Modeling and Estimation of

Transcription

1 Modeling and of Financial and Actuarial Mathematics Christian Doppler Laboratory for Portfolio Risk Management Vienna University of Technology PRisMa

2 Outline

3 Credit ratings describe the credit-worthiness of firms. We observe dependent changes of credit ratings of different firms. Without modeling the dependence between defaults we underestimate the risk. To model dependence, we apply interacting particle systems. Advantage: Intuitive way to describe the dependence

4 Applications of Interacting Particle Systems Model assumption: Credit ratings follow a time-homogeneous Markov jump process with the dynamics of an interacting particle system. Applications in the literature: Giesecke and Weber (2004): Application of a voter model Bielecki and Vidozzi (2006), Frey and Backhaus (2007): Intensity of a credit rating transition for each firm, which depends on the configuration of the credit ratings Dai Pra, Runggaldier, Sartori and Tolotti (2007): Mean-field interaction model

5 Strongly Coupled Random Walk Process Dynamics: Independent Poisson processes with intensity λ(x) 0 for each rating class x. When Poisson process for rating x jumps, then: Rating class y is chosen with probability P(x, y). Every firm with rating x tosses a coin with probability p x of heads, independently of the other firms. If head occurs, then the firm changes the rating class from x to y.

6 Outline 1 2 State Space S n State Space S Preserving the Markov Property Embedding of a Model with Fewer Firms State Space S n State Space S Preserving MP Embedding 3 4 5

7 Notation and State Spaces Notation: Credit rating classes: S = {1,..., K}, where K means the firm is in default and 1 is the best rating class Firms: F = {1,..., n} Possible state spaces: 1 Assigning a rating to each individual firm: State space S n If the firms are indistinguishable, we use the state space: 2 Counting the number of firms in the rating classes: State space: S = { η {0,..., n} S : η(x) = n } x S η S: η(x) is the number of firms in rating class x State Space S n State Space S Preserving MP Embedding

8 Parameters of the Process Parameters: µ = (µ xy ) x,y S : Matrix of transition intensities (Q-matrix) of an individual firm p = (p x ) x S [0, 1] S : Dependence vector If p x (0, 1], define: The jump intensity of the Poisson process: λ(x) = µ x p x State Space S n State Space S Preserving MP Embedding µ x = µ xx : Intensity of a single firm to leave rating x The probability of a rating change from x to y: P(x, y) = µ xy µ x, x y, µ x > 0

9 Process with State Space S n Process with state space S n : (X t ) t 0 : Markov jump process with state space S n, describing the individual credit ratings of the n firms. X has the dynamics of the strongly coupled random walk. Transition intensities: Independent Poisson processes at the rating classes Intensity of a change of k 2 firms with different ratings is zero. Intensity of a change of k 2 firms to different ratings is zero. State Space S n State Space S Preserving MP Embedding

10 Definition of Feasible of (X t ) t 0 Feasible transition: The credit rating of B firms changes from one rating class x to another rating class y x, where A firms had originally rating x. Intensity of such a feasible transition: λ(x)p(x, y)p B x (1 p x ) A B Using the matrix µ of transition intensities of a single firm: State Space S n State Space S Preserving MP Embedding µ xy px B 1 (1 p x ) A B If p x = 0, then the intensity of a change of exactly one firm is µ xy and zero otherwise. The firms with rating x move independently.

11 Q-Matrix of the Process with State Space S n Notation: z, z S n : Rating configurations Transition z z feasible: B firms change from rating class x to rating class y x, where A firms had originally rating x. A u : Number of firms with rating u in z Q-Matrix: µ xy px B 1 (1 p x ) A B, if z z is feasible, Q n (z, z) = A u 1 µ u (1 p x ) j, if z = z, u S j=0 0 otherwise. State Space S n State Space S Preserving MP Embedding DSP

12 Process with State Space S (Indist. Firms) (η t ) t 0 : Markov jump process with state space S, which describes the number of firms in the rating classes η k xy: In configuration η, k firms with rating x change to y x. Intensity of a change from η to η k xy: ( ) η(x) Q S (η, ηxy) k = µ xy px k 1 (1 p x ) η(x) k k Q-matrix of the process: Q S (η, ηxy), k Q S (η, η ) = x,y S x y η(x) k=1 Q S (η, η k xy), if η = η, 0, otherwise. if η = η k xy for x, y S and k {1,..., η(x)}, State Space S n State Space S Preserving MP Embedding

13 Preserving the Markov Property Lemma (Function of MP again MP) Assumptions: S 1, S 2 : Finite state spaces Φ : S 1 S 2 : Arbitrary function (X t ) t 0 : Markov jump process w.r.t. the filtration (F t ) t 0, generated by a Q-matrix Q 1 and state space S 1 Q 2 : Q-matrix, such that for all η, η S 2 with η η Q 2 (η, η ) = z Φ 1 (η ) Q 1 (z, z ), for all z Φ 1 (η) State Space S n State Space S Preserving MP Embedding (Φ(X t )) t 0 is a Markov jump process w.r.t. (F t ) t 0 with state space S 2, generated by the Q-matrix Q 2.

14 Illustration of the Condition Q 2 (η, η ) = Q 1 (z, z ), z Φ 1 (η ) Q 2 (η,η ) z for all z Φ 1 (η) State Space S n State Space S Preserving MP Embedding Φ 1 (η) Q 1 (z,z ) Φ 1 (η )

15 Strongly Coupled Random Walk is Dynamics of (X t ) t 0 Theorem Assumptions: (X t ) t 0 : Markov jump process with Q-matrix Q n and state space S n Φ : S n S, where for z = (z 1,..., z n ) S n (Φz)(x) = n 1 {zi =x}, x S i=1 State Space S n State Space S Preserving MP Embedding η t = Φ(X t ) for t 0 is a Markov jump process with state space S and Q-matrix Q S.

16 Embedding of a Model with Fewer Firms Theorem Assumptions: m, n N with m < n: Number of firms (X t ) t 0 : Markov jump process with state space S n and Q-matrix Q n π : S n S m : Projection with π(x) = x S m State Space S n State Space S Preserving MP Embedding Y t = π(x t ) for t 0 is a Markov jump process with state space S m, generated by Q-matrix Q m. Intensity of a rating change of m firms in a model with n firms = Intensity of a change in a model with m firms

17 Outline Portfolio of the Profit and Loss Portfolio 4 5

18 Portfolio and Profit and Loss Function Portfolio: n defaultable zero-coupon bonds issued by n firms with face value f i and maturity T i for i = 1,..., n Simplification: No recovery and zero default-free interest rate Portfolio value at future time t min{t 1,..., T n }: V (t) = = n f i P[X i (T i ) K X t ] i=1 n f i (1 ( exp{µ(t i t)} ) ) X i (t),k i=1 Portfolio Profit and loss during (0, t]: P&L(t) = V (t) V (0)

19 of the Loss of the Portfolio Model assumptions: K = 8 credit rating classes Initial rating: All firms have rating 4. Movement according to our model with generator µ based on data of Standard & Poor s Dependence vector (p x ) x S : p x = p for all x S Portfolio Portfolio assumptions: Number of bonds: n = 100 Face value: f i = 1 for every bond Common maturity: T i = 10 years

20 Distribution Function of Profit and Loss P(P&L(5) L) p=0 p=0.3 p=0.7 p=1 Portfolio L Figure: Distribution of the profit and loss of the portfolio at time t = 5 for n = 100 firms (1 000 simulations). The dependence parameter p x equals p for each rating class. Initial value of the portfolio is V (0) = 95.

21 Distribution Function of Profit and Loss P(P&L(5) L) p=0 p=0.3 p=0.7 p=1 Portfolio L Figure: Distribution of the profit and loss of the portfolio at time t = 5 for n = 100 firms (1 000 simulations). The dependence parameter p x equals p for each rating class. Initial value of the portfolio is V (0) = 95.

22 Outline Sample Paths Function Maximum Estimator Sample Paths Function MLE 5

23 Rating Process with Varying Number of Firms Notation: (X t,n ) t 0 for n N: Markov jump process with state space S n and Q-matrix Q n X 0,n : Initial rating, independent of the parameters p and µ N: Random number of firms, independent of (X t,n ) t 0 for all n N and independent of the parameters p and µ Credit rating process with N firms: Sample Paths Function MLE (Y t ) t 0 = (X t,n ) t 0 We observe independent samples of the credit rating process (Y t ) t [0,T ).

24 Space of the Sample Paths of X t,n Take modification of Y with piecewise constant sample paths with finite number of jumps Sample path of the process (X t,n ) t [0,T ), which jumps m times in [0, T ): That means: ((z 0, t 0 ),..., (z m 1, t m 1 ), z m ) Process starts in z 0 and remains there for time t 0. Then it jumps to z 1 and remains there for time t 1. Sample Paths Function MLE. Finally it reaches z m and stays there until T.

25 Space of the Sample Paths of Y Space of the sample paths of (X t,n ) t [0,T ) with m jumps: S n,m = (S n R) m S n S n,0 = S n for m = 0 Space of sample paths of (X t,n ) t [0,T ) : S n = m=0 S n,m for m N Sample Paths Function MLE Space of sample paths of the process Y = X N : S Y = n=1 S n

26 Measure on Space of Sample Paths V: Lebesgue Borel measure on R N n : Counting measure on S n σ n,m : Product measure on S n,m with σ n,m = (N n V) m N n B: Smallest σ-algebra containing all sets of S Y whose intersection with S n,l is a Borel set for each pair (n, l) N N 0 Define the measures: σ n (C S n ) = σ n,m (C S n,m ) m=0 for C B Sample Paths Function MLE and σ(c) = σ n (C S n ) n=1 for C B

27 Feasible Path of X t,n Feasible path of (X t,n ) t [0,T ) : Start in z 0 is possible, i.e. P(X 0,n = z 0 ) > 0. All transitions are feasible. Define for feasible path ω: : Number of simultaneous transitions of B firms from rating class x to y x, A firms originally with rating x N (ω) x,y,a,b T (ω) x,a : Total time that exactly A firms have rating x Sample Paths Function MLE

28 Density of a Sample Path Theorem Let E be a measurable subset of S Y. Then for n N with P(N = n) > 0 P ( (Y t ) t [0,T ) E N = n ) = f Qn (ω) σ n (dω), E S n where f Qn (ω) = 0, if the path is not feasible, and for a feasible path ω S n { f Qn (ω) = P(X 0,n = z 0 ) exp x,y S x y n A,B=1 A B ( n T (ω) x,a µ x A=1 x S µ xy p B 1 x A 1 } (1 p x ) j j=0 (1 p x ) A B) (ω) N x,y,a,b. Sample Paths Function MLE Qn

29 Function ω 1,..., ω k S Y : Feasible sample paths n j : Number of firms in path ω j n := max j {1,...,k} n j : Maximal number of observed firms N x,y,a,b : Total number of simultaneous rating changes of B firms from x to y x, A firms originally with rating x T x,a : Total time that exactly A firms have rating x Conditional likelihood function, given N 1 = n 1,..., N k = n k : Sample Paths Function MLE L(ω 1,..., ω k ) = x,y S x y n A,B=1 A B ( ( k ) P(X 0,nj = z j 0 ) exp j=1 µ xy p B 1 x { x S (1 p x ) A B) N x,y,a,b µ x n A=1 T x,a A 1 } (1 p x ) j j=0

30 Sets Subsets of the rating set S = {1,..., K}, depending on the observed feasible paths of Y : x S T >0, iff at least one firm has rating x temporarily during the observation period Parameter estimation is possible x S p>0, iff at least two firms with rating x change the rating simultaneously at some time Firms change the rating x not independently Sample Paths Function MLE x S p<1, iff at least at one time not all firms with rating x change this rating together Firms are not totally dependent

31 Maximum Estimator for the Q-Matrix µ Theorem (MLE for µ) A maximum likelihood estimator (MLE) of the parameter vector [p, µ] is given as follows. (a) For (x, y) S T >0 S with x y the MLE of entry µ xy : ˆµ xy = n A N x,y,a,b A=1 B=1 n A 1 T x,a A=1 j=0 (1 ˆp x ) j, Sample Paths Function MLE where ˆp x is the MLE of p x, which is given in the next slides. If p x = 0, then we get the same MLE in this model for µ as in the independent model: n ˆµ ind A=1 xy = N x,y,a,1 n A=1 A T x,a

32 Max. Estimator for Dep. Parameter p x Theorem (MLE for p) (b) For the rating classes x S p>0 S p<1 the MLE of p x is uniquely determined. The MLE is the unique root in (0, 1) of the polynomial of degree n with coefficients c 0 = n A,B=1 A B c j = ( 1) j (B 1)Ñx,A,B n A,B=1 A B Ñ x,a,b n i T x,i i=1 n i=j ( ) i ( i j ) j j + 1 B + A i T x,i for j {1,..., n}. Sample Paths Function MLE Ñ x,a,b : Total number of rating changes of B firms with rating x, A firms originally with rating x

33 Max. Estimator for Dep. Parameter p x Theorem (MLE for p) (c) For x S p>0 \ S p<1 the MLE is ˆp x = 1. In the remaining case of insufficient observations: (d) If x S p<1 \ S p>0, the MLE for p x is one of the following: ˆp x = 0 or the unique root of the polynomial with coefficients as above. (e) For x S T >0 \ (S p>0 S p<1 ) the MLE is ˆp x = 1, if there is a change, where only one firm has rating x and is leaving x, and at least two firms have rating x for a time interval. Sample Paths Function MLE

34 Further Research Asymptotic properties of the estimator: Consistency? Asymptotical normality? Extension of the model: Observations of the credit ratings only at discrete time points of the parameters using historical rating transitions Independent copies of the process, each one symbolizing an sector of industry Sample Paths Function MLE

35 T. Bielecki, A. Vidozzi and L. Vidozzi. An efficient approach to valuation of credit basket products and ratings triggered step-up bonds, Working paper, Illinois Institute of Technology, 2006 R. Frey and J. Backhaus. Pricing and hedging of portfolio credit derivatives with interacting default intensities, Working paper, University of Leipzig, 2007 K. Giesecke and S. Weber. Cyclical correlation, credit contagion, and portfolio losses, Journal of Banking and Finance, 2004

36 P. Dai Pra, W. Runggaldier, E. Sartori and M. Tolotti. Large portfolio losses; a dynamic contagion model, Working paper, University of Padova, F. Spitzer. Infinite systems with locally interacting components, The Annals of Probability, 1981.