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 = T 0 < T 1 < . . . < T n = T ∗ , with δ k = T k + 1 − T k ◮ 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 ] ✒ B ( t , T k ) ✓ L ( t , T k ) = 1 B ( t , T k + 1 ) − 1 , δ k

The driving process Let (Ω , F T ∗ , F = ( F t ) 0 ≤ t ≤ T ∗ , I P T ∗ ) be a complete stochastic basis evy process X T ∗ with values in R d , i.e We work with a time-inhomogeneous L´ ag processs with X T ∗ an adapted, c´ adl´ = 0 and such that 0 (1) X T ∗ has independent increments (2) the law of X T ∗ is given by its characteristic function t ✒❩ t ✓ E [ exp ( i � u , X T ∗ � )] = exp θ s ( i u ) d s I , t 0 ❩ ✏ ✑ θ s ( i u ) = i � u , b s � − 1 e i � u , x � − 1 − i � u , x � F s ( d x ) 2 � u , c s u � + . R d Here b s ∈ R d , c s a symmetric, nonneg. definite d × d -matrix and F T ∗ a L´ evy s measure ( s ∈ [ 0 , T ∗ ] ). Assume the following integrability conditions: ✒ ✓ ❩ R d ( | x | 2 ∧ 1 ) F s ( d x ) | b s | + || c s || + < ∞ sup 0 ≤ s ≤ T ∗ and ❩ ( u ∈ [ − ( 1 + ε ) M , ( 1 + ε ) M ] d ) . sup exp � u , x � F s ( d x ) < ∞ 0 ≤ s ≤ T ∗ | x | > 1

X T ∗ is a special semimartingale with canonical decomposition ❩ t ❩ t ❩ t ❩ √ c s d W T ∗ R d x ( µ X − ν T ∗ )( d s , d x ) , X T ∗ = b s d s + + t s 0 0 0 where W T ∗ denotes standard Brownian motion and µ X is the random measure of jumps of X T ∗ with the compensator ν T ∗ . The semimartingale characteristics of X T ∗ is given by ❩ t ❩ t ν T ∗ ( d s , d x ) = F T ∗ B t = b s d s , C t = c s d s , s ( d x ) d s 0 0 We assume from now on that b s = 0 .

Construction The model is constructed by a backward induction procedure. Assumptions: ( L .1) For every T k there is a deterministic function σ ( · , T k ) : [ 0 , T ∗ ] → R d + , which represents the volatility of the forward Libor rate L ( · , T k ) . We assume that n − 1 ❳ σ j ( s , T k ) ≤ M , k = 1 for all s ∈ [ 0 , T ∗ ] and every coordinate j ∈ { 1 , . . . , d } , where M > 0 is a constant from the integrability condition for X T ∗ . If s > T k , then σ ( s , T k ) = 0 . ( L .2) The initial term structure B ( 0 , T k ) is strictly positive and strictly decreasing in k .

Start by specifying the dynamics of the most distant Libor rate under I P T ∗ (forward measure associated with date T ∗ ) as ✒❩ t ❩ t ✓ σ ( s , T n − 1 ) d X T ∗ b L ( s , T n − 1 , T ∗ ) d s + L ( t , T n − 1 ) = L ( 0 , T n − 1 ) exp , s 0 0 where the drift is chosen in such a way that L ( · , T n − 1 ) becomes a I P T ∗ -martingale, namely b L ( s , T n − 1 , T ∗ ) = − 1 2 � σ ( s , T n − 1 ) , c s σ ( s , T n − 1 ) � ❩ ✏ ✑ e � σ ( s , T n − 1 ) , x � − 1 − � σ ( s , T n − 1 ) , x � F T ∗ − s ( d x ) . R d Next define a forward measure I P T n − 1 associated with the date T n − 1 via dI P T n − 1 = 1 + δ n − 1 L ( T n − 1 , T n − 1 ) 1 + δ n − 1 L ( 0 , T n − 1 ) dI P T ∗ and proceed with modeling of L ( · , T n − 2 ) .

For each k : (1) define the forward measure I P T k + 1 via n − 1 dI P T k + 1 1 + δ l L ( T k + 1 , T l ) ❨ = . dI P T ∗ 1 + δ l L ( 0 , T l ) l = k + 1 (2) the dynamics of the Libor rate L ( · , T k ) under this measure ✒❩ t ❩ t ✓ T k + 1 b L ( s , T k , T k + 1 ) d s + L ( t , T k ) = L ( 0 , T k ) exp σ ( s , T k ) d X , (1) s 0 0 where ❩ t ❩ t ❩ √ c s d W R d x ( µ X − ν T k + 1 )( d s , d x ) T k + 1 T k + 1 = + X t s 0 0 with n − 1 δ l L ( s − , T l ) ✏ ✑ ❨ ( e σ ( s , T l ) x − 1 ) + 1 ν T ∗ ( d s , d x ) ν T k + 1 ( d s , d x ) = 1 + δ l L ( s − , T l ) l = k + 1 and the drift b L ( s , T k , T k + 1 ) is chosen such that it becomes a P T k + 1 -martingale. I

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

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 : C t = K }

Defaultable bonds with credit ratings ◮ defaultable bonds whose credit migration process is denoted by C and with zero recovery upon default: B C ( · , T 1 ) , . . . , B C ( · , T n ) ◮ time- t price of such a defaultable bond can be expressed as B C ( t , T k ) = 1 { C t = 1 } B 1 ( t , T k ) + · · · + 1 { C t = K − 1 } B K − 1 ( t , T k ) , where B i ( t , T k ) 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 K − 1 ❳ B C ( T k , T k ) = 1 { C Tk = i } = 1 { τ> T k } i = 1 and it holds B i ( T k , T k ) = 1 , for all i .

Libor rates for different credit ratings ◮ The Libor rates for credit rating class i ✒ B i ( t , T k ) ✓ L i ( t , T k ) := 1 B i ( t , T k + 1 ) − 1 i = 1 , 2 , . . . , K − 1 . , δ k We put L 0 ( t , T k ) := L ( t , T k ) (default-free Libor rates) ◮ The forward credit spreads between two successive classes S i ( t , T k ) := L i ( t , T k ) − L i − 1 ( t , T k ) , i = 1 , 2 , . . . , K − 1 . ◮ The corresponding forward credit spread intensities H i ( t , T k ) := L i ( t , T k ) − L i − 1 ( t , T k ) 1 + δ k L i − 1 ( t , T k )

Observe that Libor rates 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 )) ⑤ ④③ ⑥ ⑤ ④③ ⑥ j = 1 default-free Libor intensity between j and j − 1 Idea: model H j ( · , T k ) as exponential processes and therefore 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

Assumptions: ( RBL .1) For every i ∈ { 1 , . . . , K − 1 } and every maturity T k there is a deterministic function γ i ( · , T k ) : [ 0 , T ∗ ] → R d + , which represents the volatility of the forward spread intensity H i ( · , T k ) for the rating i . We assume that γ i ( s , T k ) = 0 for s > T k and that n − 1 ❳ ( σ j ( s , T k ) + γ j 1 ( s , T k ) + · · · + γ j K − 1 ( s , T k )) ≤ M , k = 1 for all s ∈ [ 0 , T ∗ ] and every coordinate j ∈ { 1 , . . . , d } . ( RBL .2) The initial term structure L i ( 0 , T k ) of Libor rates satisfies L i ( 0 , T k ) > L i − 1 ( 0 , T k ) , for all k = 1 , . . . , n − 1 , i.e. B i ( 0 , T k ) B i − 1 ( 0 , T k ) B i ( 0 , T k + 1 ) > B i − 1 ( 0 , T k + 1 ) .

Construction of rating-based Libor rates For rating 1 and all settlement dates T k we model H 1 ( · , T k ) as ✒❩ t ❩ t ✓ b H 1 ( s , T k , T k + 1 ) d s + T k + 1 H 1 ( t , T k ) = H 1 ( 0 , T k ) exp γ 1 ( s , T k ) d X s 0 0 with initial condition ✒ B 1 ( 0 , T k ) B ( 0 , T k + 1 ) ✓ H 1 ( 0 , T k ) = 1 B ( 0 , T k ) B 1 ( 0 , T k + 1 ) − 1 . δ k X T k + 1 is defined as earlier and b H 1 ( s , T k , T k + 1 ) is the drift term (we assume b H 1 ( s , T k , T k + 1 ) = 0 , for s ≥ T k ⇒ H 1 ( t , T k ) = H 1 ( T k , T k ) , for t ≥ T k ). ✏◗ k ✑ 1 Choose drift term in such a way that the process l = 0 1 + δ l H 1 ( t , T l ) t ≤ T k becomes a I P T k + 1 -martingale (Kluge (2005)).

Hence we can introduce the following change of measure ◗ k 1 k dI P 1 , T k + 1 = B ( 0 , T k + 1 ) ❨ 1 l = 0 1 + δ l H 1 ( T l , T l ) := 1 + δ l H 1 ( T l , T l ) . ◗ k 1 B 1 ( 0 , T k + 1 ) dI P T k + 1 l = 0 1 + δ l H 1 ( 0 , T l ) l = 0 P 1 , T k + 1 - forward measure associated with the rating 1 and the settlement I date T k + 1 Use Girsanov’s theorem for semimartingales. Denote δ l H 1 ( s , T l ) h 1 ( s , T l , T l + 1 ) := s ≤ T l . 1 + δ l H 1 ( s , T l ) , Then ❩ t k h 1 ( s − , T l , T l + 1 ) γ 1 ( s , T l ) √ c s d s ❳ 1 , T k + 1 T k + 1 := W + W t t 0 l = 1 is a I P 1 , T k + 1 - Brownian motion and k ✏ ✏ ✑✑ − 1 ❨ e � γ 1 ( s , T l ) , x � − 1 ν T k + 1 ( d t , d x ) ν 1 , T k + 1 ( d t , d x ) := 1 + h 1 ( s − , T l , T l + 1 ) l = 1 1 , T k + 1 =: F ( d x ) d t t P 1 , T k + 1 -compensator of µ X . is the I