Dynamic Correlation Modeling Giuseppe Di Graziano ∗ Deutsche Bank AG joint work with Prof. L.C.G. Rogers September 21, 2007 ∗ e-mail = giuseppe.di-graziano@db.com).
1. Motivation • Deficiencies of the copula approach: 1. Unable to explain market quotes: correlation structure imposed exogenously and in a arbi- trary fashion. 2. Inconsistency across time/maturities. 3. Lack of dynamics: not suitable to price exotic products such as option on tranche spreads. • Ideally we would like to employ a model which is: 1. Flexible enough to fit market data (i.e. calibrate to standard tranche spreads across the capital structure and all liquid maturities). 2. Dynamically consistent. 3. Tractable and computationally efficient. • The key idea behind the dynamic approach developed by Di Graziano and Rogers is to use a stochastic process (and not a random variable as in the traditional factor model approach) to drive the common dynamics of the various credits in the portfolio. • In order to retain tractability and computational efficiency the process chosen to drive the portfolio common dynamics is a continuous time Markov chain.
2. . . . Some standard result about continuous time Markov chains • Let ( ξ t ) t ≥ 0 be a continuous time K -dimensional, Markov chain with infinitesimal generator Q taking values in the set K ≡ { 1 , 2 , . . . , K } • Transition probabilities are given by: � e Qt � P ij ( t ) ≡ P ( ξ t + s = j | ξ t = i ) = (1) ij • Since the process ξ t takes values in the finite set K , any function of the chain, f : K → R can be represented as a K -dimensional vector with i th component given by f i ≡ f ( i ), i.e. f 1 f 2 ... (2) ... f K • Let α ( ξ ) and f ( ξ ) be K dimensional vectors, and let J ik ( t ) represent the number of jumps from state i to state j up to time t . We have that � t � f ( ξ t ) | ξ 0 = ξ exp − V t ( ξ ) ≡ E α ( ξ u ) du − w ij J ij ( t ) 0 i � = j � � ˜ Qt f = e ( ξ )
where ˜ Q i = Q jj − α j ( j = k ); jk = exp( − w jk ) Q jk ( j � = k ) .
3. Model set up • The common dynamics of the names in the porfolio are driven by a continuous time, K - dimensional, Markov chain ( ξ t ) t ≥ 0 with infinitesimal generator (Q-matrix) Q . • Conditional on the process ξ , i.e. on F ξ t ≡ σ ( ξ s , s ≤ t ) default times τ i are independent. • The key modeling ingredient of our approach is given by the conditional survival probability of the single names � � � � τ i ≥ t | F ξ q i − C i t = P = exp , (3) t t where C i t is some additive functional of the chain of the form � t � C i λ i ( ξ u ) du + w i t = jk J jk ( t ) . (4) 0 j � = k . • The short rate is assumed to be a function of the chain. The discount factor is given by � � � t B − 1 ≡ exp − r ( ξ u ) du . (5) t 0 • The survival probability is given by � � q i t ( ξ 0 ) ≡ E 1 { τ i ≥ t } | ξ 0 exp( t ˜ Q i ) 1 ( ξ 0 ) , = (6)
where ˜ Q i Q jj − λ i = ( j = k ); (7) jk j = Q jk exp( − w jk ) ( j � = k ) . (8)
4. CDS • The CDS’s Default leg can be calculated explicitly as, � � B − 1 DL T ≡ E τ ; τ ≤ T (9) �� T � � Q ξ u k θ ξ u k } B − 1 = E { λ ( ξ u ) + exp( − C u ) du (10) u 0 k Q − 1 (exp( ˆ ˆ QT ) − I )˜ = λ ( ξ 0 ) (11) λ i = λ i + � where ˜ k Q ik θ ik . • The PV01 of the CDS is given by �� T � � T I { τ>u } B − 1 exp( u ˆ PL T ≡ E u du = Q ) 1 du 0 0 Q − 1 (exp( ˆ ˆ = QT ) − I ) 1 ( ξ 0 ) where ˆ Q i Q jj − r j − λ i = ( j = k ); jk j exp( − w i = jk ) Q jk ( j � = k ) . • The above calculations allow us to calibrate our model to single name CDS and index spreads.
5. Survival correlation • Default and survival correlations can be calculated explicitly in this model. • The pairwise joint survival probability of name i and j is P ( τ i ≥ T, τ j ≥ t | ξ t ) q ij ˜ T ( ξ t ) ≡ exp( ˜ Q ij ( T − t ))( ξ t ) , = where Q ij ˜ k − λ j Q kk − λ i = ( k = l ); kl k kl − w j exp( − w i = kl ) Q kl ( k � = l ) . • ...and the survival correlation at time t for maturity T is q ij q j q i ˜ T ( ξ t ) − ˜ T ( ξ t )˜ T ( ξ t ) ρ T ( ξ t ) = (12) � � q j q j q i q i ˜ T ( ξ t )(1 − ˜ T ( ξ t )) ˜ T ( ξ t )(1 − ˜ T ( ξ t )) where T ( ξ t ) = exp( ˜ q i Q i ( T − t ))( ξ t ) . ˜ (13) • In this set-up, the survival (default) correlation is a stochastic process driven by ξ . We are in front of a dynamic correlation approach. • Note that the correlation of defaults is obtained endogenously from the model, rather than being exogenously imposed as in the copula approach.
6. Portfolio loss distribution • We defined the portfolio loss distribution as N � L t ≡ ℓ i I { τ i ≤ t } . (14) i =1 • Let now the loss at default of the i th reference entity be given by l i = A i (1 − R i ). • The Laplace transform of the (discounted) loss process is � t � t N � E exp( − r ( ξ s ) ds − αL t ) = E exp( − r ( ξ s ) ds − α ℓ i I { τ i ≤ t } ) (15) 0 0 i =1 � �� � t N � � e − αℓ i 1 { τi ≤ t } | F ξ = E exp( − r ( ξ s ) ds ) E (16) t 0 i =1 � � � t N � � � (1 − q i t ) ζ i ( α ) + q i = E exp( − r ( ξ s ) ds ) (17) t 0 i =1 where ζ i ( α ) = Ee − αℓ i and � t � q i λ i ( ξ u ) du − w i . − t ≡ exp jk J jk ( t ) (18) 0 j � = k • Equation (17) allows us to derive the law of L t and to price a range of multi-name credit derivatives.
7. Computational approaches • How do we compute (17)? 1 . Exact method; 2 . Poisson approximation; (19) 3 . Monte Carlo. • Exact method: multiply out the product on the RHS of (17). The individual terms of the resulting sum are exponentials of some additive functional of the chain, and can be computed explicitly. However for large portfolios this method is inefficient as we need to sum over 2 N terms. • Poisson approximation: • Conditional on the path of the chain, L t is approximately compound Poisson with parameter N N � � � � 1 − exp( − C i C i Λ t = t ) ≈ (20) t i =1 i =1 • The discounted Laplace transform of L t can be approximated by � � � t � t N � r ( ξ s ) ds − α ¯ ( ζ i ( α ) − 1) C i E exp( − L t ) = E exp − r ( ξ s ) ds + t 0 0 i =1 QT 1 ( ξ 0 ) , ¯ = e where ¯ Q jk = Q jj − ν j ( j = k ); = exp( − w jk ) Q jk ( j � = k ) .
where N � (1 − ζ i ( α )) λ i ν ≡ r + i =1 N � w i w jk ≡ jk i =1 • ...which is a simple and rapid calculation. • Monte Carlo • Calculating (17) boils down to simulating the path of the chain up to T , the maturity of the claim: 1. Let i be the current state of the chain. Generate an exponential(1) random variable z , 2. let τ denote the time elapsed from the last jump: set τ = z/q i , 3. if τ ≥ T , stop otherwise go to step 4,. 4. sample ξ ( τ ) according to probabilities ( q ij /q i ), where j � = i and j ≤ M , 5. go to step 1, and set i = ξ ( τ ). • It is enough to simulate the paths of the chain once. • A new simulation is needed only when the generator Q is altered.
8. Pricing synthetic CDOs • We have now all the ingredients to price a CDO. • The PV01 of a CDO can be seen as a portfolio of puts P t ( K ) on L t . � � � t � � M � PV 01 = ∆ i E exp − r ( ξ u ) du Φ( L T j ) , (21) 0 j =1 where �� � + � � + − � 1 L + − x L − − x Φ( x ) = , (22) L + − L − • define... � ( K − L t ) + � B − 1 P t ( K ) = E (23) t (24) • Instead of computing the Laplace transform of default distribution, we can calculate the transform of P t ( K ) directly, which saves us a (time consuming) numerical integration step � ∞ ˆ e − αx P t ( x ) dx P t ( α ) ≡ 0 � ∞ � � e − αx E B − 1 = T ( x − L t ) dx L t � t 1 = α 2 E exp( − r ( ξ u ) du − αL t ) . 0
• The default leg equals the expected present value of the tranche’s losses �� T � B − 1 DL = E u d Ξ( L u ) , (25) 0 where Ξ( x ) = 1 − Φ( x ). • Integrating by parts we can simplify (25) to � � B − 1 DL = 1 − E T Φ( L T ) �� T � r ( ξ u ) B − 1 − E u Φ( L u ) du 0 ...which can be computed using the results of the previous section.
9. A note on calibration • How do we choose the functions λ i ( · ), matrices w i and the infinitesimal generator Q ? • ... the market does it for us!! • λ i , w i and Q are used to match volatility CDS quotes and index quotes as well as tranche spreads. • The number of state of the chain can be adjusted to take into account the availability of market data.
10. Example: CDX tranche spread calibration Table 1: Market and model spreads - November 1st Market Model 5Y 7Y 10Y 5Y 7Y 10Y CDX 35 45 57 36 . 5 46 . 23 56 . 39 0 − 3% 2438 4044 5125 2438 . 4 4008 . 9 5125 3 − 7% 90 209 471 86 222 . 4 470 . 8 7 − 10% 19 46 112 19 . 1 45 . 8 99 . 7 10 − 15% 7 20 53 7 20 . 4 53 . 2 15 − 30% 3 . 5 5 . 75 14 3 . 5 5 . 0 14 . 0 30 − 100% 1 . 73 3 . 12 4 1 . 7 2 . 6 3 . 8 Table 2: Calibration error - November 1st Index Traches Absolute Error 1 . 11bp 3 . 77bp Percentage Error 2 . 70% 3 . 47%
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