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Modeling Default Correlation and Clustering: A Time Change Approach

Rafael Mendoza-Arriaga1 Joint work with: Vadim Linetsky2

1- McCombs School of Business (IROM) 2- Northwestern University

The Third Western Conference in Mathematical Finance Santa Barbara, CA November, 2009

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 1 / 26

Introduction

Corporate defaults are not independent

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 2 / 26

Introduction

Corporate defaults are not independent Default dependence structure is complex and multi-faceted. For example:

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 2 / 26

Introduction

Corporate defaults are not independent Default dependence structure is complex and multi-faceted. For example:

Customer-Supplier: Delphi vs GM and Visteon vs Ford

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 2 / 26

Introduction

Corporate defaults are not independent Default dependence structure is complex and multi-faceted. For example:

Customer-Supplier: Delphi vs GM and Visteon vs Ford Counterparty Risk: Lehman vs. other financial institutions

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 2 / 26

Introduction

Corporate defaults are not independent Default dependence structure is complex and multi-faceted. For example:

Customer-Supplier: Delphi vs GM and Visteon vs Ford Counterparty Risk: Lehman vs. other financial institutions Regional risk factors: automotive bankruptcies affecting credit-worthiness of many Michigan firms

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 2 / 26

Introduction

Corporate defaults are not independent Default dependence structure is complex and multi-faceted. For example:

Customer-Supplier: Delphi vs GM and Visteon vs Ford Counterparty Risk: Lehman vs. other financial institutions Regional risk factors: automotive bankruptcies affecting credit-worthiness of many Michigan firms Industry sector-specific risk factors: high oil prices affecting credit-worthiness

• f the airline industry

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 2 / 26

Introduction

Corporate defaults are not independent Default dependence structure is complex and multi-faceted. For example:

Customer-Supplier: Delphi vs GM and Visteon vs Ford Counterparty Risk: Lehman vs. other financial institutions Regional risk factors: automotive bankruptcies affecting credit-worthiness of many Michigan firms Industry sector-specific risk factors: high oil prices affecting credit-worthiness

• f the airline industry

Systematic Risk: recession affecting credit-worthiness of nearly all firms in the nation

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 2 / 26

Introduction

Corporate defaults are not independent Default dependence structure is complex and multi-faceted. For example:

Customer-Supplier: Delphi vs GM and Visteon vs Ford Counterparty Risk: Lehman vs. other financial institutions Regional risk factors: automotive bankruptcies affecting credit-worthiness of many Michigan firms Industry sector-specific risk factors: high oil prices affecting credit-worthiness

• f the airline industry

Systematic Risk: recession affecting credit-worthiness of nearly all firms in the nation Systemic Risk: high degree of interconnectedness in the financial system through counterparty risk could have lead to a catastrophic failure of the whole system if another major financial institution of the size of Lehman or larger were allowed to fail

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 2 / 26

Introduction

Corporate defaults are not independent Default dependence structure is complex and multi-faceted. For example:

Customer-Supplier: Delphi vs GM and Visteon vs Ford Counterparty Risk: Lehman vs. other financial institutions Regional risk factors: automotive bankruptcies affecting credit-worthiness of many Michigan firms Industry sector-specific risk factors: high oil prices affecting credit-worthiness

• f the airline industry

Systematic Risk: recession affecting credit-worthiness of nearly all firms in the nation Systemic Risk: high degree of interconnectedness in the financial system through counterparty risk could have lead to a catastrophic failure of the whole system if another major financial institution of the size of Lehman or larger were allowed to fail

Default events sometimes happen simultaneously (default clustering). For example (CDS settlement events in 2008):

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 2 / 26

Introduction

Corporate defaults are not independent Default dependence structure is complex and multi-faceted. For example:

Customer-Supplier: Delphi vs GM and Visteon vs Ford Counterparty Risk: Lehman vs. other financial institutions Regional risk factors: automotive bankruptcies affecting credit-worthiness of many Michigan firms Industry sector-specific risk factors: high oil prices affecting credit-worthiness

• f the airline industry

Systematic Risk: recession affecting credit-worthiness of nearly all firms in the nation Systemic Risk: high degree of interconnectedness in the financial system through counterparty risk could have lead to a catastrophic failure of the whole system if another major financial institution of the size of Lehman or larger were allowed to fail

Default events sometimes happen simultaneously (default clustering). For example (CDS settlement events in 2008):

Oct (2 week span): Freddie and Fannie (6th), LEH (10th), WaMu (23rd)

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 2 / 26

Introduction

Corporate defaults are not independent Default dependence structure is complex and multi-faceted. For example:

Customer-Supplier: Delphi vs GM and Visteon vs Ford Counterparty Risk: Lehman vs. other financial institutions Regional risk factors: automotive bankruptcies affecting credit-worthiness of many Michigan firms Industry sector-specific risk factors: high oil prices affecting credit-worthiness

• f the airline industry

Systematic Risk: recession affecting credit-worthiness of nearly all firms in the nation Systemic Risk: high degree of interconnectedness in the financial system through counterparty risk could have lead to a catastrophic failure of the whole system if another major financial institution of the size of Lehman or larger were allowed to fail

Default events sometimes happen simultaneously (default clustering). For example (CDS settlement events in 2008):

Oct (2 week span): Freddie and Fannie (6th), LEH (10th), WaMu (23rd) Nov (1 week span): Landsbanki (4th), Glitnir (5th), Kaupthing Bank (6th)

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 2 / 26

Default Correlation and Clustering via Multivariate L´ evy Subordination

. . . . . . . . .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 3 / 26

Default Correlation and Clustering via Multivariate L´ evy Subordination

Consider n obligors with each default time, τi, defined by τi = inf{t ≥ 0 : Λi

t ≥ Ei}, i = 1, ..., n,

. . . . . . . . .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 3 / 26

Default Correlation and Clustering via Multivariate L´ evy Subordination

Consider n obligors with each default time, τi, defined by τi = inf{t ≥ 0 : Λi

t ≥

Ei

• exp(1) iid

}, i = 1, ..., n, . . . . . . . . .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 3 / 26

Default Correlation and Clustering via Multivariate L´ evy Subordination

Consider n obligors with each default time, τi, defined by τi = inf{t ≥ 0 : Λi

t

≥ Ei}, i = 1, ..., n, .

Default Hazard Process Λi

t

. . . . . . . . Consider the following processes:

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 3 / 26

Default Correlation and Clustering via Multivariate L´ evy Subordination

Consider n obligors with each default time, τi, defined by τi = inf{t ≥ 0 : Λi

t

≥ Ei}, i = 1, ..., n, .

Default Hazard Process Λi

t

. . . . . . . . Consider the following processes:

X a

s , a = 1, ..., d; are d independent one-dimensional, non-negative Markov

process starting from X a

0 = xa ≥ 0

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 3 / 26

Default Correlation and Clustering via Multivariate L´ evy Subordination

Consider n obligors with each default time, τi, defined by τi = inf{t ≥ 0 : Λi

t

≥ Ei}, i = 1, ..., n, .

Default Hazard Process Λi

t

. . . . . . . . Consider the following processes:

X a

s , a = 1, ..., d; are d independent one-dimensional, non-negative Markov

process starting from X a

0 = xa ≥ 0

Y a

t =

R t

0 X a s ds, the corresponding integrated processes

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 3 / 26

Default Correlation and Clustering via Multivariate L´ evy Subordination

Consider n obligors with each default time, τi, defined by τi = inf{t ≥ 0 : Λi

t

≥ Ei}, i = 1, ..., n, .

Default Hazard Process Λi

t

. . . . . . . . Consider the following processes:

X a

s , a = 1, ..., d; are d independent one-dimensional, non-negative Markov

process starting from X a

0 = xa ≥ 0

Y a

t =

R t

0 X a s ds, the corresponding integrated processes

Z a

t = Y a T a

t ; time changed of process Y a

t with a d-dimensional L´

evy subordinator Tt (we time change the integral)

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 3 / 26

Default Correlation and Clustering via Multivariate L´ evy Subordination

Consider n obligors with each default time, τi, defined by τi = inf{t ≥ 0 : Λi

t

≥ Ei}, i = 1, ..., n, .

Default Hazard Process Λi

t

. . . . . . . . Consider the following processes:

X a

s , a = 1, ..., d; are d independent one-dimensional, non-negative Markov

process starting from X a

0 = xa ≥ 0

Y a

t =

R t

0 X a s ds, the corresponding integrated processes

Z a

t = Y a T a

t ; time changed of process Y a

t with a d-dimensional L´

evy subordinator Tt (we time change the integral) then define the Default Hazard Process Λi

t as:

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 3 / 26

Default Correlation and Clustering via Multivariate L´ evy Subordination

Consider n obligors with each default time, τi, defined by τi = inf{t ≥ 0 : Λi

t

≥ Ei}, i = 1, ..., n, .

Default Hazard Process Λi

t

. . . . . . . . Consider the following processes:

X a

s , a = 1, ..., d; are d independent one-dimensional, non-negative Markov

process starting from X a

0 = xa ≥ 0

Y a

t =

R t

0 X a s ds, the corresponding integrated processes

Z a

t = Y a T a

t ; time changed of process Y a

t with a d-dimensional L´

evy subordinator Tt (we time change the integral) then define the Default Hazard Process Λi

t as:

Λi

t := Pd a=1 Ai,a Z a t , i = 1, ..., n,

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 3 / 26

Default Correlation and Clustering via Multivariate L´ evy Subordination

Consider n obligors with each default time, τi, defined by τi = inf{t ≥ 0 : Λi

t

≥ Ei}, i = 1, ..., n, .

Default Hazard Process Λi

t

. . . . . . . . Consider the following processes:

X a

s , a = 1, ..., d; are d independent one-dimensional, non-negative Markov

process starting from X a

0 = xa ≥ 0

Y a

t =

R t

0 X a s ds, the corresponding integrated processes

Z a

t = Y a T a

t ; time changed of process Y a

t with a d-dimensional L´

evy subordinator Tt (we time change the integral) then define the Default Hazard Process Λi

t as:

Λi

t := Pd a=1 Ai,a Z a t , i = 1, ..., n,

A is an n × d matrix with non-negative entries Ai,a ≥ 0

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 3 / 26

d-dimensional subordinator

.

d-dimensional subordinator, Tt

. . . . . . . . L´ evy process in Rd

+ = [0, ∞)d that is non-decreasing in each of its

• coordinates. That is, each of its coordinates is a one-dimensional L´

evy subordinator.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 4 / 26

d-dimensional subordinator

.

d-dimensional subordinator, Tt

. . . . . . . . L´ evy process in Rd

+ = [0, ∞)d that is non-decreasing in each of its

• coordinates. That is, each of its coordinates is a one-dimensional L´

evy subordinator. The (d-dimensional) Laplace transform of a d-dimensional subordinator is given by (here ua ≥ 0 and ⟨u, v⟩ = ∑d

a=1 uava)

E[e−⟨u,Tt⟩] = e−tφ(u)

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 4 / 26

d-dimensional subordinator

.

d-dimensional subordinator, Tt

. . . . . . . . L´ evy process in Rd

+ = [0, ∞)d that is non-decreasing in each of its

• coordinates. That is, each of its coordinates is a one-dimensional L´

evy subordinator. The (d-dimensional) Laplace transform of a d-dimensional subordinator is given by (here ua ≥ 0 and ⟨u, v⟩ = ∑d

a=1 uava)

E[e−⟨u,Tt⟩] = e−tφ(u) The Laplace exponent φ(u) is given by the L´ evy-Khintchine formula: φ(u) = ⟨γ, u⟩ + ∫

Rd

+(1 − e−⟨u,s⟩)ν(ds),

where (drift) γ ∈ Rd

+ and the L´

evy measure ν is a sigma-finite measure on Rd concentrated on Rd

+\{0} such that

∫

Rd

+(|s| ∧ 1)ν(ds) < ∞. Rafael Mendoza (McCombs) Default Correlation WCMF 2009 4 / 26

d-dimensional subordinator (special case)

.

Linear Tranformations of Independent Subordinators

. . . . . . . . Let Sp

t beN independent one-dimensional subordinators and B a

d × N matrix with non-negative entries Ba,p. Define T a

t = ∑N p=1 Ba,pSp t ,

a = 1, ..., d.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 5 / 26

d-dimensional subordinator (special case)

.

Linear Tranformations of Independent Subordinators

. . . . . . . . Let Sp

t beN independent one-dimensional subordinators and B a

d × N matrix with non-negative entries Ba,p. Define T a

t = ∑N p=1 Ba,pSp t ,

a = 1, ..., d. Then the Rd

+-valued process Tt is a d-dimensional subordinator with

Laplace exponent given by: φ(u) = ∑N

p=1 φp(vp), vp = ∑d a=1 Ba,pua

where φp(v) are Laplace exponents of N independent one-dimensional subordinators Sp, p = 1, ..., N.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 5 / 26

Two Firms Case : no time changes

Consider the following setup for analyzing two firms: Two CIR processes X i

t , i ∈ {1, 2}:

dXt = κ(θ − Xt)dt + σ√XtdBt, X0 = x

x0 θ κ σ X 1

t

0.005 0.08 0.13 0.07 X 2

t

0.035 0.013 0.21 0.055

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 6 / 26

Two Firms Case : no time changes

Consider the following setup for analyzing two firms: Two CIR processes X i

t , i ∈ {1, 2}:

dXt = κ(θ − Xt)dt + σ√XtdBt, X0 = x

x0 θ κ σ X 1

t

0.005 0.08 0.13 0.07 X 2

t

0.035 0.013 0.21 0.055

Let the matrix A be defined as:

A = „ 0.15 0.85 0.65 0.35 «

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 6 / 26

Two Firms Case : no time changes

Consider the following setup for analyzing two firms: Two CIR processes X i

t , i ∈ {1, 2}:

dXt = κ(θ − Xt)dt + σ√XtdBt, X0 = x

x0 θ κ σ X 1

t

0.005 0.08 0.13 0.07 X 2

t

0.035 0.013 0.21 0.055

Let the matrix A be defined as:

A = „ 0.15 0.85 0.65 0.35 «

The Default Hazard process Λt := A Zt:

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 6 / 26

Two Firms Case : no time changes

Consider the following setup for analyzing two firms: Two CIR processes X i

t , i ∈ {1, 2}:

dXt = κ(θ − Xt)dt + σ√XtdBt, X0 = x

x0 θ κ σ X 1

t

0.005 0.08 0.13 0.07 X 2

t

0.035 0.013 0.21 0.055

Let the matrix A be defined as:

A = „ 0.15 0.85 0.65 0.35 «

The Default Hazard process Λt := A Zt: Λ1

t := 0.15

∫ t

0 X 1 u du + 0.85

∫ t

0 X 2 u du

Λ2

t := 0.65

∫ t

0 X 1 u du + 0.35

∫ t

0 X 2 u du

(Zt = Y (t) = ∫ t

0 Xudu, since we are not using time changes yet)

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 6 / 26

Two Firms Case : no time changes

.

CIR processes

. . . . . . . .

5 10 15 20 25 30 35 t yrs 0.02 0.04 0.06 0.08 0.10 Xt

CIR 1

5 10 15 20 25 30 35 t yrs 0.01 0.02 0.03 0.04 0.05 0.06 Xt

CIR 2

.

Integrated processes

. . . . . . . .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 7 / 26

Two Firms Case : no time changes

.

CIR processes

. . . . . . . .

5 10 15 20 25 30 35 t yrs 0.02 0.04 0.06 0.08 0.10 Xt

CIR 1

5 10 15 20 25 30 35 t yrs 0.01 0.02 0.03 0.04 0.05 0.06 Xt

CIR 2

.

Integrated processes

. . . . . . . .

5 10 15 20 25 30 35 t yrs 0.5 1.0 1.5 2.0 Yt

Yt

t

Xuu, CIR 1

5 10 15 20 25 30 35 t yrs 0.1 0.2 0.3 0.4 0.5 0.6 Yt

Yt

t

Xuu, CIR 2

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 7 / 26

Two Firms Case : no time changes

.

Integrated processes

. . . . . . . .

5 10 15 20 25 30 35 t yrs 0.5 1.0 1.5 2.0 Yt

Yt

t

Xuu, CIR 1

5 10 15 20 25 30 35 t yrs 0.1 0.2 0.3 0.4 0.5 0.6 Yt

Yt

t

Xuu, CIR 2

.

Default Hazard processes

. . . . . . . .

A = „ 0.15 0.85 0.65 0.35 « Rafael Mendoza (McCombs) Default Correlation WCMF 2009 8 / 26

Two Firms Case : no time changes

.

Integrated processes

. . . . . . . .

5 10 15 20 25 30 35 t yrs 0.5 1.0 1.5 2.0 Yt

Yt

t

Xuu, CIR 1

5 10 15 20 25 30 35 t yrs 0.1 0.2 0.3 0.4 0.5 0.6 Yt

Yt

t

Xuu, CIR 2

.

Default Hazard processes

. . . . . . . .

5 10 15 20 25 30 35 t yrs 0.2 0.4 0.6 0.8 1t

1 A11 Y1t A12 Y2t

5 10 15 20 25 30 35 t yrs 0.5 1.0 1.5 2t

2 A21 Y1t A22 Y2t

A = „ 0.15 0.85 0.65 0.35 « Rafael Mendoza (McCombs) Default Correlation WCMF 2009 8 / 26

Two Firms Case : no time changes

.

Integrated processes

. . . . . . . .

5 10 15 20 25 30 35 t yrs 0.5 1.0 1.5 2.0 Yt

Yt

t

Xuu, CIR 1

5 10 15 20 25 30 35 t yrs 0.1 0.2 0.3 0.4 0.5 0.6 Yt

Yt

t

Xuu, CIR 2

.

Default Hazard processes

. . . . . . . .

5 10 15 20 25 30 35 t yrs 0.2 0.4 0.6 0.8 1t

1 A11 Y1t A12 Y2t, E0.274387

5 10 15 20 25 30 35 t yrs 0.5 1.0 1.5 2t

2 A21 Y1t A22 Y2t, E0.102775

A = „ 0.15 0.85 0.65 0.35 « Rafael Mendoza (McCombs) Default Correlation WCMF 2009 9 / 26

Two Firms Case : no time changes

.

Integrated processes

. . . . . . . .

5 10 15 20 25 30 35 t yrs 0.5 1.0 1.5 2.0 Yt

Yt

t

Xuu, CIR 1

5 10 15 20 25 30 35 t yrs 0.1 0.2 0.3 0.4 0.5 0.6 Yt

Yt

t

Xuu, CIR 2

.

Default Hazard processes

. . . . . . . .

5 10 15 20 25 30 35 t yrs 0.2 0.4 0.6 0.8 1t

1 A11 Y1t A12 Y2t, E0.274387, Τ16.00794 yrs

5 10 15 20 25 30 35 t yrs 0.5 1.0 1.5 2t

2 A21 Y1t A22 Y2t, E0.102775, Τ23.64286 yrs

A = „ 0.15 0.85 0.65 0.35 « Rafael Mendoza (McCombs) Default Correlation WCMF 2009 10 / 26

Two Firms Case: Time Changes

Consider the following CPP subordinators with characteristic triplet (γ, 0, ν) defined by the drift γ and the three parameter density ν(s) = Cs−(Y +1)e−ηs . . . . . . . . . . . . . . . . . .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 11 / 26

Two Firms Case: Time Changes

Consider the following CPP subordinators with characteristic triplet (γ, 0, ν) defined by the drift γ and the three parameter density ν(s) = Cs−(Y +1)e−ηs .

Frequent small jumps

. . . . . . . .

0.2 0.4 0.6 0.8 1.0 t yrs 10 20 30 S1t

S1

γ Y η C 1

• 1

1 10

0.005 E [S1yr] σ [S1yr] α = C

η (arrival) 1 η (E.j.size)

1.5 3.16 0.05 10

. . . . . . . . .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 11 / 26

Two Firms Case: Time Changes

Consider the following CPP subordinators with characteristic triplet (γ, 0, ν) defined by the drift γ and the three parameter density ν(s) = Cs−(Y +1)e−ηs .

Frequent small jumps

. . . . . . . .

0.2 0.4 0.6 0.8 1.0 t yrs 10 20 30 S1t

S1

γ Y η C 1

• 1

1 10

0.005 E [S1yr] σ [S1yr] α = C

η (arrival) 1 η (E.j.size)

1.5 3.16 0.05 10

.

Infrequent large jumps

. . . . . . . .

0.2 0.4 0.6 0.8 1.0 t yrs 50 100 150 200 250 300 S2t

S2

γ Y η C 1

• 1

1 100

0.00005 E [S1yr] σ [S1yr] α = C

η (arrival) 1 η (E.j.size)

1.5 10. 0.005 100

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 11 / 26

Two Firms Case: Subordination

.

S1

t (freq. small jumps)

. . . . . . . .

1 2 3 4 5 t yrs 1 2 3 4 5 6 7 St yrs

jump time1.47222 yrs, St1.65393 yrs

Recall: Λt = A Zt with Z i

t = Y i T i

t

and Tt = B St In this particular case: A = „ 0.15 0.85 0.65 0.35 « , B = „ 1 1 «

.

A11 Y 1

t + A12 Y 2 t

. . . . . . . .

5 10 15 20 25 30 35 St yrs 0.2 0.4 0.6 0.8 Yt

1tA11 Y1t A12 Y2t

.

A21 Y 1

t + A22 Y 2 t

. . . . . . . .

5 10 15 20 25 30 35 St yrs 0.5 1.0 1.5 Yt

2tA21 Y1t A22 Y2t

.

Λ1

t = A11 Y 1 S1

t + A12 Y 2

S1

t

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 0.30 YSt

1tA11 Y1S1t A12 Y2S1t

.

Λ2

t = A21 Y 1 S1

t + A22 Y 2

S1

t

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 YSt

2tA21 Y1S1t A22 Y2S1t

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 12 / 26

Two Firms Case: Subordination

.

S1

t (freq. small jumps)

. . . . . . . .

1 2 3 4 5 t yrs 1 2 3 4 5 6 7 St yrs

jump time1.47222 yrs, St1.65393 yrs

Recall: Λt = A Zt with Z i

t = Y i T i

t

and Tt = B St In this particular case: A = „ 0.15 0.85 0.65 0.35 « , B = „ 1 1 «

.

A11 Y 1

t + A12 Y 2 t

. . . . . . . .

5 10 15 20 25 30 35 St yrs 0.2 0.4 0.6 0.8 Yt

1tA11 Y1t A12 Y2t

.

A21 Y 1

t + A22 Y 2 t

. . . . . . . .

5 10 15 20 25 30 35 St yrs 0.5 1.0 1.5 Yt

2tA21 Y1t A22 Y2t

.

Λ1

t = A11 Y 1 S1

t + A12 Y 2

S1

t

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 0.30 YSt

1tA11 Y1S1t A12 Y2S1t

.

Λ2

t = A21 Y 1 S1

t + A22 Y 2

S1

t

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 YSt

2tA21 Y1S1t A22 Y2S1t

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 12 / 26

Two Firms Case: Subordination

.

S1

t (freq. small jumps)

. . . . . . . .

1 2 3 4 5 t yrs 1 2 3 4 5 6 7 St yrs

jump time3.90873 yrs, St0.621614 yrs

Recall: Λt = A Zt with Z i

t = Y i T i

t

and Tt = B St In this particular case: A = „ 0.15 0.85 0.65 0.35 « , B = „ 1 1 «

.

A11 Y 1

t + A12 Y 2 t

. . . . . . . .

5 10 15 20 25 30 35 St yrs 0.2 0.4 0.6 0.8 Yt

1tA11 Y1t A12 Y2t

.

A21 Y 1

t + A22 Y 2 t

. . . . . . . .

5 10 15 20 25 30 35 St yrs 0.5 1.0 1.5 Yt

2tA21 Y1t A22 Y2t

.

Λ1

t = A11 Y 1 S1

t + A12 Y 2

S1

t

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 0.30 YSt

1tA11 Y1S1t A12 Y2S1t

.

Λ2

t = A21 Y 1 S1

t + A22 Y 2

S1

t

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 YSt

2tA21 Y1S1t A22 Y2S1t

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 12 / 26

Two Firms Case: Subordination

.

S1

t (freq. small jumps)

. . . . . . . .

1 2 3 4 5 t yrs 1 2 3 4 5 6 7 St yrs

jump time3.90873 yrs, St0.621614 yrs

Recall: Λt = A Zt with Z i

t = Y i T i

t

and Tt = B St In this particular case: A = „ 0.15 0.85 0.65 0.35 « , B = „ 1 1 «

.

A11 Y 1

t + A12 Y 2 t

. . . . . . . .

5 10 15 20 25 30 35 St yrs 0.2 0.4 0.6 0.8 Yt

1tA11 Y1t A12 Y2t

.

A21 Y 1

t + A22 Y 2 t

. . . . . . . .

5 10 15 20 25 30 35 St yrs 0.5 1.0 1.5 Yt

2tA21 Y1t A22 Y2t

.

Λ1

t = A11 Y 1 S1

t + A12 Y 2

S1

t

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 0.30 YSt

1tA11 Y1S1t A12 Y2S1t

.

Λ2

t = A21 Y 1 S1

t + A22 Y 2

S1

t

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 YSt

2tA21 Y1S1t A22 Y2S1t

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 12 / 26

Two Firms Case: Subordination

.

S1

t (freq. small jumps)

. . . . . . . .

1 2 3 4 5 t yrs 1 2 3 4 5 6 7 St yrs

jump time5.

Recall: Λt = A Zt with Z i

t = Y i T i

t

and Tt = B St In this particular case: A = „ 0.15 0.85 0.65 0.35 « , B = „ 1 1 «

.

A11 Y 1

t + A12 Y 2 t

. . . . . . . .

5 10 15 20 25 30 35 St yrs 0.2 0.4 0.6 0.8 Yt

1tA11 Y1t A12 Y2t

.

A21 Y 1

t + A22 Y 2 t

. . . . . . . .

5 10 15 20 25 30 35 St yrs 0.5 1.0 1.5 Yt

2tA21 Y1t A22 Y2t

.

Λ1

t = A11 Y 1 S1

t + A12 Y 2

S1

t

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 0.30 YSt

1tA11 Y1S1t A12 Y2S1t

.

Λ2

t = A21 Y 1 S1

t + A22 Y 2

S1

t

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 YSt

2tA21 Y1S1t A22 Y2S1t

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 12 / 26

Two Firms Case: Subordination

.

S1

t (freq. small), B =

„ 1 1 «

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 0.30 YSt

1tA11 Y1S1t A12 Y2S1t

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 YSt

2tA21 Y1S1t A22 Y2S1t

. . .. . . . . .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 13 / 26

Two Firms Case: Subordination

.

S1

t (freq. small), B =

„ 1 1 «

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 0.30 1t

1t A11 Y1S1t A12 Y2S1t, E0.274387

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 2t

2t A21 Y1S1t A22 Y2S1t, E0.102775

. . .. . . . . .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 13 / 26

Two Firms Case: Subordination

.

S1

t (freq. small), B =

„ 1 1 «

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 0.30 1t

1t A11 Y1S1t A12 Y2S1t, E0.274387, Τ13.90873 yrs

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 2t

2t A21 Y1S1t A22 Y2S1t, E0.102775, Τ21.99206 yrs

. . .. . . . . .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 13 / 26

Two Firms Case: Subordination

.

S1

t (freq. small), B =

„ 1 1 «

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 0.30 1t

1t A11 Y1S1t A12 Y2S1t, E0.274387, Τ13.90873 yrs

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 2t

2t A21 Y1S1t A22 Y2S1t, E0.102775, Τ21.99206 yrs

.

S2

t (infreq. large), B =

„ 0 1 1 «

. . . . . . . .

1 2 3 4 5 t yrs 0.2 0.4 0.6 0.8 1t

1t A11 Y1S2t A12 Y2S2t, E0.274387

1 2 3 4 5 t yrs 0.2 0.4 0.6 0.8 1.0 1.2 1.4 2t

2t A21 Y1S2t A22 Y2S2t, E0.102775 Rafael Mendoza (McCombs) Default Correlation WCMF 2009 13 / 26

Two Firms Case: Subordination

.

S1

t (freq. small), B =

„ 1 1 «

. . . . . . . .

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 0.30 1t

1t A11 Y1S1t A12 Y2S1t, E0.274387, Τ13.90873 yrs

1 2 3 4 5 t yrs 0.05 0.10 0.15 0.20 0.25 2t

2t A21 Y1S1t A22 Y2S1t, E0.102775, Τ21.99206 yrs

.

S2

t (infreq. large), B =

„ 0 1 1 «

. . . . . . . .

1 2 3 4 5 t yrs 0.2 0.4 0.6 0.8 1t

1t A11 Y1S2t A12 Y2S2t, E0.274387, Τ12.57143 yrs

1 2 3 4 5 t yrs 0.2 0.4 0.6 0.8 1.0 1.2 1.4 2t

2t A21 Y1S2t A22 Y2S2t, E0.102775, Τ22.57143 yrs Rafael Mendoza (McCombs) Default Correlation WCMF 2009 13 / 26

Two Firms Case: Full Model

The Default Hazard process is given by, Λt = A Zt with Z i

t = Y i T i

t and Tt = B St

. . .. . . . . . . . .. . . . . .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 14 / 26

Two Firms Case: Full Model

The Default Hazard process is given by, Λt = A Zt with Z i

t = Y i T i

t and Tt = B St

Consider the matrices A = ( 0.15 0.85 0.65 0.35 ) , B = ( 0.5 0.5 0.7 0.3 ) . . .. . . . . . . . .. . . . . .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 14 / 26

Two Firms Case: Full Model

The Default Hazard process is given by, Λt = A Zt with Z i

t = Y i T i

t and Tt = B St

Consider the matrices A = ( 0.15 0.85 0.65 0.35 ) , B = ( 0.5 0.5 0.7 0.3 ) .

Λ1

t = A11 Z 1 t + A12 Z 2 t

. . . . . . . .

5 10 15 20 25 30 35 t yrs 0.1 0.2 0.3 0.4 0.5 0.6 1t

1 A11 Z1t A12 Z2t

.

Λ2

t = A21 Z 1 t + A22 Z 2 t

. . . . . . . .

5 10 15 20 25 30 35 t yrs 0.2 0.4 0.6 0.8 1.0 2t

2 A21 Z1t A22 Z2t Rafael Mendoza (McCombs) Default Correlation WCMF 2009 14 / 26

Two Firms Case: Full Model

The Default Hazard process is given by, Λt = A Zt with Z i

t = Y i T i

t and Tt = B St

Consider the matrices A = ( 0.15 0.85 0.65 0.35 ) , B = ( 0.5 0.5 0.7 0.3 ) .

Λ1

t = A11 Z 1 t + A12 Z 2 t

. . . . . . . .

1 2 3 4 5 t yrs 0.1 0.2 0.3 0.4 0.5 0.6 1t

1t A11 Z1t A12 Z2t, E0.274387

.

Λ2

t = A21 Z 1 t + A22 Z 2 t

. . . . . . . .

1 2 3 4 5 t yrs 0.2 0.4 0.6 0.8 1.0 2t

2t A21 Z1t A22 Z2t, E0.102775 Rafael Mendoza (McCombs) Default Correlation WCMF 2009 14 / 26

Two Firms Case: Full Model

The Default Hazard process is given by, Λt = A Zt with Z i

t = Y i T i

t and Tt = B St

Consider the matrices A = ( 0.15 0.85 0.65 0.35 ) , B = ( 0.5 0.5 0.7 0.3 ) .

Λ1

t = A11 Z 1 t + A12 Z 2 t

. . . . . . . .

1 2 3 4 5 t yrs 0.1 0.2 0.3 0.4 0.5 0.6 1t

1t A11 Z1t A12 Z2t, E0.274387, Τ12.57143 yrs

.

Λ2

t = A21 Z 1 t + A22 Z 2 t

. . . . . . . .

1 2 3 4 5 t yrs 0.2 0.4 0.6 0.8 1.0 2t

2t A21 Z1t A22 Z2t, E0.102775, Τ22.57143 yrs Rafael Mendoza (McCombs) Default Correlation WCMF 2009 14 / 26

Two Firms Case: Full Model

The Default Hazard process is given by, Λt = A Zt with Z i

t = Y i T i

t and Tt = B St

Consider the matrices A = ( 0.15 0.85 0.65 0.35 ) , B = ( 0.5 0.5 0.7 0.3 ) .

Λ1

t = A11 Z 1 t + A12 Z 2 t

. . . . . . . .

1 2 3 4 5 t yrs 0.1 0.2 0.3 0.4 0.5 0.6 1t

1t A11 Z1t A12 Z2t, E0.274387, Τ12.57143 yrs

.

Λ2

t = A21 Z 1 t + A22 Z 2 t

. . . . . . . .

1 2 3 4 5 t yrs 0.2 0.4 0.6 0.8 1.0 2t

2t A21 Z1t A22 Z2t, E0.102775, Τ22.57143 yrs

In our framework dependency enters through the diffusion component, A; and through the jump component, B.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 14 / 26

Two Firms Case: Full Model

The Default Hazard process is given by, Λt = A Zt with Z i

t = Y i T i

t and Tt = B St

Consider the matrices A = ( 0.15 0.85 0.65 0.35 ) , B = ( 0.5 0.5 0.7 0.3 ) .

Λ1

t = A11 Z 1 t + A12 Z 2 t

. . . . . . . .

1 2 3 4 5 t yrs 0.1 0.2 0.3 0.4 0.5 0.6 1t

1t A11 Z1t A12 Z2t, E0.274387, Τ12.57143 yrs

.

Λ2

t = A21 Z 1 t + A22 Z 2 t

. . . . . . . .

1 2 3 4 5 t yrs 0.2 0.4 0.6 0.8 1.0 2t

2t A21 Z1t A22 Z2t, E0.102775, Τ22.57143 yrs

In our framework dependency enters through the diffusion component, A; and through the jump component, B. Non-Trivial Dependency!

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 14 / 26

Joint Survival Probability

.

Theorem 1

. . . . . . . . For βa ≥ 0, let La

xa,βa(t) denote the Laplace transforms of the integrals up to

time t of the Markov processes X a starting at xa at time zero: La

xa,βa(t) = Exa

[ e−βa

R t

0 X a s ds]

.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 15 / 26

Joint Survival Probability

.

Theorem 1

. . . . . . . . For βa ≥ 0, let La

xa,βa(t) denote the Laplace transforms of the integrals up to

time t of the Markov processes X a starting at xa at time zero: La

xa,βa(t) = Exa

[ e−βa

R t

0 X a s ds]

. For an ordered subset Ξ = {i1, ..., ik} of {1, 2, ..., n} with 1 ≤ k ≤ n define li(Ξ) ∈ {0, 1}, i = 1, ..., n, by: li(Ξ) = 1{Ξ}(i), where the indicator 1{Ξ}(i) = 1 if the integer i belongs to Ξ and 1{Ξ}(i) = 0

• therwise.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 15 / 26

Joint Survival Probability

.

Theorem 1

. . . . . . . . For βa ≥ 0, let La

xa,βa(t) denote the Laplace transforms of the integrals up to

time t of the Markov processes X a starting at xa at time zero: La

xa,βa(t) = Exa

[ e−βa

R t

0 X a s ds]

. For an ordered subset Ξ = {i1, ..., ik} of {1, 2, ..., n} with 1 ≤ k ≤ n define li(Ξ) ∈ {0, 1}, i = 1, ..., n, by: li(Ξ) = 1{Ξ}(i), where the indicator 1{Ξ}(i) = 1 if the integer i belongs to Ξ and 1{Ξ}(i) = 0

• therwise.

The joint survival probability, P(τi1 > t, ..., τik > t), is given by: PΞ(t) = Pi1,...,ik(t) = ∫

Rd

+

(∏d

a=1 La xa,βΞ

a (sa)

) πt(ds),

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 15 / 26

Joint Survival Probability

.

Theorem 1

. . . . . . . . For βa ≥ 0, let La

xa,βa(t) denote the Laplace transforms of the integrals up to

time t of the Markov processes X a starting at xa at time zero: La

xa,βa(t) = Exa

[ e−βa

R t

0 X a s ds]

. For an ordered subset Ξ = {i1, ..., ik} of {1, 2, ..., n} with 1 ≤ k ≤ n define li(Ξ) ∈ {0, 1}, i = 1, ..., n, by: li(Ξ) = 1{Ξ}(i), where the indicator 1{Ξ}(i) = 1 if the integer i belongs to Ξ and 1{Ξ}(i) = 0

• therwise.

The joint survival probability, P(τi1 > t, ..., τik > t), is given by: PΞ(t) = Pi1,...,ik(t) = ∫

Rd

+

(∏d

a=1 La xa,βΞ

a (sa)

) πt(ds), where P(T 1

t ∈ ds1, ..., T d t ∈ dsd) = πt(ds) is the transition measure of the

d-dimensional subordinator T , and βΞ

a are:

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 15 / 26

Joint Survival Probability

.

Theorem 1

. . . . . . . . For βa ≥ 0, let La

xa,βa(t) denote the Laplace transforms of the integrals up to

time t of the Markov processes X a starting at xa at time zero: La

xa,βa(t) = Exa

[ e−βa

R t

0 X a s ds]

. For an ordered subset Ξ = {i1, ..., ik} of {1, 2, ..., n} with 1 ≤ k ≤ n define li(Ξ) ∈ {0, 1}, i = 1, ..., n, by: li(Ξ) = 1{Ξ}(i), where the indicator 1{Ξ}(i) = 1 if the integer i belongs to Ξ and 1{Ξ}(i) = 0

• therwise.

The joint survival probability, P(τi1 > t, ..., τik > t), is given by: PΞ(t) = Pi1,...,ik(t) = ∫

Rd

+

(∏d

a=1 La xa,βΞ

a (sa)

) πt(ds), where P(T 1

t ∈ ds1, ..., T d t ∈ dsd) = πt(ds) is the transition measure of the

d-dimensional subordinator T , and βΞ

a are:

βΞ

a = ∑n i=1 li(Ξ)Ai,a.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 15 / 26

Joint Survival Probability: Remarks

The expression in Theorem 1 expression is not very practical. It requires,

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 16 / 26

Joint Survival Probability: Remarks

The expression in Theorem 1 expression is not very practical. It requires,

the knowledge of the transition measure of the subordinator,

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 16 / 26

Joint Survival Probability: Remarks

The expression in Theorem 1 expression is not very practical. It requires,

the knowledge of the transition measure of the subordinator, the evaluation of a d-dimensional integral numerically

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 16 / 26

Joint Survival Probability: Remarks

The expression in Theorem 1 expression is not very practical. It requires,

the knowledge of the transition measure of the subordinator, the evaluation of a d-dimensional integral numerically

In practice, we know the Laplace exponent of the subordinator, but do not know the transition measure.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 16 / 26

Joint Survival Probability: Remarks

The expression in Theorem 1 expression is not very practical. It requires,

the knowledge of the transition measure of the subordinator, the evaluation of a d-dimensional integral numerically

In practice, we know the Laplace exponent of the subordinator, but do not know the transition measure. Remarkably, the Spectral Method allows us to kill two birds with one stone:

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 16 / 26

Joint Survival Probability: Remarks

The expression in Theorem 1 expression is not very practical. It requires,

the knowledge of the transition measure of the subordinator, the evaluation of a d-dimensional integral numerically

In practice, we know the Laplace exponent of the subordinator, but do not know the transition measure. Remarkably, the Spectral Method allows us to kill two birds with one stone:

Under some additional conditions on Markov processes X, we avoid BOTH the need for the numerical integration, and we only need the Laplace exponent of the subordinator.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 16 / 26

Spectral Representation of FK Semigroups of One-Dimensional Diffusions

.

Feynman-Kac semigroup of linear operators {Pt, t ≥ 0}

. . . . . . . . Suppose Xt is a one-dimensional diffusion and consider its F-K semigroup: Ptf (x) = Ex[e−

R t

0 k(Xs)dsf (Xt)] Rafael Mendoza (McCombs) Default Correlation WCMF 2009 17 / 26

Spectral Representation of FK Semigroups of One-Dimensional Diffusions

.

Feynman-Kac semigroup of linear operators {Pt, t ≥ 0}

. . . . . . . . Suppose Xt is a one-dimensional diffusion and consider its F-K semigroup: Ptf (x) = Ex[e−

R t

0 k(Xs)dsf (Xt)]

Linear operators Pt are symmetric in this Hilbert space, that is, (Ptf , g) = (f , Ptg) ∀f , g ∈ L2((e1, e2), m) wrt the speed measure: m(dx) =

2 σ2(x) exp

“R x

x0 2µ(y) σ2(y)dy

” dx where k(x) is the killing rate; whereas, µ(x) and σ(x) are the (state dependent) drift and volatility of the diffusion process Xt, respectively.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 17 / 26

Spectral Representation of FK Semigroups of One-Dimensional Diffusions

.

Spectral Representation

. . . . . . . . Under some conditions on the behavior of µ(x), σ(x), and k(x) near the boundaries e1 and e2, the spectrum is purely discrete and the spectral expansion reduces to the eigenfunction expansion: Ptf (x) = ∑∞

n=1 cne−λntϕn(x)

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 18 / 26

Spectral Representation of FK Semigroups of One-Dimensional Diffusions

.

Spectral Representation

. . . . . . . . Under some conditions on the behavior of µ(x), σ(x), and k(x) near the boundaries e1 and e2, the spectrum is purely discrete and the spectral expansion reduces to the eigenfunction expansion: Ptf (x) = ∑∞

n=1 cne−λntϕn(x)

cn = (f , ϕn) are the expansion coefficients, whereas, −λn and ϕn are eigenvalues and eigenfunctions of the infinitesimal generator G of the semigroup P:

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 18 / 26

Spectral Representation of FK Semigroups of One-Dimensional Diffusions

.

Spectral Representation

. . . . . . . . Under some conditions on the behavior of µ(x), σ(x), and k(x) near the boundaries e1 and e2, the spectrum is purely discrete and the spectral expansion reduces to the eigenfunction expansion: Ptf (x) = ∑∞

n=1 cne−λntϕn(x)

cn = (f , ϕn) are the expansion coefficients, whereas, −λn and ϕn are eigenvalues and eigenfunctions of the infinitesimal generator G of the semigroup P: Gϕn(x) = 1

2σ2(x)ϕ′′ n (x) + µ(x)ϕ′ n(x) − k(x)ϕn(x) = −λnϕn

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 18 / 26

The Feynman-Kac semigroup of the CIR Process.

Let Xt be a CIR diffusion starting from X0 = x > 0 and solving the SDE dXt = κ (θ − Xt) dt + σ√XtdBt, and with the speed measure, m(x) =

2 σ2 xb−1e− 2κ

σ2 x.

The spectrum is discrete, and the eigenfunction expansion of the Laplace transform reads (k(x) = βx killing rate):, Lx,β(t) = Ex [ e−β

R t

0 Xsds]

= ∑∞

n=0 cne−λntϕn(x),

cn = (1, ϕn). where the eigenfunctions, eigenvalues and expansion coefficients are given by,

λn = ζn + b

2 (ζ − κ),

ϕn(x) = Nn exp “

κ−ζ σ2 x

” L(b−1)

n

“

2ζ σ2 x

” , cn = (1, ϕn) =

1 Nn

“

βσ2 κ+ζ

”b“

κ−ζ κ+ζ

”n , ζ := p κ2 + 2βσ2, b := 2κθ

σ2 ,

Nn = r

βσ2(n!) 2Γ(b+n)

“

2ζ βσ2

” b

2 ,

L(b−1)

n

are the generalized Laguerre polynomials

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 19 / 26

Spectral Expansion of the Joint Survival Probability

.

Theorem 2

. . . . . . . .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 20 / 26

Spectral Expansion of the Joint Survival Probability

.

Theorem 2

. . . . . . . . Suppose that X a are one-dimensional diffusion processes with finite speed measures and such that their Feynman-Kac semigroups with k(x) = βx with β ≥ 0 have purely discrete spectra.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 20 / 26

Spectral Expansion of the Joint Survival Probability

.

Theorem 2

. . . . . . . . Suppose that X a are one-dimensional diffusion processes with finite speed measures and such that their Feynman-Kac semigroups with k(x) = βx with β ≥ 0 have purely discrete spectra. Then the joint survival probability, P(τi1 > t, ..., τik > t), has the eigenfunction expansion:

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 20 / 26

Spectral Expansion of the Joint Survival Probability

.

Theorem 2

. . . . . . . . Suppose that X a are one-dimensional diffusion processes with finite speed measures and such that their Feynman-Kac semigroups with k(x) = βx with β ≥ 0 have purely discrete spectra. Then the joint survival probability, P(τi1 > t, ..., τik > t), has the eigenfunction expansion: PΞ(t) = ∑∞

n1=1 · · · ∑∞ nd=1 e−tφ(λ1,Ξ

n1 ,...,λd,Ξ nd ) {∏d

a=1 ca,Ξ na ϕa,Ξ na (xa)

}

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 20 / 26

Spectral Expansion of the Joint Survival Probability

.

Theorem 2

. . . . . . . . Suppose that X a are one-dimensional diffusion processes with finite speed measures and such that their Feynman-Kac semigroups with k(x) = βx with β ≥ 0 have purely discrete spectra. Then the joint survival probability, P(τi1 > t, ..., τik > t), has the eigenfunction expansion: PΞ(t) = ∑∞

n1=1 · · · ∑∞ nd=1 e−tφ(λ1,Ξ

n1 ,...,λd,Ξ nd ) {∏d

a=1 ca,Ξ na ϕa,Ξ na (xa)

} where λa,Ξ

n

and ϕa,Ξ

n

are the eigenvalues and eigenfunctions of the (negative

• f) the infinitesimal generator of the Feynman-Kac semigroup for the process

X a with k(x) = βΞ

a x

xa is the initial value of the process X a

0 = xa, ca,Ξ n

= (1, ϕa,Ξ

n ), and φ(u) is

the Laplace exponent of the subordinator T .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 20 / 26

Default Correlation and Clustering Measures

With the explicit expressions for joint survival probabilities (spectral expansions), we can explicitly compute a variety of dependence measures among default events and times in this class of models. For instance,

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 21 / 26

Default Correlation and Clustering Measures

With the explicit expressions for joint survival probabilities (spectral expansions), we can explicitly compute a variety of dependence measures among default events and times in this class of models. For instance, Correlation matrix for default indicators: ρD

ij (t) := corr(1{τi≤t}, 1{τj≤t}) = Pij(t)−Pi(t)Pj(t)

√

Pi(t)(1−Pi(t))Pj(t)(1−Pj(t))

where Pi(t) are the single-name survival probabilities and Pij(t) are the joint survival probabilities for the pairs of names.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 21 / 26

Default Correlation and Clustering Measures

With the explicit expressions for joint survival probabilities (spectral expansions), we can explicitly compute a variety of dependence measures among default events and times in this class of models. For instance, Correlation matrix for default indicators: ρD

ij (t) := corr(1{τi≤t}, 1{τj≤t}) = Pij(t)−Pi(t)Pj(t)

√

Pi(t)(1−Pi(t))Pj(t)(1−Pj(t))

where Pi(t) are the single-name survival probabilities and Pij(t) are the joint survival probabilities for the pairs of names. Correlation matrix for default times: ρτ

ij := corr(τi, τj) = E[τiτj]−µτ

i µτ i

στ

i στ j

where µτ

i = E[τi] and στ i =

√ E[τ 2

i ] − (µτ i )2 are the mean and standard

deviation of single-name default times.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 21 / 26

Correlation of default indicators, ρD

ij (t) .

Moving the diffusion component, A

. . . . . . . .

Fixing Matrix B and moving A

0.0 0.5 1.0 A11 0.0 0.5 1.0 A21 0.05 0.10 0.15 0.20 0.25 Ρ

Moving the Matrix A: A = ( A11 1 − A11 A21 1 − A21 ) Fixing the Matrix B: B = ( 0.5 0.5 0.7 0.3 ) . . . . . . . . .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 22 / 26

Correlation of default indicators, ρD

ij (t) .

Moving the diffusion component, A

. . . . . . . .

Fixing Matrix B and moving A

0.0 0.5 1.0 A11 0.0 0.5 1.0 A21 0.05 0.10 0.15 0.20 0.25 Ρ

Moving the Matrix A: A = ( A11 1 − A11 A21 1 − A21 ) Fixing the Matrix B: B = ( 0.5 0.5 0.7 0.3 ) .

Moving the jump component, B

. . . . . . . .

Fixing Matrix A and moving B

0.0 0.5 1.0 B11 0.0 0.5 1.0 B21 0.09 0.10 0.11 0.12 Ρ

Fixing the Matrix A: A = ( 0.15 0.85 0.65 0.35 ) Moving the Matrix B: B = ( B11 1 − B11 B21 1 − B21 )

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 22 / 26

Higher Correlation, ρD

ij (t) Changing the L´ evy density, ν(s) = Cs−(Y +1)e−ηs and parameterizing C such that E[S1yr ] = 1.5 yrs. .

S1

t = γt + CPP, (γ = 1)

. . . . . . . .

TC CPP With diffusion 1. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.00 0.05 0.10 0.15 Ρ

Single Subordinator B = ( 1 1 ) . . . . . . . . .

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 23 / 26

Higher Correlation, ρD

ij (t) Changing the L´ evy density, ν(s) = Cs−(Y +1)e−ηs and parameterizing C such that E[S1yr ] = 1.5 yrs. .

S1

t = γt + CPP, (γ = 1)

. . . . . . . .

TC CPP With diffusion 1. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.00 0.05 0.10 0.15 Ρ

Single Subordinator B = ( 1 1 ) .

S1

t = CPP, (γ = 0). Extreme case

. . . . . . . .

TC CPP Without diffusion 0. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.0 0.5 1.0 Ρ

Single Subordinator B = ( 1 1 )

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 23 / 26

Higher Correlation, ρD

ij (t)

η controls decay rate of large size jumps, Y the small size jumps . S1

t = γt + CPP, (γ = 1)

. . . . . . . .

TC CPP With diffusion 1. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.00 0.05 0.10 0.15 Ρ

. S1

t = CPP, (γ = 0). Extreme case

. . . . . . . .

TC CPP Without diffusion 0. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.0 0.5 1.0 Ρ

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 24 / 26

Higher Correlation, ρD

ij (t)

η controls decay rate of large size jumps, Y the small size jumps The smaller η the larger the jump size . S1

t = γt + CPP, (γ = 1)

. . . . . . . .

TC CPP With diffusion 1. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.00 0.05 0.10 0.15 Ρ

. S1

t = CPP, (γ = 0). Extreme case

. . . . . . . .

TC CPP Without diffusion 0. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.0 0.5 1.0 Ρ

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 24 / 26

Higher Correlation, ρD

ij (t)

η controls decay rate of large size jumps, Y the small size jumps The smaller η the larger the jump size The more negative Y the less frequent the smaller jumps become . S1

t = γt + CPP, (γ = 1)

. . . . . . . .

TC CPP With diffusion 1. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.00 0.05 0.10 0.15 Ρ

. S1

t = CPP, (γ = 0). Extreme case

. . . . . . . .

TC CPP Without diffusion 0. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.0 0.5 1.0 Ρ

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 24 / 26

Higher Correlation, ρD

ij (t)

η controls decay rate of large size jumps, Y the small size jumps The smaller η the larger the jump size The more negative Y the less frequent the smaller jumps become Parameterizing C the overall arrival rate of jumps is modified so that E[S1yr ] = 1.5 . S1

t = γt + CPP, (γ = 1)

. . . . . . . .

TC CPP With diffusion 1. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.00 0.05 0.10 0.15 Ρ

. S1

t = CPP, (γ = 0). Extreme case

. . . . . . . .

TC CPP Without diffusion 0. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.0 0.5 1.0 Ρ

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 24 / 26

Higher Correlation, ρD

ij (t)

η controls decay rate of large size jumps, Y the small size jumps The smaller η the larger the jump size The more negative Y the less frequent the smaller jumps become Parameterizing C the overall arrival rate of jumps is modified so that E[S1yr ] = 1.5 . S1

t = γt + CPP, (γ = 1)

. . . . . . . .

TC CPP With diffusion 1. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.00 0.05 0.10 0.15 Ρ

. S1

t = CPP, (γ = 0). Extreme case

. . . . . . . .

TC CPP Without diffusion 0. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.0 0.5 1.0 Ρ

Requiring larger jumps (η small) when γ = 1 makes the jumps so infrequent (because of C) that it is more likely to default (independently before a jump arrives) by diffusion.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 24 / 26

Higher Correlation, ρD

ij (t)

η controls decay rate of large size jumps, Y the small size jumps The smaller η the larger the jump size The more negative Y the less frequent the smaller jumps become Parameterizing C the overall arrival rate of jumps is modified so that E[S1yr ] = 1.5 . S1

t = γt + CPP, (γ = 1)

. . . . . . . .

TC CPP With diffusion 1. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.00 0.05 0.10 0.15 Ρ

. S1

t = CPP, (γ = 0). Extreme case

. . . . . . . .

TC CPP Without diffusion 0. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.0 0.5 1.0 Ρ

Requiring larger jumps (η small) when γ = 1 makes the jumps so infrequent (because of C) that it is more likely to default (independently before a jump arrives) by diffusion. When γ = 0 the only way to default is via jumps. Therefore, larger jumps (η small) will trigger simultaneous defaults increasing the correlation up to 1

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 24 / 26

Higher Correlation, ρD

ij (t)

η controls decay rate of large size jumps, Y the small size jumps The smaller η the larger the jump size The more negative Y the less frequent the smaller jumps become Parameterizing C the overall arrival rate of jumps is modified so that E[S1yr ] = 1.5 . S1

t = γt + CPP, (γ = 1)

. . . . . . . .

TC CPP With diffusion 1. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.00 0.05 0.10 0.15 Ρ

. S1

t = CPP, (γ = 0). Extreme case

. . . . . . . .

TC CPP Without diffusion 0. Correlation, such that at t1, ETt1.5 yrs

6 4 2 logΗ 4 3 2 1 Y 0.0 0.5 1.0 Ρ

Requiring larger jumps (η small) when γ = 1 makes the jumps so infrequent (because of C) that it is more likely to default (independently before a jump arrives) by diffusion. When γ = 0 the only way to default is via jumps. Therefore, larger jumps (η small) will trigger simultaneous defaults increasing the correlation up to 1 Any correlation level can be achieved by using linear combinations of subordinators with different L´ evy specifications

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 24 / 26

Cloncluding Remarks

Our modeling framework allows us to induce default dependence through the diffusion component of the hazard process, (A); and through its jump component, (B).

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 25 / 26

Cloncluding Remarks

Our modeling framework allows us to induce default dependence through the diffusion component of the hazard process, (A); and through its jump component, (B). This modeling structure is flexible enough to capture the variety default dependence, such as counterparty risk, systematic and systemic risks, regional and sectorial risks, etc.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 25 / 26

Cloncluding Remarks

Our modeling framework allows us to induce default dependence through the diffusion component of the hazard process, (A); and through its jump component, (B). This modeling structure is flexible enough to capture the variety default dependence, such as counterparty risk, systematic and systemic risks, regional and sectorial risks, etc. We are able to produce arbitrarily high default correlation by choosing appropriate linear combinations of L´ evy subordinators

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 25 / 26

Cloncluding Remarks

Our modeling framework allows us to induce default dependence through the diffusion component of the hazard process, (A); and through its jump component, (B). This modeling structure is flexible enough to capture the variety default dependence, such as counterparty risk, systematic and systemic risks, regional and sectorial risks, etc. We are able to produce arbitrarily high default correlation by choosing appropriate linear combinations of L´ evy subordinators Using the Spectral Representation of the one-dimensional F-K semigroups, we are able to produce analytical tractable formulas for joint survival probabilities, default and clustering measures, pricing of credit assets, etc.

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 25 / 26

Cloncluding Remarks

Our modeling framework allows us to induce default dependence through the diffusion component of the hazard process, (A); and through its jump component, (B). This modeling structure is flexible enough to capture the variety default dependence, such as counterparty risk, systematic and systemic risks, regional and sectorial risks, etc. We are able to produce arbitrarily high default correlation by choosing appropriate linear combinations of L´ evy subordinators Using the Spectral Representation of the one-dimensional F-K semigroups, we are able to produce analytical tractable formulas for joint survival probabilities, default and clustering measures, pricing of credit assets, etc. This is a work in progress and we are currently generating numerical examples for the pricing of Credit Swap Baskets and CDO’s

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 25 / 26

Questions?

Thank you

Rafael Mendoza (McCombs) Default Correlation WCMF 2009 26 / 26