 + b 31 0 0 (a 3 + b 31 +b 32 + b 34 ) +b 32 + b 33 ) 0 0 0 0 0 (a 2 + b 21 0 (a 2 + b 21 +b 23 + b 24 ) +b 23 + a 24 0 0 0 0 0 0 (a 1 + b 12 (a 1 + b 12 +b 13 + b 14 ) +b 13 + b 14 ) 0 0 0 0 0 0 0 0 For example, consider the transition rate from State 2 : (1 0 0) to State 5 : (1 1 0) and State 6 : (1 0 1): Λ 25 = a 2 + b 21 Λ 26 = a 3 + b 31 Λ 21 = Λ 23 = Λ 24 = Λ 27 = Λ 28 = 0 and Λ 22 = 1 Λ 25 Λ 26. 17
Simulation algorithm Total hazard construction (Yu, 2007) Define Λ i (t m) = τm +t τ m λ i (u m) du which is the total hazard accumulated by firm i between the m th default and t, assuming no observed default between τ m and t. The total hazards accumulated by the firms until they defaulted are iid unit exponential random variables. One can map a set of iid unit exponentials back to the original default stopping times Λ 1 i (x n) = inf{t : Λ i (t n) x}, x 0. A simulation procedure can be constructed for defining a collection of default times based on unit exponentials. 18
Example λ A t = (a 1 + a 2 1 {t τ B } )1 {t<τ A }, λ B t = (b 1 + b 2 1 {t τ A } )1 {t<τ B }. With no default up to t, the total hazards accumulated from 0 to t are Λ A (t) = a 1 t and Λ B (t) = b 1 t. If firm A has defaulted at t 1, the total hazard accumulated by firm B from t 1 to t 1 + t is Λ B (t A, t 1 ) = (b 1 + b 2 )t. Similarly, if firm B has defaulted at t 2, one can set Λ A (t B, t 2 ) = (a 1 + a 2 )t. Using the inverse of these functions, two default times τ A and τ B can be constructed from two independent unit exponential random variables E A and E B as: 19
τ A = τ B = E A a 1, E B b 1 + 1 a 1 +a 2 ( E A a 1 b1 E B), E A a 1 + 1 b 1 +b 2 ( E B b 1 a1 E A), E B b 1, E A a 1 EB b 1, E A a 1 > EB b 1, E A a 1 EB b 1, E A a 1 > EB b 1. The joint density of τ A and τ B, which is also shared by the original default times τ A and τ B, can be derived: f(t 1, t 2 ) = { a1 (b 1 + b 2 )e (a 1 b 2 )t 1 (b 1 +b 2 )t 2, t 1 t 2, b 1 (a 1 + a 2 )e (b 1 a 2 )t 2 (a 1 +a 2 )t 1, t 1 > t 2. 20
Multiple-firm model with simple contagion structure λ i t = [ a 1 + a 2 1 {t τf }] 1{t<τ i }, i = 1,2,, N and τ F = min(τ 1, τ N ) is the first-to-default time. A single credit event defined as a large monthly change in an individual corporate bond yield spread can affect the entire corporate bond market index. The coefficient a 2 can be estimated from the average price reaction of a large cross-section of bonds to typical credit events. 21
The first-to-default time τ F can be written as ( ) E 1 τ F = min, E2,, EN. a 1 a 1 a 1 The individual default times are given through the respective unit exponential random variables by τ i = Ei a 1 τ F a 1 + a 2 + τ F, i = 1,2,, N. Using τ F = min(e 1 /a 1, τ F ), where τ F = min(e2, E N )/a 1, τ F is an exponential random variable with rate (N 1)a 1 independent of E, we obtain the marginal density of τ : f 1 (t 1 ) = (N 1)a 1(a 1 + a 2 )e (a 1+a 2 )t 1 Na 1 a 2 e Na 1t 1 (N 1)a 1 a 2. 22
Simulation scheme The total hazard accumulated by obligor i by time t as ψ i (t I n, T n ) = where n m=1 Λ i (t m t m 1 I m 1, T m 1 ) + Λ i (t t n I n, T n ), Λ i (s I m, T m ) = tm +s t m λ i (u I m, T m ) du is the total hazard accumulated by obligor i for a period of s following the m th default. It is assumed that there is no default between t n and t. The total hazards accumulated by τ = (τ 1,, τ I ) by the time they occur are independent unit exponential random variables. 23
We construct an inverse mapping which generates a set of random times from a collection of independent unit exponential random variables. Λ 1 i (x I n, T n ) = inf{s : Λ i (s I n, T n ) x}, x 0. The following recursive procedure constructs a new collection of random variables τ = ( τ 1,, τ I ): 1. Draw a collection of i.i.d.unit exponential random variables E = (E 1,, E I ). 2. Let and let k 1 = arg min 1 i I {Λ 1 i (E i )}, τ k 1 = Λ 1 k 1 (E k 1). 24
3. Assume that the values of ( τ k 1,, τ k m 1) are already determined as T m 1 = (t 1,, t m 1 ), where m 2. Define the defaulted set I m 1 = (k 1,, k m 1 ) and the remaining set I m 1 = (1,, n)\i m 1. Recall that ψ i (t I m 1, T m 1 ) is the total hazard accumulated by firm i to time t given the first m 1 defaults, let k m = and let arg min 1 Ĩ m 1 {Λ 1 i (E i ψ i (t m 1 I m 1, T m 1 ) I m 1, T m 1 )}, τ k m = t m 1 + Λ 1 k m (E k m ψ km (t m 1 I m 1, T m 1 ) I m 1, T m 1 ). 4. If m = I, then stop. Otherwise, increase m by 1 and go to Step 3. 25
Counterparty risk of credit default swap Protection Buyer CDS swap premium (fee payment up to ) T contingent payment (credit loss if < T) Protection Seller C = default time of Asset R T = maturity date of swap Reference Asset R 3 parties: Protection Seller (Counterparty), Protection Buyer, Reference Obligor 26
Protection Buyer pays periodic swap premium (insurance fee) to Protection Seller (counterparty) to acquire protection on a risky reference asset (compensation upon default). Before the 1997 crisis in Korea, Korean financial institutions are willing to offer protection on Korean bonds. The financial melt down caused failure of compensation payment on defaulting Korean bonds by the Korean Protection Sellers. Given that the counterparty (Protection Seller) may default on the contingent compensation payment, what is the impact of the counterparty risk on the swap premium? 27
How does the inter-dependent default risk structure between the Protection Seller and the Reference Obligor affect the swap rate? 1. Replacement cost (Protection Seller defaults earlier) If the Protection Seller C defaults prior to the Reference Entity, then the Protection Buyer renews the CDS with a new counterparty. Supposing that the default risks of the Protection Seller C and Reference Entity R are positively correlated, then there will be an increase in the swap rate of the new CDS. 2. Settlement risk (Reference Entity defaults earlier) The Protection Seller defaults during the settlement period after the default of Reference Entity. 28
Common external shock Assume the default intensity of the protection seller and reference entity to be subject to a positive jump in value upon the occurrence of an external shock event. The default intensities of the shock event and the default of R and C are given by λ R t = a R (t)[(α R 1)1 {τs t} + 1] λ C t = a C (t)[(α C 1)1 {τs t} + 1] λ S t = λ S. Here, the proportional factors α C and α R are assumed to be positive constants, with α C > 1 and α R > 1. Also, a R (t) and a C (t) are the default intensity of R and C prior to the arrival of the shock event S. 29
Write H t = (H R t H C t H S t ) and assign State 1:(0 0 0), State 3:(0 1 0), State 4:(0 0 1), State 7:(0 1 1). The marginal distribution for τ R is given by P r [τ R > T F t ] = p 11 (t, T) + p 13 (t,t) + p 14 (t, T) + p 17 (t, T) = e T t a R (u) du + λ S T t [ e λ S(T t) where the transition probabilities are given by p 11 (t, T) = e T t [a R(u)+a C (u)+λ S ] du e T s (α R 1)a R (u) du λ S (s t) ds ]. p 13 (t, T) = e T t [a R(u)+λ S ] du [1 e T t a C (u) du ] p 14 (t, T) = λ S e T t [a R(u)+a C (u)] du T t e T s [(α R 1)a R (u)+(α C 1)a C (u)] du λ S s ds. 30
Credit swap premium The swap premium payments are made continuously at a constant swap rate C(T). Let ρ to be the deterministic recovery rate of the reference asset upon default. The contingent compensation payment of 1 ρ is made by the protection seller during (t, t + dt] provided that there has been no default during (0, t) and default of the reference asset occurs during the infinitesimal time interval (t,t + dt]. The expected present value of contingent compensation payment over (t,t + dt] is (1 ρ)e rt [p 11 (0, t)a R (t) + p 14 (0, t)α R a R (t)] dt. 31
The probability of no default up to time t is given by p 11 (0, t) + p 14 (0, t) and the expected present value of the swap premium payment over (t, t + dt] is C(T)e rt [p 11 (0, t) + p 14 (0, t)] dt. By equating the expected present value of the swap premium payment and contingent compensation payment upon default over the whole period [0, T], we obtain C(T) = (1 ρ) T 0 e rt [p 11 (0, t)a R (t) + p 14 (0, t)α R a R (t)] dt T0 e rt, [p 11 (0,t) + p 14 (0, t)] dt where ρ is the recovery rate. 32
Calibration of the parameter functions The parameter function α R (t) in the intensity of α R t can be calibrated using the term structure of prices of defaultable bonds issued by the reference entity. Let B R (t, T) denote the time-t price of the defaultable bond with unit par and zero recovery upon default. B R (t, T) = e r(t t) E P [1 {τr >T } F t] = e r(t t) P r [τ R > T F t ]. Relation between the parameter function a R (t) and the term structure of B R (t, T): B R T (t,t) = rb R(t, T) + a R (T)B R (t, T) + a R (T)(α R 1)B R (t,t) a R (T)(α R 1)e T t a R (u) du e λ s (T t) r(t t). This gives an integral equation for a R (t). 33
Summary Contagion models with interacting default intensity processes Upward jumps in the default intensity of a non-defaulted firm upon the default of one of the default-correlated firms. Various formulations have been considered Matrix-analytic approach Total hazard construction Application to the analysis of counterparty risk in credit default swaps. 34