Synthetic CDO Pricing Using the Student t Factor Model with Random Recovery

Transcription

1 Synthetic CDO Pricing Using the Student t Factor Model with Random Recovery UNSW Actuarial Studies Research Symposium 2006 University of New South Wales Tom Hoedemakers Yuri Goegebeur Jurgen Tistaert Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 1/28

2 Overview 1. Introduction Motivation 2. Setting and Notations 3. Modelling the Joint Default Time Distribution with Copulas 4. Random Recovery and Portfolio Distribution 5. Large Portfolio Approximation 6. Numerical Illustration 7. Conclusion and Possible Extensions Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 2/28

3 Single-Name CDS Cash Flows Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 3/28

4 What is a CDO? A CDO (collateralized debt obligation) is a portfolio of debt securities that is split into tranches of debt subordination. Each tranche protects the tranches senior to it from losses on the underlying portfolio. Typical tranches are equity tranche (no subordination) mezzanine tranche senior tranche super senior tranche Individual tranches can be rated by a rating agency. Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 4/28

5 CDO: Protection Buyers and Sellers Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 5/28

6 CDO: Credit Market Size Global Credit Derivatives Outstanding Global Issuance of Collateralized Debt Obligations: (In trillions of U.S. dollars) Cash Versus Synthetic (In billions of U.S. dollars) Corporate bonds and loans Credit derivatives Cash Synthetic Global Issuance of Collateralized Debt Obligations by Underlying Assets (in billions of U.S. dollars) Tranche Notional Value Versus Economic Risk Transfer (as a percent of the reference pool) Asset-backet securities and other structured products Noninvestment-grade bonds and leveraged loans Investment-grade bonds Bank balance sheets assets Other assets Notional Risk Transfer 0% - 3% 3%-7% 7%-10% 10%-15%15%-30% 30%-100% Equity tranche Mezzanine tranches Senior tranches Super-senior tranche Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 6/28

7 Literature Benchmark model: one factor Gaussian model Li (2000) Used by all major investments banks to communicate quotes The oponents Within the copula family Student t (O Kane & Schloegl, Lindskog & McNeil),Clayton (Schönbucher), double t (Hull & White), multifactor Gaussian, Marshall-Olkin (Giesecke, Lindskog & McNeil), random factor loadings (Andersen & Sidenius) Intensity models Affine Jump Diffusion (Hutt), Gamma (Joshi), Stochastic Networks (Davis, Backhaus & Frey, Giesecke), Structural models Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 7/28

8 Setting and Notations Notation i = 1,..., n credits T 1,..., T n default times N i nominal of credit i R i recovery rate I [0,T ] (T i ) default indicator L i = N i (1 R i ) loss given default Semi-explicit pricing for CDO tranches Default payments are based on the accumulated losses on the pool of credits: L(T ) = n L i I [0,T ] (T i ) Tranche premiums only involves call options on the accumulated losses: i=1 E[(L(T ) K) + ] Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 8/28

9 Setting and Notations Default occurs whenever a stochastic variable X i (or a stochastic process in dynamic models) lies below a critical threshold C i at the end of time period [0, T ]. In the Merton model default occurs when the value of the assets of a firm falls below the value of the firms liabilities. In order to apply these models at portfolio level we require a multivariate version of a firm-value model. In the factor copula model the critical variable X i is interpreted as the latent default time of company i. In a CDO the cash flows are functions of the whole random vector X = (X 1,..., X n ). To evaluate a CDO, all we need is today s (risk neutral) joint distribution of the X i s: P (X 1 < C 1,..., X n < C n ) Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 9/28

10 Copulas A copula captures the nonparametric, scale-invariant, distribution-free nature of the association between random variables. Theorem 1 Sklar (1959) Let X = (X 1,..., X n ) be a random vector with joint distribution function F X and marginal distribution functions F Xi, i = 1,..., n. Then there exists a copula C such that for all x R n F X (x) = C(F X1 (x 1 ),..., F Xn (x n )). (1) If F X1,..., F Xn are all continuous then C is unique. Given a copula C and marginal distribution functions F X1,..., F Xn, the function F X as defined by (1) is a joint distribution function with margins F X1,..., F Xn. The critical variables X i adopt a Gaussian/student t copula: C ΦR (u 1, u 2,..., u n ; R) = Φ R ( Φ 1 (u 1 ), Φ 1 (u 2 ),..., Φ 1 (u n ) ) C t(ν,r) (u 1, u 2,..., u n ; R) = t ν,r ( t 1 ν (u 1 ), t 1 ν (u 2 ),..., t 1 ν (u n ) ) Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 10/28

11 Gaussian vs. Student t Copula 1000 samples of Gaussian and student t copula with Kendall s τ = 0.5 Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 11/28

12 Default Time Distribution To reduce the high dimensionality of the modelling problem a set of common factors is chosen (assumed to drive the default dependency between firms) Assume that the variable Y i depends on a single common factor Z: Y i = a i Z + 1 a i 2 ε i, where Z, ε 1,..., ε n, are independent N(0,1) random variables. The critical variable X i D = Y i ν W Y i for Gaussian factor model for student t factor model with W χ 2 ν independent of Y 1,..., Y n If we didn t choose a factor structure for the X i, we would have to estimate all the 1 2n(n 1) elements of the covariance matrix of the critical variables. Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 12/28

13 Default Time Distribution The latent default times are related to the original default times in the following way T i t F Ti (T i ) F Ti (t) U i F Ti (t) Φ 1 (F Ti (t)) for Gaussian factor model X i t 1 ν (F Ti (t)) for student t factor model The model is calibrated to observable market prices of credit default swaps, i.e. the default thresholds are chosen so that they produce risk neutral default probabilities implied by quoted credit default swap spreads: C i = Φ 1 (F Ti (t)) for Gaussian factor model t 1 ν (F Ti (t)) for student t factor model Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 13/28

14 Factor Models The i th issuer defaults if X i C i or C i a i Z 1 ai ε i 2 W ν C i a i Z 1 ai 2 for Gaussian factor model for student t factor model The probability that the i th issuer defaults ( ) C i a i z P (X i C i Z = z) = Φ 1 ai 2 P (X i C i Z = z, W = w) = Φ ( w ν C ) i a i z 1 ai 2 for Gaussian factor model for student t factor model Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 14/28

15 Random Recovery The joint default time distribution describes the joint default behavior of the debtors underlying the CDO structure and hence completely determines the CDO cash flows. At default only a fraction of the nominal can be recovered. The recovery rates are assumed to be random and follow the cumulative Gaussian recovery model proposed by Andersen and Sidenius (2004): R i = Φ(µ i + b i Z + ξ i ), where ξ i N(0, σ 2 ξ i ), i = 1,..., n. The error terms ξ 1,..., ξ n are assumed to be independent from each other and also independent from Z, W and ε 1,..., ε n. The interdependence between default of the i th obligor and recovery on the j th obligor is controlled by a i b j. Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 15/28

16 Recovery Rate Density Functions with σ 2 ξ = 0.25 (a) (b) density b=0.5 b=sqrt(.5) b=sqrt(.75) b=sqrt(.9) density b=0.5 b=sqrt(.5) b=sqrt(.75) b=sqrt(.9) recovery rate recovery rate (c) density b=0.5 b=sqrt(.5) b=sqrt(.75) b=sqrt(.9) recovery rate Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 16/28

17 Portfolio Loss Distribution L(T ) The conditional portfolio loss distribution: ( ) P (L(T ) l Z = z, W = w) = F L1 z,w F Ln z,w (l), 0 l N, where N = n i=1 N i, F Li z,w (l) = P ( L i l Z = z, W = w) = 1 P (T i T Z = z, W = w)(1 P (L i l Z = z, W = w)), and P (T i T Z = z, W = w) = Φ P (L i l Z = z, W = w) = Φ ( w ν t 1 ν ) (F Ti (T )) a i z 1 ai 2 ( ) µ i + b i Z Φ 1 1 l N i σ ξi Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 17/28

18 Portfolio Loss Distribution L(T ): recursion Let K i denote the loss given default of obligor i expressed in loss units u For l = 1,..., l max i = N i /u : P (K i = 0 Z = z, W = w) = P (L i 0 Z = z, W = w) = 0 P (K i = l Z = z, W = w) = P (L i lu Z = z, W = w) P (L i (l 1)u Z = z, W = w) Let K(T ) denote the portfolio loss over [0, T ] expressed in loss units u and let P (i) denote the distribution of K(T ) for the first i obligors. Then P (i) (K(T ) = l Z = z, W = w) = min(l max,l) i k=0 P (i 1) (K(T ) = l k Z = z, W = w)p ( K i = k Z = z, W = w) The recursion starts from the boundary case of an empty portfolio for which P (0) (K(T ) = l Z = z, W = w) = δ l,0 Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 18/28

19 Portfolio Loss Distribution L(T ): FFT Characteristic function of K(T ): E ( e itk(t )) = E ( n i=1 e itk ii [0,T ] (T i ) ) = E n ( ) E e itk ii [0,T ] (T i ) Z, W with }{{} i=1 = l max i l=0 ( e itl P (T i T Z = z, W = w) + P (T i > T Z = z, W = w) ) P (K i = l Z = z, W = w) = 1 P (T i T Z = z, W = w) 1 l max i l=0 e itl P (K i = l Z = z, W = w) Apply an inverse Fourier transform to the sequence E(e i2πkk(t )/(lmax +1) ) (k = 0,..., l max = n i=1 lmax i ) Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 19/28

20 LHP Approximation Approximate the real reference credit portfolio with a portfolio consisting of a large number of equally weighted identical instruments (having the same term structure of default probabilities, recovery rates, and correlations to the common factor) Homogeneous portfolio: a i = a and C i = C for all i and the notional amounts and recovery rates R are the same for all issuers. Basel II agreement: infinitely granular portfolios LHP using a one factor Gaussian copula has become a standard model in practice. Closed-form analytic synthetic CDO pricing formula This model fails to fit the prices of different CDO tranches simultaneously: lack of tail dependence of the Gaussian copula Use student t copula Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 20/28

21 Stochastic Order Bounds Theorem: (Kaas et al., 2000) For any (X 1,, X n ) and any Z, we have that S l := n E [X i Z] cx n X i cx n F 1 X i (U) =: S c i=1 i=1 i=1 Assume that all E [X i Z] are functions of Z S l is a comonotonic sum. E[(S l d) + ] E[(S d) + ] E[(S c d) + ] The stop-loss premiums of a sum of comonotonic random variables can easily be obtained from the stop-loss premiums of the terms: E[(S c d) + ] = n E i=1 [ (Xi F 1 X i (F S c (d)) ) ], + ( F 1 S c (0) < d < F 1 S c (1) ) Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 21/28

22 Stop-loss Premium E [(X d) + ] = d F X (x)dx, =surface above the distribution function, from d on. < d < Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 22/28

23 Stop-loss Premium E [(X d) + ] = d F X (x)dx, =surface above the distribution function, from d on. < d < F (x ) X d E[(X d ) ] + Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 22/28

24 LHP Approximation X i := N i (1 R i )I [0,T ] (T i ): loss at time T associated with name i Assume X i s are independent conditionally upon Z and identically distributed: 1 n n i=1 X i E[X i Z] }{{} LP A L(T ) = X X n : aggregated loss at time T Convex order bounds: n E[X i Z] cx L(T ) cx i=1 n i=1 F 1 X i (U), with E[X i Z] = N i (1 R i )P (T i T Z). Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 23/28

25 LHP Approximation Loss on the portfolio as a percentage of the total portfolio notional: n L = (1 R) 1 n i=1 I {Xi C} The probability that the i th issuer defaults ( ) η P (X i C i η) = Φ, 1 a 2 with η = C az C W ν for Gaussian factor model az for student t factor model Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 24/28

26 LHP Approximation Law of large numbers: 1 n n ( ) η I {Xi C} Φ 1 a 2 i=1 LPH: Assume that exactly this fraction of issuers defaults for each realization of η, so that L h(η), with h : R [0, 1] and ( ) x h(x) = (1 R)Φ 1 a 2 Distribution of L: P [L θ] = F η (h 1 (θ)), θ [0, 1] CDO premiums can be easily computed! Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 25/28

27 LPH Approximation: student t factor model Distribution of the mixing variable η P [W x] = Γ ( ν 2, ) x 2 P [η t Z] = I {Z t a } + I {Z t a } Γ Cumulative distribution function F η (t): ( ) ν 2, ν(t+az)2 2C 2 P [η t] = Φ ( ) t a + 1 t a 2π Γ ( ) ν ν(t + au)2, 2 2C 2 e u2 2 du Density f η (t): f η (t) = 1 a π2 ν+1 2 Γ( ν 2 ) + 0 e 1 2a 2 (t C w ν )2 w ν 2 1 e w 2 dw Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 26/28

28 Conclusion and Possible Extensions Synthetic CDO Pricing Dependence between default times is modelled through Student t copulas. Factor approach leading to semi-analytic pricing expressions that ease model risk assessment. The loss amounts or equivalently, the recovery rates associated with defaults are random. Closed-form solutions for the loss distribution can be derived under the LHP approximation. Possible extensions: other copulas: skewed t copula, grouped t copula,... stable distributions moment matching techniques... Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 27/28

29 References [1] Andersen, L. and Sidenius, J., Extensions to the Gaussian copula: random recovery and random factor loading. Journal of Credit Risk, 1, [2] Andersen, L., Sidenius, J. and Basu, S., All your hedges in one basket. Risk, Nov [3] Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J., Statistics of extremes - theory and applications. Wiley Series in Probability and Statistics. [4] Demarta, S. and McNeil, A.J., The t copula and related copulas. To appear International Statistical Review. [5] Gibson, M.S., Understanding the risk of synthetic CDOs. Technical report Federal Reserve Board. [6] Joe, H., Multivariate models and dependence concepts. Chapman & Hall. [7] Kotz, S., Balakrishnan, N. and Johnson, N., Continuous multivariate distributions. Wiley. [8] Li, D., On Default Correlation: a Copula Approach. Journal of Fixed Income. 9, [9] Nelsen, R.B., An introduction to copulas. Springer. [10] Sklar, A., Fonctions de repartition a n dimensions et leurs marges. Publications de l Institut de Statistique de l Université de Paris, 8, Tom Hoedemakers, November 13, 2006 CDO Pricing - p. 28/28