Credit-rating based L evy Libor model (joint work with Ernst - - PowerPoint PPT Presentation

Credit-rating based L´ evy Libor model

(joint work with Ernst Eberlein)

Zorana Grbac

Department of Math. Stochastics, University of Freiburg

Concluding Workshop, RICAM, Linz, 03. 12. 2008

Outline

L´ evy Libor model without risk Credit-rating based L´ evy Libor model Rating-dependent Libor rates Credit migration process Pricing problems Concluding remarks

L´ evy Libor model - notation

◮ T∗ > 0 fixed time horizon ◮ discrete tenor structure 0 = T0 < T1 < . . . < Tn = T∗,

with δk = Tk+1 − Tk

◮ default-free zero coupon bonds B(·, T1), . . . , B(·, Tn) ◮ forward Libor rate at time t ≤ Tk for the accrual period [Tk, Tk+1]

L(t, Tk) = 1 δk ✒ B(t, Tk) B(t, Tk+1) − 1 ✓ ,

The driving process

Let (Ω, FT∗, F = (Ft)0≤t≤T∗, I PT∗) be a complete stochastic basis We work with a time-inhomogeneous L´ evy process XT∗ with values in Rd, i.e an adapted, c´ adl´ ag processs with XT∗ = 0 and such that (1) XT∗ has independent increments (2) the law of XT∗

t

is given by its characteristic function I E[exp(iu, XT∗

t

)] = exp ✒❩ t θs(iu) ds ✓ , θs(iu) = iu, bs − 1 2u, csu + ❩

Rd

✏ eiu,x − 1 − iu, xFs(dx) ✑ . Here bs ∈ Rd, cs a symmetric, nonneg. definite d × d-matrix and FT∗

s

a L´ evy measure (s ∈ [0, T∗]). Assume the following integrability conditions: sup

0≤s≤T∗

✒ |bs| + ||cs|| + ❩

Rd(|x|2 ∧ 1)Fs(dx)

✓ < ∞ and sup

0≤s≤T∗

❩

|x|>1

expu, xFs(dx) < ∞ (u ∈ [−(1 + ε)M, (1 + ε)M]d).

XT∗ is a special semimartingale with canonical decomposition XT∗

t

= ❩ t bs ds + ❩ t √cs dWT∗

s

+ ❩ t ❩

Rd x(µX − νT∗)(ds, dx),

where WT∗ denotes standard Brownian motion and µX is the random measure of jumps of XT∗ with the compensator νT∗. The semimartingale characteristics of XT∗ is given by Bt = ❩ t bs ds, Ct = ❩ t cs ds, νT∗(ds, dx) = FT∗

s (dx) ds

We assume from now on that bs = 0.

Construction

The model is constructed by a backward induction procedure. Assumptions: (L.1) For every Tk there is a deterministic function σ(·, Tk) : [0, T∗] → Rd

+,

which represents the volatility of the forward Libor rate L(·, Tk). We assume that

n−1

❳

k=1

σj(s, Tk) ≤ M, for all s ∈ [0, T∗] and every coordinate j ∈ {1, . . . , d}, where M > 0 is a constant from the integrability condition for XT∗. If s > Tk, then σ(s, Tk) = 0. (L.2) The initial term structure B(0, Tk) is strictly positive and strictly decreasing in k.

Start by specifying the dynamics of the most distant Libor rate under I PT∗ (forward measure associated with date T∗) as L(t, Tn−1) = L(0, Tn−1) exp ✒❩ t bL(s, Tn−1, T∗) ds + ❩ t σ(s, Tn−1) dXT∗

s

✓ , where the drift is chosen in such a way that L(·, Tn−1) becomes a I PT∗-martingale, namely bL(s, Tn−1, T∗) = −1 2σ(s, Tn−1), csσ(s, Tn−1) − ❩

Rd

✏ eσ(s,Tn−1),x − 1 − σ(s, Tn−1), x ✑ FT∗

s (dx).

Next define a forward measure I PTn−1 associated with the date Tn−1 via dI PTn−1 dI PT∗ = 1 + δn−1L(Tn−1, Tn−1) 1 + δn−1L(0, Tn−1) and proceed with modeling of L(·, Tn−2).

For each k: (1) define the forward measure I PTk+1 via dI PTk+1 dI PT∗ =

n−1

❨

l=k+1

1 + δlL(Tk+1, Tl) 1 + δlL(0, Tl) . (2) the dynamics of the Libor rate L(·, Tk) under this measure L(t, Tk) = L(0, Tk) exp ✒❩ t bL(s, Tk, Tk+1) ds + ❩ t σ(s, Tk) dX

Tk+1 s

✓ , (1) where X

Tk+1 t

= ❩ t √cs dW

Tk+1 s

+ ❩ t ❩

Rd x(µX − νTk+1)(ds, dx)

with νTk+1(ds, dx) =

n−1

❨

l=k+1

δlL(s−, Tl) 1 + δlL(s−, Tl) ✏ (eσ(s,Tl)x − 1) + 1 ✑ νT∗(ds, dx) and the drift bL(s, Tk, Tk+1) is chosen such that it becomes a I PTk+1-martingale.

This construction guarantees that processes ✒ B(·, Tj) B(·, Tk) ✓ are martingales for all j = 1, . . . , n under the forward measure I PTk associated with the date Tk (k = 1, . . . , n). The t-time price of a contingent claim with payoff X at maturity Tk can be calculated as πX

t = B(t, Tk)I

EI

PTk [X|Ft].

Credit ratings

◮ 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

◮ these ratings correspond to the states of a conditional Markov chain C

with the absorbing state K

◮ the default time τ = the first time when C reaches state K

τ = inf{t > 0 : Ct = K}

Defaultable bonds with credit ratings

◮ defaultable bonds whose credit migration process is denoted by C and

with zero recovery upon default: BC(·, T1), . . . , BC(·, Tn)

◮ time-t price of such a defaultable bond can be expressed as

BC(t, Tk) = 1{Ct=1}B1(t, Tk) + · · · + 1{Ct=K−1}BK−1(t, Tk), where Bi(t, Tk) represents the bond price at time t provided that the bond has the rating i during the time interval [0, t].

◮ Payoff at maturity equals

BC(Tk, Tk) =

K−1

❳

i=1

1{CTk =i} = 1{τ>Tk} and it holds Bi(Tk, Tk) = 1, for all i.

Libor rates for different credit ratings

◮ The Libor rates for credit rating class i

Li(t, Tk) := 1 δk ✒ Bi(t, Tk) Bi(t, Tk+1) − 1 ✓ , i = 1, 2, . . . , K − 1. We put L0(t, Tk) := L(t, Tk) (default-free Libor rates)

◮ The forward credit spreads between two successive classes

Si(t, Tk) := Li(t, Tk) − Li−1(t, Tk), i = 1, 2, . . . , K − 1.

◮ The corresponding forward credit spread intensities

Hi(t, Tk) := Li(t, Tk) − Li−1(t, Tk) 1 + δkLi−1(t, Tk)

Observe that Libor rates for the rating i can be expressed as 1 + δkLi(t, Tk) = (1 + δkLi−1(t, Tk))(1 + δkHi(t, Tk)) = (1 + δkL(t, Tk)) ⑤ ④③ ⑥

default-free Libor i

❨

j=1

(1 + δkHj(t, Tk)) ⑤ ④③ ⑥

intensity between j and j−1

Idea: model Hj(·, Tk) as exponential processes and therefore ensure automatically the monotonicity of Libor rates w.r.t the credit rating: L(t, Tk) < L1(t, Tk) < · · · < LK−1(t, Tk) = ⇒ worse credit rating, higher interest rate

Assumptions: (RBL.1) For every i ∈ {1, . . . , K − 1} and every maturity Tk there is a deterministic function γi(·, Tk) : [0, T∗] → Rd

+, which represents the

volatility of the forward spread intensity Hi(·, Tk) for the rating i. We assume that γi(s, Tk) = 0 for s > Tk and that

n−1

❳

k=1

(σj(s, Tk) + γj

1(s, Tk) + · · · + γj K−1(s, Tk)) ≤ M,

for all s ∈ [0, T∗] and every coordinate j ∈ {1, . . . , d}. (RBL.2) The initial term structure Li(0, Tk) of Libor rates satisfies Li(0, Tk) > Li−1(0, Tk), for all k = 1, . . . , n − 1, i.e. Bi(0, Tk) Bi(0, Tk+1) > Bi−1(0, Tk) Bi−1(0, Tk+1).

Construction of rating-based Libor rates

For rating 1 and all settlement dates Tk we model H1(·, Tk) as H1(t, Tk) = H1(0, Tk) exp ✒❩ t bH1(s, Tk, Tk+1) ds + ❩ t γ1(s, Tk) dX

Tk+1 s

✓ with initial condition H1(0, Tk) = 1 δk ✒B1(0, Tk)B(0, Tk+1) B(0, Tk)B1(0, Tk+1) − 1 ✓ . XTk+1 is defined as earlier and bH1(s, Tk, Tk+1) is the drift term (we assume bH1(s, Tk, Tk+1) = 0, for s ≥ Tk ⇒ H1(t, Tk) = H1(Tk, Tk), for t ≥ Tk). Choose drift term in such a way that the process ✏◗k

l=0 1 1+δlH1(t,Tl)

✑

t≤Tk

becomes a I PTk+1-martingale (Kluge (2005)).

Hence we can introduce the following change of measure dI P1,Tk+1 dI PTk+1 := ◗k

l=0 1 1+δlH1(Tl,Tl)

◗k

l=0 1 1+δlH1(0,Tl)

= B(0, Tk+1) B1(0, Tk+1)

k

❨

l=0

1 1 + δlH1(Tl, Tl). I P1,Tk+1 - forward measure associated with the rating 1 and the settlement date Tk+1 Use Girsanov’s theorem for semimartingales. Denote h1(s, Tl, Tl+1) := δlH1(s, Tl) 1 + δlH1(s, Tl), s ≤ Tl. Then W

1,Tk+1 t

:= W

Tk+1 t

+ ❩ t

k

❳

l=1

h1(s−, Tl, Tl+1)γ1(s, Tl)√cs ds is a I P1,Tk+1- Brownian motion and ν1,Tk+1(dt, dx) :=

k

❨

l=1

✏ 1 + h1(s−, Tl, Tl+1) ✏ eγ1(s,Tl),x − 1 ✑✑−1 νTk+1(dt, dx) =: F

1,Tk+1 t

(dx) dt is the I P1,Tk+1-compensator of µX.

Proposition

(a) The Libor rate L1(·, Tk) is a I P1,Tk+1-martingale for all Tk. (b) The connection between two subsequent forward measures is given by dI P1,Tk dI P1,Tk+1 = B1(0, Tk+1) B1(0, Tk) (1 + δkL1(Tk, Tk)). Proceeding backwards, the connection to the terminal forward measure I P1,T∗ is given by dI P1,Tk dI P1,T∗ = B1(0, T∗) B1(0, Tk)

n−1

❨

l=k

(1 + δlL1(Tk, Tl)).

For each i ∈ K\K: (1) Remember that the forward Libor rate for the rating class i 1 + δkLi(t, Tk) = (1 + δkL(t, Tk))

i−1

❨

j=1

(1 + δkHj(t, Tk))(1 + δkHi(t, Tk)), where H1(·, Tk), . . . , Hi−1(·, Tk) are already modelled. (2) It remains to model Hi(·, Tk) as an exponential process of the form Hi(t, Tk) = Hi(0, Tk) exp ✒❩ t bHi(s, Tk, Tk+1) ds + ❩ t γi(s, Tk) dX

i−1,Tk+1 s

✓ , (2) where X

i−1,Tk+1 t

= ❩ t √cs dW

i−1,Tk+1 s

+ ❩ t ❩

Rd x(µX − νi−1,Tk+1)(ds, dx)

and the drift term bHi(s, Tk, Tk+1) = 0, for s ≥ Tk, i.e. Hi(t, Tk) = Hi(Tk, Tk) for t ≥ Tk. (3) Choose the drift term in such a way that the process ✥ k ❨

l=0

1 1 + δlHi(t, Tl) ✦

t≤Tk

becomes a I Pi−1,Tk+1-martingale.

We can define a new measure I Pi,Tk+1 called forward measure associated with the rating i and date Tk+1 on (Ω, FTk+1) via dI Pi,Tk+1 dI Pi−1,Tk+1 := ◗k

l=0 1 1+δlHi(Tl,Tl)

◗k

l=0 1 1+δlHi(0,Tl)

= Bi−1(0, Tk+1) Bi(0, Tk+1)

k

❨

l=0

1 1 + δlHi(Tl, Tl). Denoting hi(s, Tl, Tl+1) := δlHi(s, Tl) 1 + δlHi(s, Tl), s ≤ Tl, it follows that W

i,Tk+1 t

:= W

i−1,Tk+1 t

+ ❩ t

k

❳

l=1

hi(s−, Tl, Tl+1)γi(s, Tl)√cs ds is a I Pi,Tk+1- standard Brownian motion and νi,Tk+1(dt, dx) :=

k

❨

l=1

✏ 1 + hi(s−, Tl, Tl+1) ✏ eγi(s,Tl),x − 1 ✑✑−1 νi−1,Tk+1(dt, dx) =: F

i,Tk+1 t

(dx) dt is the I Pi,Tk+1-compensator of µX.

Observe that we can express the measure I Pi,Tk+1 with respect to the (default-free) forward measure I PTk+1 dI Pi,Tk+1 dI PTk+1 = ◗i

j=1

◗k

l=0 1 (1+δlHj(Tl,Tl))

◗i

j=1

◗k

l=0 1 (1+δlHj(0,Tl))

= B(0, Tk+1) Bi(0, Tk+1)

i

❨

j=1 k

❨

l=0

1 (1 + δlHj(Tl, Tl)) and we have W

i,Tk+1 t

= W

i−1,Tk+1 t

+ ❩ t

k

❳

l=1

hi(s−, Tl, Tl+1)γi(s, Tl)√cs ds = W

Tk+1 t

+ ❩ t

i

❳

j=1 k

❳

l=1

hj(s−, Tl, Tl+1)γj(s, Tl)√cs ds and νi,Tk+1(dt, dx) =

k

❨

l=1

✏ 1 + hi(s−, Tl, Tl+1) ✏ eγi(s,Tl),x − 1 ✑✑−1 νi−1,Tk+1(dt, dx) =

i

❨

j=1 k

❨

l=1

✏ 1 + hj(s−, Tl, Tl+1) ✏ eγj(s,Tl),x − 1 ✑✑−1 νTk+1(dt, dx).

L(t, Tn−1)

• Li(t, Tn−1)
• Li+1(t, Tn−1)
• L(t, Tk)
• Li(t, Tk)

dI Pi,Tk dI Pi,Tk+1

• dI

Pi+1,Tk+1 dI Pi,Tk+1

Li+1(t, Tk)

• L(t, Tk−1)
• Li(t, Tk−1)
• Li+1(t, Tk−1)
• L(t, T1)

Li(t, T1) Li+1(t, T1)

Default-free Rating i Rating i + 1

Theorem

Assume that (L.1), (L.2), (RBL.1) and (RBL.2) are in force and that L(·, Tk) and Hi(·, Tk) are given by (1) and (2). The credit-rating based Libor model has the following properties: (a) For every Tk and t ≤ Tk it holds L(t, Tk) < L1(t, Tk) < · · · < LK−1(t, Tk), i.e. Libor rates are monotone with respect to the credit rating. (b) The forward measure I Pi,Tk associated with the rating i and the date Tk is defined via dI Pi,Tk dI PTk = B(0, Tk) Bi(0, Tk)

i

❨

j=1 k−1

❨

l=0

1 (1 + δlHj(Tl, Tl)). (c) For each rating i and each date Tk the Libor rate Li(·, Tk) is a martingale with respect to the forward measure I Pi,Tk+1. (d) For every i it holds dI Pi,Tk dI Pi,Tk+1 = Bi(0, Tk+1) Bi(0, Tk) (1 + δkLi(Tk, Tk)).

Credit migration process

Bielecki and Rutkowski (2002)

◮ Let Λ = (Λt)0≤t≤T∗ be a matrix-valued F-adapted stochastic process on

(Ω, FT∗, I PT∗) Λ(t) = ✷ ✻ ✻ ✻ ✹ λ11(t) λ12(t) . . . λ1K(t) λ21(t) λ22(t) . . . λ2K(t) . . . . . . ... . . . . . . ✸ ✼ ✼ ✼ ✺ where λij are nonnegative processes, integrable on every [0, t] and λii(t) = − P

j∈K\{i} λij(t).

◮ Let µ = (δij, j ∈ K) be a probability distribution on Ω = K. ◮ Define

(˜ Ω, G, QT∗) = (Ω × ΩU × Ω, FT∗ ⊗ F U ⊗ 2Ω, I PT∗ ⊗ I PU ⊗ µ),

◮ On (ΩU, F U, I

PU) a sequence (Ui,j), i, j ∈ N, of mutually independent random variables, uniformly distributed on [0, 1].

Construction of the migration process

◮ The jump times τk are constructed recursively as

τk := τk−1 + inf ✭ t ≥ 0 : exp ✥❩ τk−1+t

τk−1

λCk−1,Ck−1(u) du ✦ ≤ U1,k ✮ , where we set τ0 := 0.

◮ The new state at the jump time τk is defined as

Ck := C(U2,k, Ck−1, τk), where we set C0(ω, ωU, 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 Leb ({u ∈ [0, 1] : C(u, i, t) = j}) = −λij(t) λii(t) if λii(t) < 0 and 0 if λii(t) = 0.

◮ Finally, we define for every t ≥ 0

Ct := Ck−1, for t ∈ [τk−1, τk) , k ≥ 1.

Process C is a conditionally Markov chain relative to F, i.e. for every 0 ≤ t ≤ s and any function h : K → R it holds I E[h(Cs)|Ft ∨ FC

t ] = I

E[h(Cs)|Ft ∨ σ(Ct)], where FC = (F C

t ) denotes the filtration generated by C and Gt = Ft ∨ FC t .

Proposition

The conditional expectations with respect to enlarged σ-algebras can be expressed in terms of Ft-conditional expectations. It holds I EQT∗ [Y|Ft ∨ σ(Ct)] =

K

❳

i=1

1{Ct=i} I EQT∗ [Y1{Ct=i}|Ft] I EQT∗ [1{Ct=i}|Ft] , for any G-measurable random variable Y.

Properties of C

(a) for every t ≤ s ≤ u and any function h : K → R a stronger version of conditional Markov property holds: I EQT∗ [h(Cs)|Fu ∨ FC

t ] = I

EQT∗ [h(Cs)|Fu ∨ σ(Ct)] (b) for every t ≤ s and B ∈ FC

t :

I EQT∗ [1B|Fs] = I EQT∗ [1B|Ft] (c) conditional Chapman-Kolmogorov equality P(t, s) = P(t, u)P(u, s), where P(t, s) = [pij(t, s)]i,j∈K and pij(t, s) := QT∗(Cs = j, Ct = i|Fs) QT∗(Ct = i|Fs) (d) conditional forward Kolmogorov equation dP(t, s) ds = P(t, s)Λ(s)

Note that the (H)-hypothesis (H) Every local F-martingale is a local G-martingale holds in this conditional Markov setting. It is equivalent to any of the following three equalities: (H1) I EQT∗ [XY|Ft] = I EQT∗ [X|Ft]I EQT∗ [Y|Ft], i.e. FT∗ and F C

t are conditionally independent given Ft

(H2) I EQT∗ [Y|FT∗] = I EQT∗ [Y|Ft], for Y bounded F C

t -measurable random variable

(H3) I EQT∗ [X|Ft ∨ FC

t ] = I

EQT∗ [X|Ft], for X bounded FT∗-measurable random variable

No-arbitrage condition for the model with migration

Remember the defaultable bond price process with zero recovery BC(t, Tk) =

K−1

❳

i=1

Bi(t, Tk)1{Ct=i}. Then we can write it as BC(t, Tk) B(t, Tk) =

K−1

❳

i=1 i

❨

j=1 k−1

❨

l=0

1 1 + δkHj(t, Tl) ⑤ ④③ ⑥

:=H(i,t,Tk)

Bi(t, 0) B(t, 0) 1{Ct=i}, (3) where we put Bi(t, 0) B(t, 0) := exp ✒❩ t λi(s) ds ✓ , for some F-adapted process (λi(·)) that is integrable on [0, T∗] (i ∈ K \ K).

Proposition

Assume that the processes (Hj(·, Tk)) are given by (2) and satisfy the drift condition, i.e. the processes (H(i, ·, Tk)) are I PTk-martingales. The process ✏

BC(·,Tk) B(·,Tk)

✑ defined in (3) is a local martingale with respect to the forward measure I PTk iff for every i ∈ K \ K and t ≤ Tk it holds H(i, t, Tk)λi,K(t) +

K−1

❳

j=1

✒ H(i, t, Tk) − H(j, t, Tk) exp( ❩ t (λj(s) − λi(s)) ds) ✓ λij(t) = H(i, t−, Tk)λi(t). Remark: In case when the recovery is not zero, a similar condition can be

• derived. The bond price process has the additional term

BC(t, Tk) =

K−1

❳

i=1

Bi(t, Tk)1{Ct=i} + δCτ−B(t, Tk)1{Ct=K} (fractional recovery of Treasury value)

Defaultable bond price

The price of a Tk-maturity defaultable bond with zero recovery at time t ≤ Tk equals BC(t, Tk) = B(t, Tk)

K−1

❳

i=1

I EQTk [1 − piK(t, Tk)|Ft]1{Ct=i}. Proof: The promised payoff of such a bond at maturity time Tk is 1{CTk =K}. Using properties of the process C, we get BC(t, Tk) = B(t, Tk)I EQTk [1{CTk =K}|Ft ∨ FC

t ]

= B(t, Tk)I EQTk [1{CTk =K}|Ft ∨ σ(Ct)] = B(t, Tk)

K−1

❳

i=1

I EQTk [1{CTk =K}1{Ct=i}|Ft] I EQTk [1{Ct=i}|Ft] 1{Ct=i} = B(t, Tk)

K−1

❳

i=1

I EQTk [1 − piK(t, Tk)|Ft]1{Ct=i}.

More generally, the price of a contingent claim with a promised FTk - measurable payoff Y at maturity Tk and zero recovery upon default equals πt(Y) = B(t, Tk)

K−1

❳

i=1

I EQTk [Y(1 − piK(t, Tk))|Ft]1{Ct=i}.

◮ In a special case when no migration between time t and Tk is allowed,

the pricing formula can be simplified considerably. It can be shown that πt(Y) =

K−1

❳

i=1

Bi(t, Tk)I EI

Pi,Tk [Y|Ft]1{Ct=i},

where I Pi,Tk is the forward measure associated with rating i and date Tk. (see analogous result in Eberlein, Kluge and Sch¨

• nbucher (2006))

◮ Unfortunately if this is not the case, one has to work with the full

transition matrix of C and the forward measures cannot be used this way.

A rating-dependent Libor rate forward cap

◮ provides insurance against increasing interest rates ◮ cap is a series of caplets, i.e. call options on subsequent Libor rates

The payoff of a caplet with strike K and maturity Tk on the Libor rate for rating i is δk(Li(Tk, Tk) − K)+. . Applying the previous result, time-t price of the caplet equals Capleti(t, Tk) = δkBi(t, Tk+1)I EI

Pi,Tk+1 [(Li(Tk, Tk) − K)+|Ft].

Hence, Capi(t) =

n

❳

j=1

δj−1Bi(t, Tj)I EI

Pi,Tj [(Li(Tj−1, Tj−1) − K)+|Ft].

◮ by using Laplace transformation techniques we can calculate the price

at time 0

Credit default swap

◮ provides protection against default of an underlying asset ◮ consider a maturity date Tm and a defaultable bond with fractional

recovery of Treasury value as the underlying asset

◮ protection buyer pays a fixed amount s periodically at dates T1, . . . , Tm−1

until default

◮ protection seller promises to make a payment that covers the loss if

default happens: 1 − δCτ− is received at Tk+1 if default occurs in (Tk, Tk+1] The value of the premium leg at t ≤ T1:

m−1

❳

k=1

sB(t, Tk)I EQTk [1{τ>Tk}|Gt] The value of the default leg at t ≤ T1:

m

❳

k=2

B(t, Tk)I EQTk [(1 − δCτ−)1{Tk−1<τ≤Tk}|Gt]

The premium leg can be written as s

m−1

❳

k=1 K−1

❳

i=1

Bi(t, Tk)1{Ct=i} and the default leg

m

❳

k=2

B(t, Tk)

K−1

❳

i=1

1{Ct=i}

K−1

❳

j=1

I EQTk [(1 − δj)1{Tk−1<τ≤Tk,Cτ−=j}1{Ct=i}|Ft] I EQTk [1{Ct=i}|Ft] . Hence, the swap rate s at time 0 is equal to s =

K−1

❳

i=1

1{C0=i} Pm

k=2 B(t, Tk) PK−1 j=1 I EQTk [(1−δj)1 {Tk−1<τ≤Tk,Cτ−=j}1{C0=i}] I EQTk [1{C0=i}]

Pm−1

k=1 Bi(0, Tk)

.

Concluding remarks

◮ rating-dependent Libor rates are introduced ◮ the whole term structure of default-free and rating-dependent Libor

rates is modelled

◮ credit rating migrations are added and no-arbitrage conditions derived ◮ rating-dependent forward measures can be used for pricing only in

special cases where no migration between classes is allowed ⇒ pricing formulae for caps, floors, swaps on rating-dependent Libor rates

◮ in general, explicit pricing formulae for derivatives depending on ratings

(e.g. options on defaultable bonds, credit default swaps, credit default swaptions) require further assumptions on the migration process

• T. Bielecki and M. Rutkowski, Credit Risk: Modeling, Valuation and

Hedging, Springer, 2002.

• E. Eberlein, W. Kluge and P

. J. Sch¨

• nbucher, The L´

evy Libor model with default risk, Journal of Credit Risk 2, 3-42, 2006.

• E. Eberlein and F

. ¨ Ozkan, The L´ evy LIBOR model. Finance and Stochastics 9, 327-348, 2005.

• J. Jacod and A.N. Shiryaev, Limit theorems for stochastic processes,

Springer, 2003.

• W. Kluge, Time-inhomogeneous L´

evy processes in interest rate and credit risk models, Ph.D. thesis, University of Freiburg, 2005.