Marshall-Olkin distributions and portfolio credit risk
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1 Marshall-Olkin distributions and portfolio credit risk Moderne Finanzmathematik und ihre Anwendungen für Banken und Versicherungen, Fraunhofer ITWM, Kaiserslautern, in Kooperation mit der TU München und DEVnet GmbH & Co KG, December 4, 2009 Jan-Frederik Mai HVB-Institute for Mathematical Finance Technische Universität München Joint work with Matthias Scherer.
2 Overview 1. Motivation 2. Exchangeable Marshall-Olkin distributions 3. The Lévy-frailty default model 4. Conclusion
3 Motivation: CDO pricing Situation: > Consider a portfolio of d credit-risky assets.
4 Motivation: CDO pricing Situation: > Consider a portfolio of d credit-risky assets. > (τ 1,..., τ d ) vector of (random) default times of the underlyings.
5 Motivation: CDO pricing Situation: > Consider a portfolio of d credit-risky assets. > (τ 1,..., τ d ) vector of (random) default times of the underlyings. > Typically, d = 125, i.e. large.
6 Motivation: CDO pricing Situation: > Consider a portfolio of d credit-risky assets. > (τ 1,..., τ d ) vector of (random) default times of the underlyings. > Typically, d = 125, i.e. large. > Constant and identical recovery rates, and equally weighted portfolio.
7 Motivation: CDO pricing Situation: > Consider a portfolio of d credit-risky assets. > (τ 1,..., τ d ) vector of (random) default times of the underlyings. > Typically, d = 125, i.e. large. > Constant and identical recovery rates, and equally weighted portfolio. > L t := 1 d d k=1 1 {τ k <t}= percentage of defaults up to time t 0.
8 Motivation: CDO pricing Situation: > Consider a portfolio of d credit-risky assets. > (τ 1,..., τ d ) vector of (random) default times of the underlyings. > Typically, d = 125, i.e. large. > Constant and identical recovery rates, and equally weighted portfolio. > L t := 1 d d k=1 1 {τ k <t}= percentage of defaults up to time t 0. CDO = financial contract whose payment streams depend on L t.
9 Motivation: CDO pricing Pricing of a CDO requires one to compute expectations of the form E [ f(l t ) ] = f(x) P(L t dx), f complicated. [0,1]
10 Motivation: CDO pricing Pricing of a CDO requires one to compute expectations of the form E [ f(l t ) ] = f(x) P(L t dx), f complicated. [0,1] Goal: Model (τ 1,..., τ d ) such that... (1)... one can approximate P(L t dx) efficiently (without Monte Carlo). (2)... dependence between the τ k is (economically) intuitive.
![[0,1] Goal: Model (τ 1,..., τ d ) such that... (1).](/docs-images/45/10137537/images/page_10.jpg ".. one can approximate P(L t dx) efficiently (without Monte Carlo). (2).")
11 Latent One-Factor Copula Models Large Homogeneous Portfolio Assumption: > Infinite portfolio size, i.e. (τ 1,..., τ d ) {τ k } k N. > {τ k } k N is i.i.d. conditioned on a common market factor M.
12 Latent One-Factor Copula Models Large Homogeneous Portfolio Assumption: > Infinite portfolio size, i.e. (τ 1,..., τ d ) {τ k } k N. > {τ k } k N is i.i.d. conditioned on a common market factor M. Advantage: P(L t dx) P(M dx).
13 Latent One-Factor Copula Models Large Homogeneous Portfolio Assumption: > Infinite portfolio size, i.e. (τ 1,..., τ d ) {τ k } k N. > {τ k } k N is i.i.d. conditioned on a common market factor M. Advantage: P(L t dx) P(M dx). Examples: > [Li 2000]: M normal r.v. (τ 1,..., τ d ) Gaussian copula. > [Albrecher et al. 2007]: M ID r.v. (τ 1,..., τ d )???-copula. > [Schönbucher 2002]: M positive r.v. (τ 1,..., τ d ) Archimedean copula.
![Advantage: P(L t dx) P(M dx). Examples: > [Li 2000]: M normal r.v. (τ 1,..., τ d ) Gaussian copula.](/docs-images/45/10137537/images/page_13.jpg "> [Albrecher et al. 2007]: M ID r.v. (τ 1,..., τ d )???-copula. > [Schönbucher 2002]: M positive r.v. (τ 1,..., τ d ) Archimedean copula.")
14 Latent One-Factor Copula Models Large Homogeneous Portfolio Assumption: > Infinite portfolio size, i.e. (τ 1,..., τ d ) {τ k } k N. > {τ k } k N is i.i.d. conditioned on a common market factor M. Advantage: P(L t dx) P(M dx). Examples: > [Li 2000]: M normal r.v. (τ 1,..., τ d ) Gaussian copula. > [Albrecher et al. 2007]: M ID r.v. (τ 1,..., τ d )???-copula. > [Schönbucher 2002]: M positive r.v. (τ 1,..., τ d ) Archimedean copula. Problem: (economically) intuitive?
![Examples: > [Li 2000]: M normal r.v. (τ 1,..., τ d ) Gaussian copula. > [Albrecher et al. 2007]: M ID r.v. (τ 1,..., τ d )???-copula.](/docs-images/45/10137537/images/page_14.jpg "> [Schönbucher 2002]: M positive r.v. (τ 1,..., τ d ) Archimedean copula. Problem: (economically) intuitive?")
15 Our idea: Marshall-Olkin copula Assume (τ 1,..., τ d ) Marshall-Olkin copula. Proposed by [Giesecke 2003; Embrechts et al. 2001; Lindskog, McNeil 2003].
16 Our idea: Marshall-Olkin copula Assume (τ 1,..., τ d ) Marshall-Olkin copula. Proposed by [Giesecke 2003; Embrechts et al. 2001; Lindskog, McNeil 2003]. Advantage: intuitive shock interpretation.
17 Our idea: Marshall-Olkin copula Assume (τ 1,..., τ d ) Marshall-Olkin copula. Proposed by [Giesecke 2003; Embrechts et al. 2001; Lindskog, McNeil 2003]. Advantage: intuitive shock interpretation. Problem: P(L t dx) intractable, since... > No latent market factor no large homogeneous portfolio assumption. > The model for (τ 1,..., τ d ) cannot be extended to {τ k } k N.
18 Our idea: Marshall-Olkin copula Assume (τ 1,..., τ d ) Marshall-Olkin copula. Proposed by [Giesecke 2003; Embrechts et al. 2001; Lindskog, McNeil 2003]. Advantage: intuitive shock interpretation. Problem: P(L t dx) intractable, since... > No latent market factor no large homogeneous portfolio assumption. > The model for (τ 1,..., τ d ) cannot be extended to {τ k } k N. Solution: We show how to define {τ k } k N appropriately.
19 Overview 1. Motivation 2. Exchangeable Marshall-Olkin distributions 3. The Lévy-frailty default model 4. Conclusion
20 Exchangeable Marshall-Olkin distributions Definition (in dimension 3) On a probability space (Ω, F, P) define the following independent random variables (= arrival times of exogeneous shocks ): E {1} E {2} E {3} Exp(λ 1 ) E {1,2} E {1,3} E {2,3} Exp(λ 2 ) E {1,2,3} Exp(λ 3 )
21 Exchangeable Marshall-Olkin distributions Definition (in dimension 3) On a probability space (Ω, F, P) define the following independent random variables (= arrival times of exogeneous shocks ): E {1} E {2} E {3} Exp(λ 1 ) E {1,2} E {1,3} E {2,3} Exp(λ 2 ) E {1,2,3} Exp(λ 3 ) Define the vector of extinction times ( τ 1, τ 2, τ 3) via τ 1 : = min { E {1} } { E {1,2}, E {1,3} } { E {1,2,3} } τ 2 : = min { E {2} } { E {1,2}, E {2,3} } { E{1,2,3} } τ 3 : = min { E {3} } { E{1,3}, E {2,3} } { E{1,2,3} } Exp(λ λ 2 + λ 3 ) Exp(λ λ 2 + λ 3 ) Exp(λ λ 2 + λ 3 )
22 Exchangeable Marshall-Olkin distributions General case Method of construction: [Marshall, Olkin (1967)] > Given: parameters λ 1,..., λ d > 0. > Consider (Ω, F, P) where E I Exp(λ I ) are independent, I {1,..., d}. > (τ 1,..., τ d ) has a so-called Marshall-Olkin distribution, where { } τ k := min EI, k = 1,..., d. I:k I
23 Exchangeable Marshall-Olkin distributions General case Method of construction: [Marshall, Olkin (1967)] > Given: parameters λ 1,..., λ d > 0. > Consider (Ω, F, P) where E I Exp(λ I ) are independent, I {1,..., d}. > (τ 1,..., τ d ) has a so-called Marshall-Olkin distribution, where { } τ k := min EI, k = 1,..., d. I:k I Intuitive Shock interpretation: E I = arrival time of an exogenous economy shock, destroys all components k I.
24 Exchangeable Marshall-Olkin copula Theorem: [Marshall, Olkin 1967] The exch. Marshall-Olkin distribution with parameters λ 1,..., λ d > 0 is given by ) ( ) P(τ 1 > t 1,..., τ d > t d ) = C d (P(τ 1 > t 1 ),..., P(τ d > t d ) = C d e O t 1,..., e O t d, where O := d 1 i=0 ( d 1 ) i λi+1 and C d (u 1,..., u d ) := d k=1 u 1 O d k i=0 ( d k i )λ i+1 (k), 0 u (1)... u (d) 1.
25 Exchangeable Marshall-Olkin copula Theorem: [Marshall, Olkin 1967] The exch. Marshall-Olkin distribution with parameters λ 1,..., λ d > 0 is given by ) ( ) P(τ 1 > t 1,..., τ d > t d ) = C d (P(τ 1 > t 1 ),..., P(τ d > t d ) = C d e O t 1,..., e O t d, where O := d 1 i=0 ( d 1 ) i λi+1 and C d (u 1,..., u d ) := d k=1 u 1 O d k i=0 ( d k i )λ i+1 (k), 0 u (1)... u (d) 1. C d is the survival copula of the exchangeable Marshall-Olkin distribution.
26 Suitable for portfolio credit risk? Problems: > P(L t dx) is complicated, especially for d 2.
27 Suitable for portfolio credit risk? Problems: > P(L t dx) is complicated, especially for d 2. > For large d 2, even Monte Carlo is impossible (O(2 d )).
28 Suitable for portfolio credit risk? Problems: > P(L t dx) is complicated, especially for d 2. > For large d 2, even Monte Carlo is impossible (O(2 d )). > Multiple shock model latent factor representation?
29 Suitable for portfolio credit risk? Problems: > P(L t dx) is complicated, especially for d 2. > For large d 2, even Monte Carlo is impossible (O(2 d )). > Multiple shock model latent factor representation? > We do not want exponential margins.
30 Suitable for portfolio credit risk? Problems: > P(L t dx) is complicated, especially for d 2. > For large d 2, even Monte Carlo is impossible (O(2 d )). > Multiple shock model latent factor representation? > We do not want exponential margins. Our contribution: We identify a very tractable subclass, where everything works out and interesting mathematical byproducts are obtained.
31 Overview 1. Motivation 2. Exchangeable Marshall-Olkin distributions 3. The Lévy-frailty default model 4. Conclusion
32 Lévy Subordinators (1) A Lévy subordinator Λ = {Λ t } t 0 is a non-decreasing Lévy process, i.e. > Λ 0 = 0 and non-decreasing > independent and stationary increments > stochastically continuous
33 Lévy Subordinators (1) A Lévy subordinator Λ = {Λ t } t 0 is a non-decreasing Lévy process, i.e. > Λ 0 = 0 and non-decreasing > independent and stationary increments > stochastically continuous Theorem: [Lévy 1937, Khinchin 1938] Λ Lévy subordinator Ψ : [0, ) [0, ) : E[e x Λ t ] = e t Ψ(x), x 0, t 0.
34 Lévy Subordinators (1) A Lévy subordinator Λ = {Λ t } t 0 is a non-decreasing Lévy process, i.e. > Λ 0 = 0 and non-decreasing > independent and stationary increments > stochastically continuous Theorem: [Lévy 1937, Khinchin 1938] Λ Lévy subordinator Ψ : [0, ) [0, ) : E[e x Λ t ] = e t Ψ(x), x 0, t 0. Ψ is called Laplace exponent, concave, in C (0, ).
35 Lévy Subordinators (2) Example 1:(Compound Poisson subordinator) Λ t = µ t + N t i=1 Ψ(x) = µ x + E[N 1 ] ( 1 E[e x J 1 ] ) J i, N = {N t } t 0 Poisson process, {J i } i N i.i.d. > 0, Λ(t) (Compound Poisson with Drift) Λ(t) (Compound Poisson with Drift) Time t Time t
36 Lévy Subordinators (3) Example 2:(Inverse Gaussian subordinator) Λ t = inf{s > 0 : η s + W s = β t}, Ψ(x) = β ( 2 x + η2 η ) {W s } s 0 Brownian motion, Λ(t) (Inverse Gaussian) Λ(t) (Inverse Gaussian) Time t Time t
37 Lévy Subordinators (4) Example 3:(Stable subordinator) For α [0, 1] there exists a Lévy subordinator Λ with Ψ(x) = x 1 α Λ(t) (α stable) Λ(t) (α stable) Time t Time t
38 Lévy-frailty construction Main Theorem: > G = continuous distribution function,, G(0) = 0, G(t) < 1 for all t > 0.
39 Lévy-frailty construction Main Theorem: > G = continuous distribution function,, G(0) = 0, G(t) < 1 for all t > 0. > Λ = {Λ t } t 0 Lévy subordinator satisfying Ψ(1) = 1.
40 Lévy-frailty construction Main Theorem: > G = continuous distribution function,, G(0) = 0, G(t) < 1 for all t > 0. > Λ = {Λ t } t 0 Lévy subordinator satisfying Ψ(1) = 1. > {E k } k N i.i.d. with E 1 Exp(1), independent of Λ.
41 Lévy-frailty construction Main Theorem: > G = continuous distribution function,, G(0) = 0, G(t) < 1 for all t > 0. > Λ = {Λ t } t 0 Lévy subordinator satisfying Ψ(1) = 1. > {E k } k N i.i.d. with E 1 Exp(1), independent of Λ. Define: τ k := inf { t > 0 : Λ log(1 G(t)) > E k }, k N.
42 Lévy-frailty construction Main Theorem: > G = continuous distribution function,, G(0) = 0, G(t) < 1 for all t > 0. > Λ = {Λ t } t 0 Lévy subordinator satisfying Ψ(1) = 1. > {E k } k N i.i.d. with E 1 Exp(1), independent of Λ. Define: τ k := inf { t > 0 : Λ log(1 G(t)) > E k }, k N. Then: P(τ 1 > t 1,..., τ d > t d ) = C d ( 1 G(t1 ),..., 1 G(t d ) ), where C d is the survival copula of a certain Marshall-Olkin distribution.
43 Some Remarks G(t) = P(τ k t)= default probability at time t 0 (known beforehand).
44 Some Remarks G(t) = P(τ k t)= default probability at time t 0 (known beforehand). Above theorem constitutes a (proper) subclass of exch. Marshall-Olkin copulas.
45 Some Remarks G(t) = P(τ k t)= default probability at time t 0 (known beforehand). Above theorem constitutes a (proper) subclass of exch. Marshall-Olkin copulas. Is this alternative construction useful?
46 Some Remarks G(t) = P(τ k t)= default probability at time t 0 (known beforehand). Above theorem constitutes a (proper) subclass of exch. Marshall-Olkin copulas. Is this alternative construction useful? Yes. Now we have a latent one-factor model: > G := σ(λ t : t 0) path of a Lévy subordinator. > (τ 1,..., τ d ) are i.i.d. conditioned on the latent factor G.
47 The Portfolio Loss Distribution Exact portfolio loss distribution: ( ) d m ( ) m P(L t = m) = ( 1) k (1 G(t)) Ψ(d+k m) m k k=0 Example: Ψ(x) = x 1 α, G(t) 26%, d = α= α= α=
48 Approximation of Portfolio Loss Corollary: As the portfolio size d, it holds that {L t } t [0,T ] L 2 { } 1 e Λ log(1 G(t)) t [0,T ].
49 Approximation of Portfolio Loss Corollary: As the portfolio size d, it holds that {L t } t [0,T ] L 2 In particular, provided Λ t has a density f Λt { } 1 e Λ log(1 G(t)) t [0,T ]. for all t > 0, we have that ( ) 1 f Lt (x) f Λ log(1 G(t)) log(1 x), x (0, 1). 1 x
50 Approximation of Portfolio Loss Corollary: As the portfolio size d, it holds that {L t } t [0,T ] L 2 In particular, provided Λ t has a density f Λt { } 1 e Λ log(1 G(t)) t [0,T ]. for all t > 0, we have that ( ) 1 f Lt (x) f Λ log(1 G(t)) log(1 x), x (0, 1). 1 x Thus, we can efficiently compute E [ f(l t ) ] = f(x) P(L t dx) [0,1] [0,1] f(x) f Λ log(1 G(t)) ( log(1 x) ) 1 1 x dx.
51 Inverse Gaussian density of the portfolio loss One-parametric Inverse Gaussian Lévy subordinator (parameter η > 0): ) ( ( f Lt (x) log(1 G(t)) η log(1 G(t)) log(1 G(t)) log 2 (1 x) 2 π ( 2+η 2 η) exp ( 2+η 2 η x) exp 2 ( log(1 x)) ( 2+η η2 log(1 x) 2 η) 2 2 ), x 0. density value η = 10 η = 1 η = loss percentage
52 Overview 1. Motivation 2. Exchangeable Marshall-Olkin distributions 3. The Lévy-frailty default model 4. Conclusion
53 Conclusion Latent factor representation for certain Marshall-Olkin distributions
54 Conclusion Latent factor representation for certain Marshall-Olkin distributions Very useful to study Marshall-Olkin distributions
55 Conclusion Latent factor representation for certain Marshall-Olkin distributions Very useful to study Marshall-Olkin distributions Interesting coherences between copulas, completely monotone sequences, Lévy subordinators
56 Conclusion Latent factor representation for certain Marshall-Olkin distributions Very useful to study Marshall-Olkin distributions Interesting coherences between copulas, completely monotone sequences, Lévy subordinators Efficient approximation of portfolio loss distribution quick calibration to market data
57 Conclusion Latent factor representation for certain Marshall-Olkin distributions Very useful to study Marshall-Olkin distributions Interesting coherences between copulas, completely monotone sequences, Lévy subordinators Efficient approximation of portfolio loss distribution quick calibration to market data The model is more dynamic than state-of-the-art one-factor models, since latent factor is a stochastic process
58 References with M. Scherer: A tractable multivariate default model based on a stochastic time-change. International Journal of Theoretical and Applied Finance, 12:2, pp (2009). with M. Scherer: Lévy-Frailty Copulas. Journal of Multivariate Analysis, 100:7, pp (2009). with M. Scherer: Reparameterizing Marshall-Olkin copulas with applications to sampling. Journal of Statistical Computation and Simulation (in press) (2009).
59 Thank you for your attention.
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