Credit Risk in Lévy Libor Modeling: Rating Based Approach

Credit Risk in Lévy Libor Modeling: Rating Based Approach Zorana Grbac Department of Math. Stochastics, University of Freiburg Joint work with Ernst Eberlein Croatian Quants Day University of Zagreb, 9th April 2010 () 1 / 36

Libor market models introduction Discrete tenor structure: 0 = T 0 < T 1 <... < T n = T, with δ k = T k+1 T k T 0 T 1 T 2 T 3 T n 1 T n = T () 2 / 36

Libor market models introduction Discrete tenor structure: 0 = T 0 < T 1 <... < T n = T, with δ k = T k+1 T k T 0 T 1 T 2 T 3 T n 1 T n = T Default-free zero coupon bonds: B(, T 1),..., B(, T n) () 2 / 36

Libor market models introduction Discrete tenor structure: 0 = T 0 < T 1 <... < T n = T, with δ k = T k+1 T k T 0 T 1 T 2 T 3 T n 1 T n = T Default-free zero coupon bonds: B(, T 1),..., B(, T n) Forward Libor rate at time t T k for the accrual period [T k, T k+1 ] L(t, T k ) = 1 ( ) B(t, Tk ) δ k B(t, T k+1 ) 1 () 2 / 36

Libor market models introduction Discrete tenor structure: 0 = T 0 < T 1 <... < T n = T, with δ k = T k+1 T k T 0 T 1 T 2 t T 3 T k T k+1 T n 1 T n = T Default-free zero coupon bonds: B(, T 1),..., B(, T n) Forward Libor rate at time t T k for the accrual period [T k, T k+1 ] L(t, T k ) = 1 ( ) B(t, Tk ) δ k B(t, T k+1 ) 1 () 3 / 36

Libor market models introduction Discrete tenor structure: 0 = T 0 < T 1 <... < T n = T, with δ k = T k+1 T k T 0 T 1 T 2 t T 3 T k T k+1 T n 1 T n = T Defaultable zero coupon bonds with credit ratings: B C(, T 1),..., B C(, T n) Defaultable forward Libor rate at time t T k for the accrual period [T k, T k+1 ] L C(t, T k ) = 1 ( ) BC(t, T k ) δ k B C(t, T k+1 ) 1 () 4 / 36

Libor modeling modeling under forward martingale measures, i.e. risk-neutral measures that use zero-coupon bonds as numeraires on a given stochastic basis, construct a family of Libor rates L(, T k ) and a collection of mutually equivalent probability measures P Tk such that ( ) B(t, Tj) B(t, T k ) are P Tk -local martingales 0 t T k T j () 5 / 36

Credit risk with ratings Credit risk: risk associated to any kind of credit-linked events (default, changes in the credit quality etc.) Credit rating: measure of the credit quality (i.e. tendency to default) of a company Credit ratings identified with elements of a finite set K = {1, 2,..., K}, where 1 is the best possible rating and K is the default event Credit migration is modeled by a conditional Markov chain C with state space K, where K is the absorbing state Default time τ: the first time when C reaches state K, i.e. τ = inf{t > 0 : C t = K} () 6 / 36

Defaultable bonds with ratings Consider defaultable bonds with credit migration process C and fractional recovery of Treasury value q = (q 1,..., q K 1) upon default Payoff of such a bond at maturity equals B C(T k, T k ) = 1 {τ>tk } + q Cτ 1 {τ Tk } K 1 = 1 {CTk =i} + q Cτ 1 {CTk =K}, i=1 where C τ denotes the pre-default rating. time-t price of such a defaultable bond can be expressed as K 1 B C(t, T k ) = B i(t, T k )1 {Ct=i} + q Cτ B(t, T k )1 {Ct=K}, i=1 where B i(t, T k ) represents the bond price at time t provided that the bond has rating i during the time interval [0, t]. We have B i(t k, T k ) = 1, for all i. () 7 / 36

Canonical construction of C Let (Ω, F T, F = (F t) 0 t T, P T ) be a given complete stochastic basis. Let Λ = (Λ t) 0 t T be a matrix-valued F-adapted stochastic process λ 11(t) λ 12(t)... λ 1K(t) λ 21(t) λ 22(t)... λ 2K(t) Λ(t) =........ 0 0... 0 which is the stochastic infinitesimal generator of C. Enlarge probability space (Ω, F T, P T ) ( Ω, G T, Q T ) and use canonical construction to construct C (Bielecki and Rutkowski, 2002) The process C is a conditional Markov chain relative to F if for every 0 t s and any function h : K R E QT [h(c s) F t F C t ] = E QT [h(c s) F t σ(c t)], where F C = (F C t ) denotes the filtration generated by C. (skip details) () 8 / 36

Canonical construction - details Let Λ = (Λ t) 0 t T be a matrix-valued F-adapted stochastic process on (Ω, F T, P T ) λ 11(t) λ 12(t)... λ 1K(t) λ 21(t) λ 22(t)... λ 2K(t) Λ(t) =........ 0 0... 0 where λ ij are nonnegative processes, integrable on every [0, t] and λ ii(t) = j K\{i} λij(t). Let µ = (δ ij, j K) be a probability distribution on Ω = K. Define ( Ω, G T, Q T ) = (Ω Ω U Ω, F T F U 2 Ω, P T P U µ), On (Ω U, F U, P U ) a sequence (U i,j), i, j N, of mutually independent random variables, uniformly distributed on [0, 1]. () 9 / 36

The jump times τ k are constructed recursively as { ( ) } τk 1 +t τ k := τ k 1 + inf t 0 : exp λ Ck 1 C k 1 (u)du U 1,k, with τ 0 := 0. τ k 1 The new state at the jump time τ k is defined as C k := C(U 2,k, C k 1, τ k ), with C 0(ω, ω U, ω) = ω and where C : [0, 1] K R + Ω K is any mapping such that for any i, j K, i j, it holds if λ ii(t) < 0 and 0, if λ ii(t) = 0. Finally, for every t 0 Leb ({u [0, 1] : C(u, i, t) = j}) = λij(t) λ ii(t), C t := C k 1, for t [τ k 1, τ k ), k 1. () 10 / 36

Definition The process C is a conditional Markov chain relative to F, i.e. for every 0 t s and any function h : K R it holds E QT [h(c s) F t F C t ] = E QT [h(c s) F t σ(c t)], where F C = (F C t ) denotes the filtration generated by C. Proposition The conditional expectations with respect to enlarged σ-algebras can be expressed in terms of F t-conditional expectations. It holds E QT [Y F t σ(c t)] = K i=1 E QT [Y1 {Ct=i} F t] 1 {Ct=i} E QT [1, {Ct=i} F t] for any G-measurable random variable Y. () 11 / 36

Properties of C (a) for every t s u and any function h : K R a stronger version of conditional Markov property holds: E QT [h(c s) F u F C t ] = E QT [h(c s) F u σ(c t)] (b) for every t s and B F C t : E QT [1 B F s] = E QT [1 B F t] (c) F-conditional Chapman-Kolmogorov equation where P(t, s) = [p ij(t, s)] i,j K and p ij(t, s) := P(t, s) = P(t, u)p(u, s), QT (Cs = j, Ct = i Fs) Q T (C t = i F s) (d) F-conditional forward Kolmogorov equation dp(t, s) ds = P(t, s)λ(s) () 12 / 36

The progressive enlargement of filtration G = (G t) 0 t T, where satisfies the (H)-hypothesis: G t := F t F C t, (H) Every local F-martingale is a local G-martingale. () 13 / 36

The progressive enlargement of filtration G = (G t) 0 t T, where satisfies the (H)-hypothesis: G t := F t F C t, (H) Every local F-martingale is a local G-martingale. It is well-known that (H) is equivalent to (H1) E QT [Y F T ] = E QT [Y F t], for any bounded, F C t -measurable random variable Y. But this follows easily from property E QT [1 B F s] = E QT [1 B F t], t s, B F C t, which is proved as a consequence of the canonical construction. () 13 / 36

Risk-free Lévy Libor model (Eberlein and Özkan, 2005) Let (Ω, F T, F = (F t) 0 t T, P T ) be a complete stochastic basis. as driving process take a time-inhomogeneous Lévy process X = (X 1,..., X d ) whose Lévy measure satisfies certain integrability conditions, i.e. an adapted, cádlág process with X 0 = 0 and such that (1) X has independent increments (2) the law of X t is given by its characteristic function ( t ) E[exp(i u, X t )] = exp θ s(iu)ds with θ s(iu) = i u, b s 1 2 u, csu + R d 0 ) (e i u,x 1 i u, x Fs T (dx). () 14 / 36

X is a special semimartingale with canonical decomposition t t t X t = b sds + csdws T + x(µ ν T )(ds, dx), 0 0 0 R d where W T denotes a P T -standard Brownian motion and µ is the random measure of jumps of X with P T -compensator ν T. We assume that b = 0. () 15 / 36

Construction of Libor rates Begin by specifying the dynamics of the most distant Libor rate under P T (regarded as the forward measure associated with date T ) ( t t L(t, T n 1) = L(0, T n 1) exp b L (s, T n 1)ds + σ(s, T n 1)dX s ), 0 where the drift is chosen in such a way that L(, T n 1) becomes a P T -martingale: b L (s, T n 1) = 1 σ(s, Tn 1), csσ(s, Tn 1) 2 ( ) e σ(s,tn 1),x 1 σ(s, T n 1), x F T (dx). R d 0 s () 16 / 36

General step: for each T k (i) define the forward measure P Tk+1 via dp Tk+1 n 1 1 + δ l L(t, T l ) = dp T 1 + δ l L(0, T l ) = B(0, T ) B(t, T k+1 ) B(0, T k+1 ) B(t, T ). Ft l=k+1 (ii) the dynamics of the Libor rate L(, T k ) under this measure ( t t L(t, T k ) = L(0, T k ) exp b L (s, T k )ds + where X T k+1 t = t 0 0 csdw T k+1 s + with P Tk+1 -Brownian motion W T k+1 and ν T k+1 (ds, dx) = n 1 l=k+1 t 0 0 σ(s, T k )dx T k+1 s R d x(µ ν T k+1 )(ds, dx) ), (1) ( ) δl L(s, T l ) 1 + δ l L(s, T l ) (e σ(s,t l),x 1) + 1 ν T (ds, dx). The drift term b L (s, T k ) is chosen such that L(, T k ) becomes a P Tk+1 -martingale. () 17 / 36

This construction guarantees that the forward bond price processes ( ) B(t, Tj) B(t, T k ) 0 t T j T k are martingales for all j = 1,..., n under the forward measure P Tk associated with the date T k (k = 1,..., n). The arbitrage-free price at time t of a contingent claim with payoff X at maturity T k is given by = B(t, T k )E PTk [X F t]. π X t () 18 / 36

How to include credit risk with ratings in the Lévy Libor model? () 19 / 36

How to include credit risk with ratings in the Lévy Libor model? (1) Use defaultable bonds with ratings to introduce a concept of defaultable Libor rates () 19 / 36

How to include credit risk with ratings in the Lévy Libor model? (1) Use defaultable bonds with ratings to introduce a concept of defaultable Libor rates (2) Adopt the backward construction of Eberlein and Özkan (2005) to model default-free Libor rates (3) Define and model the pre-default term structure of rating-dependent Libor rates To include credit migration between different rating classes: (4) Enlarge probability space: (Ω, F, F, P T ) ( Ω, G, G, Q T ) and construct the migration process C (5) The (H)-hypothesis X remains a time-inhomogeneous Lévy process with respect to Q T and G with the same characteristics () 19 / 36

Conditional Markov chain C under forward measures Note that dq Tk dq T = ψ k, where ψ k is an F Tk -measurable random variable with expectation 1. () 20 / 36

Conditional Markov chain C under forward measures Note that dq Tk dq T = ψ k, where ψ k is an F Tk -measurable random variable with expectation 1. Theorem Let C be a canonically constructed conditional Markov chain with respect to Q T. Then C is a conditional Markov chain with respect to every forward measure Q Tk and p Q T k ij (t, s) = p Q T ij (t, s) i.e. the matrices of transition probabilities under Q T and Q Tk are the same. () 20 / 36

Rating-dependent Libor rates The forward Libor rate for credit rating class i L i(t, T k ) := 1 ( ) Bi(t, T k ) δ k B i(t, T k+1 ) 1, i = 1, 2,..., K 1 We put L 0(t, T k ) := L(t, T k ) (default-free Libor rates). The corresponding discrete-tenor forward inter-rating spreads H i(t, T k ) := Li(t, T k) L i 1(t, T k ) 1 + δ k L i 1(t, T k ) () 21 / 36

Observe that the Libor rate for the rating i can be expressed as 1 + δ k L i(t, T k ) = (1 + δ k L i 1(t, T k ))(1 + δ k H i(t, T k )) i = (1 + δ k L(t, T k )) (1 + δ k H j(t, T k )) }{{}}{{} j=1 default-free Libor spread j 1 j Idea: model H j(, T k ) as exponential semimartingales and thus ensure automatically the monotonicity of Libor rates w.r.t. the credit rating: L(t, T k ) L 1(t, T k ) L K 1(t, T k ) = worse credit rating, higher interest rate () 22 / 36

Pre-default term structure of rating-dependent Libor rates For each rating i and tenor date T k we model H i(, T k ) as ( t H i(t, T k ) = H i(0, T k ) exp b H i (s, T k )ds + with initial condition 0 t 0 γ i(s, T k )dx T k+1 s H i(0, T k ) = 1 ( ) Bi(0, T k )B i 1(0, T k+1 ) δ k B i 1(0, T k )B i(0, T k+1 ) 1. X T k+1 is defined as earlier and b H i (s, T k ) is the drift term (we assume b H i (s, T k ) = 0, for s > T k H i(t, T k ) = H i(t k, T k ), for t T k ). ) (2) () 23 / 36

Theorem Assume that L(, T k ) and H i(, T k ) are given by (1) and (2). Then: (a) The rating-dependent forward Libor rates satisfy for every T k and t T k L(t, T k ) L 1(t, T k ) L K 1(t, T k ), i.e. Libor rates are monotone with respect to credit ratings. (b) The dynamics of the Libor rate L i(, T k ) under P Tk+1 is given by where ( t L i(t, T k ) = L i(0, T k ) exp b L i (s, T k )ds + + 0 t 0 t 0 csσ i(s, T k )dw T k+1 s ), R d S i(s, x, T k )(µ ν T k+1 )(ds, dx) () 24 / 36

σ i(s, T k ) := l i(s, T k ) 1( ) l i 1(s, T k )σ i 1(s, T k ) + h i(s, T k )γ i(s, T k ) = l i(s, T k ) 1[ l(s, T k )σ(s, T k ) + i j=1 ] h j(s, T k )γ j(s, T k ) represents the volatility of the Brownian part and ( ) S i(s, x, T k ) := ln 1 + l i(s, T k ) 1 (β i(s, x, T k ) 1) controls the jump size. Here we set and h i(s, T k ) := δ kh i(s, T k ) 1 + δ k H i(s, T k ), l i(s, T k ) := δ kl i(s, T k ) 1 + δ k L i(s, T k ), ( ) β i(s, x, T k ) := β i 1(s, x, T k ) 1 + h i(s, T k )(e γ i(s,t k ),x 1) ( ) = 1 + l(s, T k )(e σ(s,tk),x 1) i j=1 ( ) 1 + h j(s, T k )(e γ j(s,t k ),x 1). () 25 / 36

L(t, T n 1) L i 1(t, T n 1) H i (t,t n 1 ) L i(t, T n 1) L(t, T k ) L i 1(t, T k ) H i (t,t k ) L i(t, T k ) L(t, T k 1 ) L i 1(t, T k 1 ) H i (t,t k 1 ) L i(t, T k 1 ) L(t, T 1) L i 1(t, T 1) H i (t,t 1 ) L i(t, T 1) Default-free Rating i 1 Rating i Figure: Connection between subsequent Libor rates () 26 / 36

No-arbitrage condition for the rating based model Recall the defaultable bond price process with fractional recovery of Treasury value q K 1 B C(t, T k ) = B i(t, T k )1 {Ct=i} + q Cτ B(t, T k )1 {Ct=K}. i=1 No-arbitrage: the forward bond price process B C(, T k ) B(, T j) must be a Q Tj -local martingale for every k, j = 1,..., n 1. () 27 / 36

No-arbitrage condition for the rating based model Recall the defaultable bond price process with fractional recovery of Treasury value q K 1 B C(t, T k ) = B i(t, T k )1 {Ct=i} + q Cτ B(t, T k )1 {Ct=K}. i=1 Or equivalently: the forward bond price process B C(, T k ) B(, T = BC(, T k) B(, T j) j) B(, T k ) B(, T k ) }{{} dq Tk dq Tj must be a Q Tk -local martingale for every k = 1,..., n 1. () 28 / 36

We postulate that the forward bond price process is given by B C(t, T k ) B(t, T k ) := = K 1 i k 1 i=1 j=1 l=0 i=1 1 1 + δ l H j(t, T l ) } {{ } :=H(t,T k,i) t e 0 λ i (s)ds 1 {Ct=i} + q Cτ 1 {Ct=K} K 1 t H(t, T k, i)e 0 λ i (s)ds 1 {Ct=i} + q Cτ 1 {Ct=K}, (3) where λ i is some F-adapted process that is integrable on [0, T ]. (go to DFM) Note that this specification is consistent with the definition of H i which implies the following connection of bond prices and inter-rating spreads: B j(t, T k ) B j 1(t, T k ) = Bj(t, T k 1) 1 B j 1(t, T k 1 ) 1 + δ k 1 H j(t, T k 1 ) and relation B i(t, T k ) B(t, T k ) = B1(t, T k) B(t, T k ) i j=2 B j(t, T k ) B j 1(t, T k ). () 29 / 36

Lemma Let T k be a tenor date and assume that H j(, T k ) are given by (2). The process H(, T k, i) has the following dynamics under P Tk H(t, T k, i) = H(0, T k, i) ( E t b H (s, T k, i)ds 0 + 0 R d j=1 l=1 cs 0 i k 1 h j(s, T l )γ j(s, T l )dw T k s j=1 l=1 ( i ) k 1 ( ) 1 1 + h j(s, T l )(e γ j(s,t l ),x 1) 1 (µ ν T k )(ds, dx) ), where b H (s, T k, i) is the drift term. () 30 / 36

No-arbitrage condition Theorem Let T k be a tenor date. Assume that the processes H j(, T k ), j = 1,..., K 1, are given by (2). Then the process B C(,T k ) B(,T k defined in (3) is a local martingale with respect to the ) forward measure Q Tk and filtration G iff: for almost all t T k on the set {C t K} ( b H e t ) 0 λ C t (s)ds (t, T k, C t) + λ Ct (t) = 1 q Ct λ CtK(t) (4) H(t, T k, C t) K 1 (1 H(t, T k, j)e ) t 0 λ j (s)ds + j=1,j C t H(t, T k, C t)e t 0 λ C t (s)ds λ Ctj(t). Sketch of the proof: Use the fact that the jump times of the conditional Markov chain C do not coincide with the jumps of any F-adapted semimartingale, use some martingales related to the indicator processes 1 {Ct=i}, i K, and stochastic calculus for semimartingales. () 31 / 36

Defaultable forward measures Assume that B C(,T k ) is a true martingale w.r.t. forward measure B(,T k ) QT k. (back to DFP) () 32 / 36

Defaultable forward measures Assume that B C(,T k ) is a true martingale w.r.t. forward measure B(,T k ) QT k. (back to DFP) The defaultable forward measure Q C,Tk for the date T k is defined on (Ω, G Tk ) by dq C,Tk := B(0, T k) B C(t, T k ) dq Tk B C(0, T k ) B(t, T k ). Gt This corresponds to the choice of B C(, T k ) as a numeraire. Proposition The defaultable Libor rate L C(, T k ) is a martingale with respect to Q C,Tk+1 and dq C,Tk = BC(0, T k+1) B C(0, T k ) (1 + δ kl C(t, T k )). Gt dq C,Tk+1 () 32 / 36

Pricing problems I: Defaultable bond Proposition The price of a defaultable bond with maturity T k and fractional recovery of Treasury value q at time t T k is given by [ K 1 B C(t, T k )1 {Ct K} = B(t, T k ) E QTk [1 p ik(t, T k ) F t] i=1 1 {Ct=i} K 1 + j=1 ] E QTk [1 {t<τ Tk }1 {Ct=i}1 {Cτ =j}q j F t]. E QTk [1 {Ct=i} F t] () 33 / 36

Pricing problems II: Credit default swap consider a maturity date T m and a defaultable bond with fractional recovery of Treasury value q as the underlying asset protection buyer pays a fixed amount S periodically at tenor dates T 1,..., T m 1 until default protection seller promises to make a payment that covers the loss if default happens: 1 q Cτ has to paid at T k+1 if default occurs in (T k, T k+1 ] Proposition The swap rate S at time 0 is equal to m k=2 B(0, T k) K 1 j=1 E Q Tk [(1 q j)1 {Tk 1 <τ T k,c τ =j} ] S = m 1 k=1 B(0, T, k)e QTk [1 p ik(0, T k )] if the observed class at time zero is i. () 34 / 36

Pricing problems III: use of defaultable measures Proposition Let Y be a promised G Tk -measurable payoff at maturity T k of a defaultable contingent claim with fractional recovery q upon default and assume that Y is integrable with respect to Q Tk. The time-t value of such a claim is given by π t (Y) = B C(t, T k )E QC,Tk [Y G t]. Example: a cap on the defaultable forward Libor rate The time-t price of a caplet with strike K and maturity T k on the defaultable Libor rate is given by C t(t k, K) = δ k B C(t, T k+1 )E QC,Tk+1 [(L C(T k, T k ) K) + G t] and the price of the defaultable forward Libor rate cap at time t T 1 is given as a sum C t(k) = n δ k 1 B C(t, T k )E QC,Tk [(L C(T k 1, T k 1 ) K) + G t]. k=1 () 35 / 36

T. Bielecki and M. Rutkowski, Credit Risk: Modeling, Valuation and Hedging, Springer, 2002. E. Eberlein, W. Kluge, and P. J. Schönbucher, The Lévy Libor model with default risk, Journal of Credit Risk 2, 3-42, 2006. E. Eberlein and F. Özkan, The Lévy LIBOR model. Finance and Stochastics 9, 327-348, 2005. R. Elliott, M. Jeanblanc, and M. Yor, On models of default risk, Mathematical Finance 10, 179-195, 2000. Z. Grbac, Credit Risk in Lévy Libor Modeling: Rating Based Approach, Ph.D. Thesis, University of Freiburg, 2009 J. Jacod and A.N. Shiryaev, Limit theorems for stochastic processes, Springer, 2003. P. Protter, Stochastic Integration and Differential Equations, Springer, 2005. K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999. () 36 / 36