Dynamic Modelling of CDOs Thorsten Schmidt Technische Universit¨ at Chemnitz www.tu-chemnitz.de/mathematik/fima thorsten.schmidt@mathematik.tu-chemnitz.de Vienna, July 2010 joint work with J. Zabczyk Thorsten Schmidt, TU Chemnitz 1
The top-down approach Introduction Essentials of securitization Consider a CDO as a pool of m defaultable entities. Default i occurs at τ i with associated loss q i Cumulative loss m � A t = q i 1 { τ i ≤ t } . i =1 Normalize the total nominal to 1, set I := [0 , 1]. Loss is split into tranches: a tranche refers to an interval ( x i , x i − 1 ] ⊂ I , 0 = x 0 < x 1 < · · · < x k = 1 Thorsten Schmidt, TU Chemnitz 2
The top-down approach Partition of losses into tranches Examples: Traded indicies (iTraxx, CDX) Thorsten Schmidt, TU Chemnitz 3
The top-down approach Single tranche CDOs A STCDO is specified by a number of future dates T 0 < T 1 < · · · < T m , a tranche with lower and upper detachment points x 1 < x 2 , a fixed spread S . We write � H ( x ) := ( x 2 − x ) + − ( x 1 − x ) + = 1 { x ≤ y } dy . ( x 1 , x 2 ] An investor in this STCDO receives SH ( A T k ) at T k , k = 1 , . . . , m − 1 (payment leg), pays H ( A t ) − H ( A t − ) at any time when ∆ A t � = 0. (default leg) Thorsten Schmidt, TU Chemnitz 4
The top-down approach Filipovi´ c, Overbeck and Schmidt (2009) A security which pays 1 { A T ≤ x } at T is called ( T , x )- bond . Its price at time t ≤ T is denoted by P ( t , T , x ). Proposition The value of the STCDO at time t ≤ T 1 is � n � � � Γ( t , S ) = S P ( t , T i , y ) + P ( t , T n , y ) − P ( t , T 0 , y ) + γ ( t , y ) dy ( x 1 , x 2 ] i =1 with � T n � u � � r u e − t r s ds 1 { A u ≤ y } | F t γ ( t , y ) = I E du . T 0 Solving Γ = 0 for S gives the fair spread. Thorsten Schmidt, TU Chemnitz 5
The top-down approach Drift condition (A1) A t = � s ≤ t ∆ A s is an increasing marked point process with compensator ν A ( t , dx ) dt and values in [0 , 1]. Consider λ ( t , x ), such that � t M x t = 1 { A t ≤ x } + 1 { A s ≤ x } λ ( s , x ) ds 0 is a martingale. E( e −� u , Z t � ) = e tJ ( u ) Consider a d -dimensional L´ evy process Z such that I u ∈ R d with J ( u ) = � m , u � + 1 � e −� u , z � − 1 + 1 {| z |≤ 1 } ( z ) � u , z � � � 2 � Σ u , u � + ν ( dz ) . ˜ R d (1) Thorsten Schmidt, TU Chemnitz 6
The top-down approach Forward-rate approach: We consider � T � � P ( t , T , x ) = 1 { A t ≤ x } exp − f ( t , u , x ) du t where � t � t f ( t , T , x ) = f (0 , T , x ) + a ( s , T , x ) ds + � b ( s , T , x ) , dZ s � 0 0 (2) � t � c ( s , T , x ; y ) µ A ( ds , dy ) + 0 I Thorsten Schmidt, TU Chemnitz 7
The top-down approach No-arbitrage condition � t e − 0 r s ds P ( t , T , x ) are local martingales for all ( T , x ). (3) Under some technical assumptions, we have that Theorem (3) is equivalent to � s � � s � a ( t , u , x ) du = J b ( t , u , x ) du t t � � s � t c ( t , u , x ; y ) du − 1 � e − 1 { L t + y ≤ x } ν A ( t , dy ) + (4) I f ( t , t , x ) = r t + λ ( t , x ) , (5) where (4) and (5) hold on { A t ≤ x } , Q ⊗ dt-a.s. Thorsten Schmidt, TU Chemnitz 8
The top-down approach In Filipovi´ c, Overbeck, Schmidt (2009) also tractable affine models are 1 developed. Variance-Minimizing Hedging Strategies are developed in Filipovi´ c, 2 Schmidt (2010) which lead to explicit strategies in a affine one-factor model Thorsten Schmidt, TU Chemnitz 9
Market models Market models Thorsten Schmidt, TU Chemnitz 10
Market models Forward rate modelling Denote T := { T 0 , . . . , T n } , δ k := T k +1 − T k and let P ( t , T , x ) = p ( t , T , x )1 { A t ≤ x } , (6) ( p ( t , T , x )) 0 ≤ t ≤ T a strictly positive special semimartingale with p ( T , T , x ) = 1. Definition The ( T k , T k +1 , x ) -spread is given by P ( t , T k , x ) F ( t , T k , T k +1 , x ) := P ( t , T k +1 , x ) . (7) Thorsten Schmidt, TU Chemnitz 11
Market models Proposition. Forward spreads given on { A t ≤ x i } by dF ( t , T k , T k +1 , x i ) F ( t − , T k , T k +1 , x i ) = α ki ( t ) dt + � β ki ( t ) , dW ( t ) � � � � e � β ki ( t ) , z � − 1 � � e γ ki ( t , A t − ; y ) − 1 � 1 { A t − + y ≤ x i } µ A ( dt , dy ) , + µ ( dt , dz ) + R d I ( T k , x i ) ∈ S , k < n and zero on { A t > x i } constitute an arbitrage-free market if k � α ki ( t ) = − λ ( t , x i ) + � β ji ( t ) , Σ β ki ( t ) � j = η ( t ) � k − 1 � � � e � β ki ( t ) , z � + e −� β ki ( t ) , z � − 1 e −� β ji ( t ) , z � − 1 � � + ν ( dz ) R d j = η ( t ) � k − 1 � � � e γ ki ( t , A t − ; y ) + � e − γ ki ( t , A t − ; y ) − 1 � e − γ ji ( t , A t − ; y ) 1 { A t + y ≤ x i } ν A ( t , dy ) + I j = η ( t ) for all ( T k , x i ) , ( T k +1 , x i ) ∈ S . Thorsten Schmidt, TU Chemnitz 12
Market models Similar techniques as in interest rate markets can be applied to value CDOs and options on CDOs. Grbac, Eberlein, Schmidt (2010) study directly discrete rates. Statistical results show that in shorter time periods affine models are appropriate. Further issues Model risk Measuring the risk of credit and market risk simultaneously. Thorsten Schmidt, TU Chemnitz 13
Market models [1] A Brace, D. Gatarek, M. Musiela: The market model of interest rate dynamics (1995). [2] E Eberlein, Z Grbac, T Schmidt: Market Models for CDOs driven by L´ evy processes (2010). arXiv:1006.2012 [3] D Filipovi´ c, L Overbeck, T Schmidt: Dynamic CDO term structure modelling (2009). [4] D Filipovi´ c, T Schmidt: ”Pricing and Hedging of CDOs: A Top-Down Approach”, (2010). Contemporary Quantitative Finance,Chiarella, C. and Novikov, A. (Eds.) [5] T Schmidt, J Zabczyk: CDO term structure modelling with L´ evy processes and the relation to market models (2010). arXiv:1007.1706 Thorsten Schmidt, TU Chemnitz 14
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