Credit Risk in Lévy Libor Modeling: Rating Based Approach

Transcription

1 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

2 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

3 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

4 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

5 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

6 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

7 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

8 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 model additionally defaultable Libor rates L C(, T k ) such that ( ) BC(t, T j) B(t, T k ) are P Tk -local martingales 0 t T k T j () 5 / 36

9 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 () 6 / 36

10 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

11 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. () 7 / 36

12 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

13 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) = which is the stochastic infinitesimal generator of C. () 8 / 36

14 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) = 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) () 8 / 36

15 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) = 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

16 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) = 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

17 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

18 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

19 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

20 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

21 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. () 13 / 36

22 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

23 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. () 14 / 36

24 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

25 X is a special semimartingale with canonical decomposition t t t X t = b sds + csdws T + x(µ ν T )(ds, dx), 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

26 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

27 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 Next, define the forward measure P Tn 1 associated with date T n 1 via dp Tn δn 1L(t, Tn 1) = dp T 1 + δ n 1L(0, T n 1) Ft and proceed with modeling of L(, T n 2)... () 16 / 36

28 General step: for each T k (i) define the forward measure P Tk+1 via dp Tk+1 n 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 + t 0 0 σ(s, T k )dx T k+1 s R d x(µ ν T k+1 )(ds, dx) ), (1) () 17 / 36

29 General step: for each T k (i) define the forward measure P Tk+1 via dp Tk+1 n 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

30 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

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

32 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

33 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 () 19 / 36

34 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 () 19 / 36

35 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: () 19 / 36

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 () 19 / 36

37 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

38 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 (6) Define on this space the forward measures Q Tk by: for each tenor date T k Q Tk is obtained from Q T in the same way as P Tk from P T (k = 1,..., n 1) () 19 / 36

39 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

40 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

41 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. Theorem The (H)-hypothesis holds under all Q Tk, i.e. every (F, Q Tk )-local martingale is a (G, Q Tk )-local martingale. () 20 / 36

42 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). () 21 / 36

43 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

44 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 () 22 / 36

45 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

46 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

47 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 ). the forward Libor rate L i(, T k ) is obtained from relation 1 + δ k L i(t, T k ) = (1 + δ k L(t, T k )) i (1 + δ k H j(t, T k )). j=1 ) (2) () 23 / 36

48 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 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

49 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

50 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

51 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 () 27 / 36

52 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

53 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

54 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= 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) () 29 / 36

55 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= 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

56 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 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 ( ) 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

57 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). () 31 / 36

58 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

59 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

60 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. () 32 / 36

61 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

62 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

63 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 ] () 34 / 36

64 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

65 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]. () 35 / 36

66 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 () 35 / 36

67 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

68 T. Bielecki and M. Rutkowski, Credit Risk: Modeling, Valuation and Hedging, Springer, E. Eberlein, W. Kluge, and P. J. Schönbucher, The Lévy Libor model with default risk, Journal of Credit Risk 2, 3-42, E. Eberlein and F. Özkan, The Lévy LIBOR model. Finance and Stochastics 9, , R. Elliott, M. Jeanblanc, and M. Yor, On models of default risk, Mathematical Finance 10, , 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, P. Protter, Stochastic Integration and Differential Equations, Springer, K.-I. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, () 36 / 36