Modeling Default Correlation and Clustering: A Time Change Approach Rafael Mendoza-Arriaga 1 Joint work with: Vadim Linetsky 2 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 of 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 of 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 of 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 of 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 of 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 of 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, τ i , defined by τ i = inf { t ≥ 0 : Λ i t ≥ E i } , 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, τ i , defined by τ i = inf { t ≥ 0 : Λ i t ≥ } , i = 1 , ..., n , E i ���� exp (1) iid 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, τ i , defined by τ i = inf { t ≥ 0 : Λ i ���� ≥ E i } , i = 1 , ..., n , t . Default Hazard Process Λ i . 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, τ i , defined by τ i = inf { t ≥ 0 : Λ i ���� ≥ E i } , i = 1 , ..., n , t . Default Hazard Process Λ i . t . . Consider the following processes: X a s , a = 1 , ..., d ; are d independent one-dimensional, non-negative Markov 0 = x a ≥ 0 process starting from X a . . . . . 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, τ i , defined by τ i = inf { t ≥ 0 : Λ i ���� ≥ E i } , i = 1 , ..., n , t . Default Hazard Process Λ i . t . . Consider the following processes: X a s , a = 1 , ..., d ; are d independent one-dimensional, non-negative Markov 0 = x a ≥ 0 process starting from X a R t Y a 0 X a t = 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, τ i , defined by τ i = inf { t ≥ 0 : Λ i ���� ≥ E i } , i = 1 , ..., n , t . Default Hazard Process Λ i . t . . Consider the following processes: X a s , a = 1 , ..., d ; are d independent one-dimensional, non-negative Markov 0 = x a ≥ 0 process starting from X a R t Y a 0 X a t = s ds , the corresponding integrated processes Z a t = Y a t ; time changed of process Y a t with a d -dimensional L´ evy T a subordinator T t (we time change the integral) . . . . . Rafael Mendoza (McCombs) Default Correlation WCMF 2009 3 / 26
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