DYNAMIC CDO TERM STRUCTURE MODELLING Damir Filipovi´ c (joint with Ludger Overbeck and Thorsten Schmidt) Vienna Institute of Finance www.vif.ac.at Special Semester on Stochastics with Emphasis on Finance Johann Radon Institute for Computational and Applied Mathematics (RICAM) Kick-off-Workshop, 11 September 2008 1
Overview 1. Collateralized Debt Obligations (CDOs) 2. ( T, x )-Bonds 3. Arbitrage-free Term Structure Movements 4. Doubly Stochastic Framework 2
Collateralized Debt Obligations (CDOs) • most important type of portfolio credit derivative • security backed by pool of reference entities ( assets ): bonds, loans, protection seller position in single name CDS, etc. • assets sold to special-purpose vehicle (SPV) • SPV issues notes on CDO tranches ( liabilities ) • important reference indices: ITraxx Europe, CDX (USA), 3
Basic Structure of a CDO Payments in a CDO structure. Payments corresponding to synthetic CDOs are in italics . 4
Literature Bennani (05): “The forward loss model: A dynamic term structure approach for the pricing of portfolio credit derivatives” Cont and Minca (08), “Recovering portfolio default intensities implied by CDO quotes” Ehlers and Sch¨ onbucher (06), “Pricing Interest Rate-Sensitive Credit Portfo- lio Derivatives” Ehlers and Sch¨ onbucher (06), “Background Filtrations and Canonical Loss Processes for Top-Down Models of Portfolio Credit Risk” Filipovi´ c, Overbeck and Schmidt (08), “Dynamic CDO Term Structure Mod- elling” Sch¨ onbucher (05), “Portfolio losses and the term structure of loss transition rates: A new methodology for the pricing of portfolio credit derivatives” Sidenius, Piterbarg and Andersen (IJTAF 08), “A new framework for dynamic credit portfolio loss modelling” 5
Overview 1. Collateralized Debt Obligations (CDOs) 2. ( T, x ) -Bonds 3. Arbitrage-free Term Structure Movements 4. Doubly Stochastic Framework 6
( T, x ) -Bonds • (Ω , F , ( F t ) , Q ), Q risk-neutral measure • CDO pool of credits normalized to 1. • Loss process L t = � s ≤ t ∆ L s [0 , 1]-valued increasing MPP with abs. continuous compensator ν ( t, dx ) dt . • ( T, x ) -bond pays 1 { L T ≤ x } at maturity T , x ∈ [0 , 1]. Its price P ( t, T, x ) at t ≤ T is decreasing in T , increasing in x . Note: P ( t, T ) = P ( t, T, 1) is risk-free zero-coupon bond. 7
Default Times of the ( T, x ) -Bonds Lemma 1: For any x ∈ [0 , 1], the process 1 { L t ≤ x } has intensity λ ( t, x ) = ν ( t, ( x − L t , 1]) . That is, � t M x t = 1 { L t ≤ x } + 0 1 { L s ≤ x } λ ( s, x ) ds is a martingale. Conversely, λ ( t, x ) uniquely determines ν ( t, dx ) via ν ( t, (0 , x ]) = λ ( t, L t ) − λ ( t, L t + x ) , x ∈ [0 , 1] . Proof. F ( L t ) − � t � 1 0 ( F ( L s + y ) − F ( L s )) ν ( s, dy ) ds is a martingale, for any 0 bounded measurable function F . For F ( L t ) = 1 { L t ≤ x } , we obtain F ( L s + y ) − F ( L s ) = − 1 { L s + y>x } 1 { L s ≤ x } . 8
( T, x ) -Bonds • Contingent claim with payoff F ( L T ) at T can be decomposed � 1 0 F ′ ( y )1 { L T ≤ y } dy F ( L T ) = F (1) − • Hence static replicating portfolio, at t ≤ T , is � 1 0 F ′ ( y ) P ( t, T, y ) dy F (1) P ( t, T ) − ⇒ ( T, x ) -bonds span all European type contingent claims 9
Single Tranche CDOs (STCDOs) Standard instrument for investing in CDO-pool (e.g. iTraxx). Specified by • a number of future dates T 0 < T 1 < · · · < T n , • a tranche with lower and upper detachment points x 1 < x 2 , • a fixed spread S . 10
Single Tranche CDOs (STCDOs) � x 2 Write H ( x ) = ( x 2 − x ) + − ( x 1 − x ) + = x 1 1 { x ≤ y } dy An investor in this STCDO • receives SH ( L T i ) at T i , i = 1 , . . . , n ( payment leg ), • pays − ∆ H ( L t ) = H ( L t − ) − H ( L t ) at any time t ∈ ( T 0 , T n ] where ∆ L t � = 0 ( default leg ). ⇒ STCDO can be priced via ( T, x )-bonds 11
Single Tranche CDOs (STCDOs) Cash-flow attributed to tranche ( x 1 , x 2 ]: 12
Overview 1. Collateralized Debt Obligations (CDOs) 2. ( T, x )-Bonds 3. Arbitrage-free Term Structure Movements 4. Doubly Stochastic Framework 13
Term Structure Movements Aim: describe ( T, x )-bond price movements explicitly by P ( t, T, x ) = 1 { L t ≤ x } e − � T t f ( t,u,x ) du P ( t, T, x ) = P ( t, T ) Q T [ L T ≤ x | F t ] is F t -conditional CDF of L T w.r.t. Q T 14
Note the Difference F t -conditional CDF of stock price S T w.r.t. Q T C ( t, T, x ) = P ( t, T ) Q T [ S T ≤ x | F t ] 15
Term Structure Movements ( T, x )-bond price P ( t, T, x ) = 1 { L t ≤ x } e − � T t f ( t,u,x ) du where f ( t, T, x ) is the ( T, x ) -forward rate � t � t 0 b ( s, T, x ) ⊤ · dW s f ( t, T, x ) = f (0 , T, x ) + 0 a ( s, T, x ) ds + Risk-free T -forward rate f ( t, T ) = f ( t, T, 1) short rate r t = f ( t, t, 1) 16
Term Structure Movements Include contagion : • direct: ∆ f ( t, T, x ) = c ( t, T, x ; ∆ L t ) • indirect: b ( t, T, x ) = b ( t, T, x ; L ), same for a , c � t � t 0 b ( s, T, x ; L ) ⊤ · dW s f ( t, T, x ) = f (0 , T, x ) + 0 a ( s, T, x ; L ) ds + � + c ( s , T , x ; ∆L s ) 1 { ∆L s > 0 } s ≤ t 17
Arbitrage-free Term Structure Movements No arbitrage (NA) : e − � t 0 r s ds P ( t, T, x ) local martingale ∀ ( T, x ) Theorem 2: NA is equivalent to 2 � T � T � � a ( t, u, x ) du = 1 � � b ( t, u, x ) du � � 2 � � t t � � � 1 e − � T � � t c ( t , u , x ; y ) du − 1 + ν ( t , dy ) , 0 λ ( t, x ) = f ( t, t, x ) − r t on { L t ≤ x } , dt ⊗ d Q -a.s. for all ( T, x ). NB: recall ν ( t, dy ) = − λ ( t, L t + dy ) = − f ( t, t, L t + dy ) 18
Single Tranche CDOs (STCDOs) Write p ( t, T, x ) = e − � T t f ( t,u,x ) du . Lemma 4: The value of the STCDO at time t ≤ T 0 is Γ( t, S ) n � � dy = ( x 1 ,x 2 ] 1 { L t ≤ y } p ( t, T i , y ) − p ( t, T 0 , y ) + p ( t, T n , y ) + γ ( t, y ) S i =1 where � T n γ ( t, y ) = f ( t, u ) p ( t, u, y ) du if f ( t, u ) and L t are independent. T 0 Forward STCDO spread S ∗ ( t ) defined by Γ( t, S ∗ ( t )) = 0. 19
STCDO swaption with strike K has payoff at maturity T 0 n � + . � K − S ∗ � � ( x 1 ,x 2 ] 1 { L t ≤ y } p ( T 0 , T i , y ) dy T 0 i =1
Martingale Problem Aim: exogenous specification of b ( t, T, x ) and c ( t, T, x ) deter- mines full ( T, x )-bond model P ( t, T, x ). Martingale problem: implicit loss process L t such that ν ( t, dx ) = − f ( t, t, L t + dx ) becomes compensator Assumption: canonical stochastic basis Ω = Ω 1 × Ω 2 , F t = G t ⊗ H t , Q ( dω 1 , dω 2 ) = Q 1 ( dω 1 ) Q 2 ( ω 1 , dω 2 ): • (Ω 1 , G , ( G t ) , Q 1 ) carrying market information, i.e. Brownian motion W ( ω ) = W ( ω 1 ), • (Ω 2 , H ) canonical space of [0 , 1]-valued increasing MPPs, loss process = coordinate process: L t ( ω ) = ω 2 ( t ) • Q 2 probability kernel from Ω 1 to H to be determined below. 20
Martingale Problem Solution: Jacod (75), “Multivariate Point Processes: Predictable Pro- jection, Radon-Nikodym Derivatives, Representation of Martingales” Theorem 3: Given vola and contagion parameters b ( ω ; t, T, x ) = b ( ω 1 , ω 2 ; t, T, x ) and c ( ω ; t, T, x, y ) = c ( ω 1 , ω 2 ; t, T, x, y ) 1. Define a ( t, T, x ) via NA drift condition. 2. Solve for f ( t, T, x ) along any loss path ω 2 . 3. Jacod (75): ∃ unique kernel Q 2 such that NA holds. 21
Martingale Problem Theorem 3 contd.: Moreover, on { τ n < ∞} , = e − � τn + t ν ( ω 1 ,ω 2 ( τ n ); s, [0 , 1]) ds � � Q τ n +1 − τ n > t | G ⊗ H τ n τn and ν ( τ n , A ) Q [∆ L τ n ∈ A | G ⊗ H τ n − ] = A ⊂ [0 , 1] ν ( τ n , [0 , 1]) , where 0 < τ 1 < τ 2 < · · · denote jump times of L . 22
Monte–Carlo algorithm Along any Brownian path ω (1) , . . . , ω ( N ) , by recursion 1 1 • solve f ( t, T, x ) with L t ≡ L τ i − 1 for t ≥ τ j − 1 � t ǫ ( j ) ∼ exp iid � τ j − 1 λ ( s, L τ j − 1 ) ds ≥ ǫ ( j ) � • set τ j = inf , t | − λ ( τ j ,L τj − 1 + dx ) • simulate ∆ L τ j ∼ , x ≥ 0 λ ( τ j ,L τj − 1 ) • restart at τ j with ∆ f ( τ j , T, x ) = c ( τ j , T, x ; ∆ L τ j ) 23
Overview 1. Collateralized Debt Obligations (CDOs) 2. ( T, x )-Bonds 3. Arbitrage-free Term Structure Movements 4. Doubly Stochastic Framework 24
Doubly Stochastic Framework No contagion b ( ω ) = b ( ω 1 ) and c = 0. Then L becomes (uniquely) G -conditional Markov. Moreover, for any G -measurable X ≥ 0: X e − � T � � t λ ( s,x ) ds | G t E [ X 1 { L T ≤ x } | F t ] = 1 { L t ≤ x } E . (This is the SPA 08 framework) 25
Affine Term Structure Models State space Z ⊂ R d , state process dZ t = µ ( Z t ) dt + σ ( Z t ) · dW t , Z 0 = z Affine term structure (ATS) f ( t, T, x ) = A ′ ( t, T, x ) + B ′ ( t, T, x ) ⊤ · Z t � T � T t A ′ ( t, u, x ) du , B ( t, T, x ) = t B ′ ( t, u, x ) du . Write A ( t, T, x ) = 26
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