Modeling and Estimation of Introduction Dependent Credit Rating - PowerPoint PPT Presentation

Modeling of Dependent Credit Rating Transitions Verena Goldammer Modeling and Estimation of Introduction Dependent Credit Rating Transitions The Model Simulation Likelihood Estimation Verena Goldammer References Financial and Actuarial Mathematics Christian Doppler Laboratory for Portfolio Risk Management Vienna University of Technology Linz, 04.12.2008

Outline Modeling of Dependent Credit Rating Transitions Verena Goldammer Introduction Introduction 1 The Model Simulation The Model 2 Likelihood Estimation References Simulation 3 Likelihood Estimation 4 References 5

Introduction Modeling of Dependent Credit Rating Transitions Verena Goldammer Introduction The Model Credit ratings describe the credit-worthiness of firms. Simulation We observe dependent changes of credit ratings of Likelihood Estimation different firms. References 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

Applications of Interacting Particle Systems Modeling of Dependent Credit Rating Transitions Verena Goldammer Model assumption: Introduction Credit ratings follow a time-homogeneous Markov jump process The Model Simulation with the dynamics of an interacting particle system. Likelihood Estimation Applications in the literature: References 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

Strongly Coupled Random Walk Process Modeling of Dependent Credit Rating Transitions Verena Goldammer Introduction Dynamics: The Model Independent Poisson processes with intensity λ ( x ) ≥ 0 Simulation for each rating class x . Likelihood Estimation When Poisson process for rating x jumps, then: References 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 .

Outline Modeling of Dependent Credit Rating Transitions Verena Goldammer Introduction 1 Introduction The Model State Space Sn The Model 2 State Space S State Space S n Preserving MP Embedding State Space S Simulation Preserving the Markov Property Likelihood Estimation Embedding of a Model with Fewer Firms References Simulation 3 Likelihood Estimation 4 References 5

Notation and State Spaces Modeling of Dependent Credit Rating Transitions Verena Goldammer Notation: Introduction Credit rating classes: S = { 1 , . . . , K } , where K means the The Model firm is in default and 1 is the best rating class State Space Sn State Space S Firms: F = { 1 , . . . , n } Preserving MP Embedding Simulation Possible state spaces: Likelihood Estimation 1 Assigning a rating to each individual firm: References State space S n If the firms are indistinguishable, we use the state space: 2 Counting the number of firms in the rating classes: � � η ∈ { 0 , . . . , n } S : � State space: S = η ( x ) = n x ∈ S η ∈ S : η ( x ) is the number of firms in rating class x

Parameters of the Credit Rating Process Modeling of Dependent Credit Rating Transitions Verena Parameters: Goldammer µ = ( µ xy ) x , y ∈ S : Matrix of transition intensities ( Q -matrix) Introduction of an individual firm The Model State Space Sn p = ( p x ) x ∈ S ∈ [0 , 1] S : Dependence vector State Space S Preserving MP Embedding If p x ∈ (0 , 1], define: Simulation The jump intensity of the Poisson process: Likelihood Estimation λ ( x ) = µ x References p x 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 , x � = y , µ x > 0 µ x

Process with State Space S n Modeling of Dependent Credit Rating Transitions Verena Goldammer Introduction Process with state space S n : The Model ( X t ) t ≥ 0 : Markov jump process with state space S n , State Space Sn State Space S describing the individual credit ratings of the n firms. Preserving MP Embedding X has the dynamics of the strongly coupled random walk. Simulation Likelihood Transition intensities: Estimation Independent Poisson processes at the rating classes References ⇒ 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.

Definition of Feasible Transitions of ( X t ) t ≥ 0 Modeling of Dependent Credit Rating Transitions Verena Feasible transition: Goldammer The credit rating of B firms changes from one Introduction rating class x to another rating class y � = x , The Model State Space Sn where A firms had originally rating x . State Space S Preserving MP Intensity of such a feasible transition: Embedding Simulation λ ( x ) P ( x , y ) p B x (1 − p x ) A − B Likelihood Estimation References Using the matrix µ of transition intensities of a single firm: µ xy p B − 1 (1 − p x ) A − B x 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.

Q -Matrix of the Process with State Space S n Modeling of Dependent Credit Rating Transitions Verena Notation: Goldammer z ∈ S n : Rating configurations z , � Introduction Transition z → � z feasible: The Model State Space Sn B firms change from rating class x to rating class y � = x , State Space S Preserving MP where A firms had originally rating x . Embedding Simulation A u : Number of firms with rating u in z Likelihood Estimation References Q -Matrix:  µ xy p B − 1 (1 − p x ) A − B , if z → � z is feasible,   x   − � A u − 1 � Q n ( z , � z ) = (1 − p x ) j , µ u if z = � z ,    u ∈ S j =0  0 otherwise. DSP

Process with State Space S (Indist. Firms) Modeling of Dependent Credit Rating Transitions ( η t ) t ≥ 0 : Markov jump process with state space S , which Verena Goldammer describes the number of firms in the rating classes Introduction η k xy : In configuration η , k firms with rating x change to The Model y � = x . State Space Sn State Space S Intensity of a change from η to η k xy : Preserving MP Embedding � η ( x ) � Simulation Q S ( η, η k µ xy p k − 1 (1 − p x ) η ( x ) − k Likelihood xy ) = x Estimation k References Q -matrix of the process:  if η ′ = η k Q S ( η, η k xy ) , xy for x , y ∈ S     and k ∈ { 1 , . . . , η ( x ) } ,    η ( x ) − � � Q S ( η, η ′ ) = Q S ( η, η k if η = η ′ , xy ) ,    x , y ∈ S k =1    x � = y  0 , otherwise.

Preserving the Markov Property Modeling of Dependent Credit Rating Transitions Verena Lemma (Function of MP again MP) Goldammer Introduction Assumptions: The Model S 1 , S 2 : Finite state spaces State Space Sn State Space S Preserving MP Φ : S 1 → S 2 : Arbitrary function Embedding Simulation ( X t ) t ≥ 0 : Markov jump process w.r.t. the Likelihood filtration ( F t ) t ≥ 0 , generated by a Q-matrix Q 1 Estimation and state space S 1 References Q 2 : Q-matrix, such that for all η, η ′ ∈ S 2 with η � = η ′ � Q 2 ( η, η ′ ) = Q 1 ( z , z ′ ) , for all z ∈ Φ − 1 ( η ) z ′ ∈ Φ − 1 ( η ′ ) ⇒ (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 .

Illustration of the Condition Modeling of Dependent Credit Rating Transitions � Verena Q 2 ( η, η ′ ) = Q 1 ( z , z ′ ) , for all z ∈ Φ − 1 ( η ) Goldammer z ′ ∈ Φ − 1 ( η ′ ) Introduction The Model State Space Sn State Space S Q 2 ( η , η ’ ) Preserving MP Embedding Simulation Likelihood Estimation z References Φ −1 ( η ) Q 1 (z,z’ ) Φ −1 ( η ’ )

Strongly Coupled Random Walk is Dynamics of Modeling of Dependent Credit Rating ( X t ) t ≥ 0 Transitions Verena Goldammer Introduction Theorem The Model State Space Sn Assumptions: State Space S Preserving MP Embedding ( X t ) t ≥ 0 : Markov jump process with Q-matrix Q n and Simulation state space S n Likelihood Φ : S n → S , where for z = ( z 1 , . . . , z n ) ∈ S n Estimation References n � (Φ z )( x ) = ✶ { z i = x } , x ∈ S i =1 ⇒ η t = Φ( X t ) for t ≥ 0 is a Markov jump process with state space S and Q-matrix Q S .

Embedding of a Model with Fewer Firms Modeling of Dependent Credit Rating Transitions Verena Goldammer Theorem Introduction Assumptions: The Model State Space Sn m , n ∈ ◆ with m < n: Number of firms State Space S Preserving MP Embedding ( X t ) t ≥ 0 : Markov jump process with state space S n Simulation and Q-matrix Q n Likelihood π : S n → S m : Projection with π ( x ) = x | S m Estimation References ⇒ 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