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                             Tail dependence of Gaussian Copula
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Copula approach to Default Correlation and the Credit Crisis of 2008/2009

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This presentation briefly explains the credit crisis and furthermore goes into more detail about credit risk valuation in CDO and CDS stuctures.

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Copula approach to Default Correlation and the Credit Crisis of 2008/2009

  1. 1. Modeling Correla- tion in Credit Modeling Correlation in Credit Risk Risk “Copula Functions” Robbin Tops “The Crisis” Robbin Tops Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion Advisor: Dr. Bas Kleijn KdV Instituut voor wiskunde Faculteit der Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam
  2. 2. Modeling Correla- tion in Credit Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  3. 3. Modeling Correla- tion in Credit Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  4. 4. Modeling Correla- tion in Credit Table of Contents Risk Robbin Tops “The Crisis” 1 “The Crisis” Credit Derivative Products 2 Credit Derivative Products Pricing Pricing model: Credit- Metrics 3 Pricing Critical View Conclusion 4 Pricing model: CreditMetrics 5 Critical View 6 Conclusion
  5. 5. Modeling Correla- tion in Credit Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion •
  6. 6. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  7. 7. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops The Credit Crisis brought two groups together “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  8. 8. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops Which represent “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  9. 9. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  10. 10. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops Investors wanted to turn their money into... “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  11. 11. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops MORE MONEY! “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  12. 12. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops Normally, investors go to the “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  13. 13. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops and get so called T-BONDS or Treasury Bond “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  14. 14. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops BUT “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  15. 15. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops Federal Reserve Chairman Alan Greenspan lowered interest rates. “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion And investors said: “Thank, but no thanks”.
  16. 16. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops On the flipside U.S. Bank could borrow for almost nothing. “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  17. 17. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  18. 18. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  19. 19. Modeling Correla- tion in Credit Leverage Risk Robbin Tops “The Crisis” Definition Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  20. 20. Modeling Correla- tion in Credit Example: Leverage Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  21. 21. Modeling Correla- tion in Credit Example: Leverage Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  22. 22. Modeling Correla- tion in Credit Example: Leverage Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  23. 23. Modeling Correla- tion in Credit Example: Leverage Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  24. 24. Modeling Correla- tion in Credit Example: Leverage Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  25. 25. Modeling Correla- tion in Credit Example: Leverage Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  26. 26. Modeling Correla- tion in Credit Example: Leverage Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  27. 27. Modeling Correla- tion in Credit Example: Leverage Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion Thus Leverage make good deals into GREAT deals.
  28. 28. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops We still have our investors sitting on a lot of money, wanting to make “The more! Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  29. 29. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops This gives Wallstreet an idea! “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  30. 30. Modeling Correla- tion in Credit What is the Credit Crisis? Risk Robbin Tops Connecting investors to homeowners through mortgages. “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  31. 31. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops “The A family wants to buy a house Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  32. 32. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops They go to a mortgage broker. “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  33. 33. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops The mortgage broker links the homeowners to a mortgage lender. “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  34. 34. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops An investment banker from Wallstreet calls the mortgage lender. “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  35. 35. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops The mortgage lender sells the mortgage to the investment banker. “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  36. 36. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops The investment banker buys many of these mortgages to make a “The deal with a lot of leverage! Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion Thus borrowing a lot of money from the federal reserve!
  37. 37. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops All these mortgages are now in a box and the investment banker “The receives all the mortgage payments from the homeowners. Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  38. 38. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops “The Crisis” This box is called a CDO. Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  39. 39. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops “The The math-wizards from Wallstreet cut this box in three slices. Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  40. 40. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops If some homeowners default the bottom tray may not get filled. “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  41. 41. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops Other banks or insurance companies (e.g. AIG) will insure the “safe” “The Crisis” slice with a CDS. Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  42. 42. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops Thus the rating agencies will give a rating according to the slices in “The the CDO. Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  43. 43. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops INTERESTING: Mortgages alone are almost never rated AAA but the “The top slice receives AAA ratings. Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  44. 44. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops Now the investment banker sells the slices individually: “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion The “Safe” slice is sold to investors only wanting safe investments. The “Okay” slice is sold to other investment bankers. The “Risky” slice is sold to hedge funds and other risk takers.
  45. 45. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops “The MORE!!! Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  46. 46. Modeling Correla- tion in Credit Connection: homeowners and investors Risk Robbin Tops The whole process repeats itself, but no more families. “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  47. 47. Modeling Correla- tion in Credit Crisis Risk Robbin Tops This gives the investment banker another idea!!! “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  48. 48. Modeling Correla- tion in Credit Crisis Risk Robbin Tops “The If a homeowner defaults on his mortgage... Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  49. 49. Modeling Correla- tion in Credit Crisis Risk Robbin Tops ...the investment banker owns the house. “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion But housing prices have been rising practically forever!
  50. 50. Modeling Correla- tion in Credit Crisis Risk Robbin Tops Thus the investment banker adds more risk to the mortgages. “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  51. 51. Modeling Correla- tion in Credit Crisis Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  52. 52. Modeling Correla- tion in Credit Crisis Risk Robbin Tops Instead of lending to responsible people... “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  53. 53. Modeling Correla- tion in Credit Crisis Risk Robbin Tops Sub-prime mortgages! “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  54. 54. Modeling Correla- tion in Credit Crisis Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  55. 55. Modeling Correla- tion in Credit Crisis Risk Robbin Tops Again the investment banker makes a CDO, now with the sub-prime “The mortgages! Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  56. 56. Modeling Correla- tion in Credit Crisis Risk Robbin Tops Now some of the sub-prime mortgages default. “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion No big deal!?
  57. 57. Modeling Correla- tion in Credit Crisis Risk Robbin Tops BUT more sub-prime mortgages defaulted! “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  58. 58. Modeling Correla- tion in Credit Crisis Risk Robbin Tops This changed the relation between supply and demand. “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  59. 59. Modeling Correla- tion in Credit Crisis Risk Robbin Tops This created an interesting situation for homeowners who did not “The default. Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  60. 60. Modeling Correla- tion in Credit Crisis Risk Robbin Tops “The Thus they walked away from their mortgages! Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  61. 61. Modeling Correla- tion in Credit Crisis Risk Robbin Tops Now the investment banker has a box full of worthless houses and no “The one wants to buy them! Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  62. 62. Modeling Correla- tion in Credit Crisis Risk Robbin Tops But he was not the only one! “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  63. 63. Modeling Correla- tion in Credit Crisis Risk Robbin Tops Because the investment banker used a lot of leverage to amplify his “The deal. Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion He was in a lot of trouble!
  64. 64. Modeling Correla- tion in Credit Crisis Risk Robbin Tops Consequently the whole financial system freezes, creating a frozen “The Crisis” credit market! Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  65. 65. Modeling Correla- tion in Credit Crisis Risk Robbin Tops “The BOOM!!! Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion Jarvis, J. , http://jonathanjarvis.com/crisis- of- credit, 6-01-2010.
  66. 66. Modeling Correla- tion in Credit Crisis Summary Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  67. 67. Modeling Correla- tion in Credit Crisis Summary Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  68. 68. Modeling Correla- tion in Credit Credit Default Swap Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing periodic payments Pricing ?? bps above LIBOR (CDS spread) model: −→ Credit- Metrics Protection Buyer Protection Seller Critical ←− View Specified payment in case Conclusion of default (Loss given Default)
  69. 69. Modeling Correla- tion in Credit ‘Cash’ Collateralized Debt Obligation Risk Robbin Tops “The Crisis” Credit Portfolio Derivative Company 1 → Bond 1 Products Company 2 → Bond 2 Periodic coupon Super Senior Tranche Pricing Periodic payments Lowest return/Residual loss . . payments ?? bps above LIBOR Pricing . . −→ −→ Senior Tranche model: . . Credit- 2nd lowest return/3rd ..% of loss Metrics SPV Mezzanine Tranche Critical 2nd highest return/2nd ..% of loss View ←− ←− ↓ ↓ Sp. payment Sp. payment Conclusion (in case of default) (in case of default) Equity Tranche Highest return/1st ..% of loss Company n → Bond n
  70. 70. Modeling Correla- tion in Credit Collateralized Debt Obligation Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  71. 71. Modeling Correla- tion in Credit Usual scenario Risk Robbin Tops “The Crisis” Credit Derivative Products Individual Bonds/Loans Pricing Bond issuer may default on the bond/loan (Credit Risk), Pricing Money that is loaned to bond issuer is illiquid (Market Risk). model: Credit- Metrics Critical View Conclusion
  72. 72. Modeling Correla- tion in Credit Usual scenario Risk Robbin Tops “The Crisis” Credit Derivative Products Individual Bonds/Loans Pricing Bond issuer may default on the bond/loan (Credit Risk), Pricing Money that is loaned to bond issuer is illiquid (Market Risk). model: Credit- Metrics CDSs Critical Protection buyer may default (Credit Risk), View Protection seller may default (Credit Risk). Conclusion Default dependence of protection buyer and seller!
  73. 73. Modeling Correla- tion in Credit CDO scenario Risk Robbin Tops “The Crisis” Cash CDOs Credit Derivative Any amount of bond issuers may default (Credit Risk), Products Money that is loaned is illiquid (Market Risk). Pricing Pricing model: Credit- Metrics Default dependence of all bond issuers! Critical View Conclusion
  74. 74. Modeling Correla- tion in Credit CDO scenario Risk Robbin Tops “The Crisis” Cash CDOs Credit Derivative Any amount of bond issuers may default (Credit Risk), Products Money that is loaned is illiquid (Market Risk). Pricing Pricing model: Credit- Metrics Default dependence of all bond issuers! Critical View Synthetic CDOs Conclusion Any amount of protection buyer may default (Credit Risk), Any amount of protection seller may default (Credit Risk). Default dependence between buyers and seller!
  75. 75. Modeling Correla- tion in Credit Introduction Risk Robbin Tops What will follow “The Li’s approach to default correlation: Crisis” Model survival time of credit entities, Credit Model asset correlation, Derivative Products Use copula and correlation to create dependence structure, Pricing Rescale marginals of joint distribution to survival time distributions, Pricing Generate from copula to calculate default correlation. model: Credit- Metrics Critical View Conclusion
  76. 76. Modeling Correla- tion in Credit Introduction Risk Robbin Tops What will follow “The Li’s approach to default correlation: Crisis” Model survival time of credit entities, Credit Model asset correlation, Derivative Products Use copula and correlation to create dependence structure, Pricing Rescale marginals of joint distribution to survival time distributions, Pricing Generate from copula to calculate default correlation. model: Credit- Metrics Diagram Critical View Survival Time Conclusion Distributions Joint Survival Times & Default Correlation Asset Value Gaussian Processes & Copula Asset −→ Function Correlation
  77. 77. Modeling Correla- tion in Credit Survival time distribution Risk Robbin Tops Definition (Time-to-default) Let Ti be the random variable time-to-default of financial entity i and “The Crisis” STi (t ) := P (Ti > t ) is the survival function of i. Credit Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion Cox, D. R. , Oake, D. , Analysis of Survival Data, London: Chapman and Hall, 1984.
  78. 78. Modeling Correla- tion in Credit Survival time distribution Risk Robbin Tops Definition (Time-to-default) Let Ti be the random variable time-to-default of financial entity i and “The Crisis” STi (t ) := P (Ti > t ) is the survival function of i. Credit Derivative Products Definition (Hazard Rate Function) Pricing If T is absolutely continuous and define fT (t ) as the density function Pricing of T , then model: f (t ) −ST (t ) Credit- Metrics h (t ) : = T = Critical ST (t ) ST (t ) View is the hazard rate function. Conclusion Cox, D. R. , Oake, D. , Analysis of Survival Data, London: Chapman and Hall, 1984.
  79. 79. Modeling Correla- tion in Credit Survival time distribution Risk Robbin Tops Definition (Time-to-default) Let Ti be the random variable time-to-default of financial entity i and “The Crisis” STi (t ) := P (Ti > t ) is the survival function of i. Credit Derivative Products Definition (Hazard Rate Function) Pricing If T is absolutely continuous and define fT (t ) as the density function Pricing of T , then model: f (t ) −ST (t ) Credit- Metrics h (t ) : = T = Critical ST (t ) ST (t ) View is the hazard rate function. Conclusion Thus: t h(s )ds ST (t ) = e 0 Cox, D. R. , Oake, D. , Analysis of Survival Data, London: Chapman and Hall, 1984.
  80. 80. Modeling Correla- tion in Credit Hazard Rate h(s ) Risk Robbin Estimation Tops Historical default information, “The Merton option theoretical approach, Crisis” Implied approach using market price of defaultable bonds or Credit asset swap spreads. Derivative Products Pricing Pricing model: Credit- Metrics Critical View Conclusion Duffie, D. and Singleton, J. , Modeling Term Structure of Defaultable Bonds. In: Rev. Financ. Stud. 12, pp. 687–720, 1999. Li, D. X. , On Default Correlation: A Copula Function Approach. In: Journal of Fixed Income 9(4), pp. 43–54, 2000.
  81. 81. Modeling Correla- tion in Credit Hazard Rate h(s ) Risk Robbin Estimation Tops Historical default information, “The Merton option theoretical approach, Crisis” Implied approach using market price of defaultable bonds or Credit asset swap spreads. Derivative Products Pricing Example (Hazard rate function for B rating) Pricing model: Credit- Metrics Critical View Conclusion Duffie, D. and Singleton, J. , Modeling Term Structure of Defaultable Bonds. In: Rev. Financ. Stud. 12, pp. 687–720, 1999. Li, D. X. , On Default Correlation: A Copula Function Approach. In: Journal of Fixed Income 9(4), pp. 43–54, 2000.
  82. 82. Modeling Correla- tion in Credit Asset correlation Risk Robbin Definition (Asset value process) Tops The asset valuation of a financial entity i is assumed to follow a “The standard geometric Brownian motion, i.e. Crisis” Credit 1 2 √ Derivative i Vti = V0 exp (µi − σi )t + σi tZt Products 2 Pricing Pricing where µi represents the mean of the rate of return, σi the volatilities model: Credit- of returns on assets and Zt ∼ N (0, 1). Metrics Critical View Conclusion Merton, R. , On the pricing of corporate debt: The risk structure of interest rates. In: Journal of Finance 28, pp. 449–470, 1974.
  83. 83. Modeling Correla- tion in Credit Asset correlation Risk Robbin Definition (Asset value process) Tops The asset valuation of a financial entity i is assumed to follow a “The standard geometric Brownian motion, i.e. Crisis” Credit 1 2 √ Derivative i Vti = V0 exp (µi − σi )t + σi tZt Products 2 Pricing Pricing where µi represents the mean of the rate of return, σi the volatilities model: Credit- of returns on assets and Zt ∼ N (0, 1). Metrics Critical View Thus Conclusion qi := P (Vti ≤ vdef ) is assumed to be the individual default probability. j ρasset := ρ(Rti , Rt ) is the (linear) correlation coefficient between the normalized asset returns of financial entities i and j. Merton, R. , On the pricing of corporate debt: The risk structure of interest rates. In: Journal of Finance 28, pp. 449–470, 1974.
  84. 84. Modeling Correla- tion in Credit Copulas Risk Robbin Definition (Copula) Tops If F is a n-dimensional joint distribution function with marginals “The F1 , F2 , . . . , Fn then a copula C is a function Crisis” Credit C : [0, 1]n −→ [0, 1] Derivative Products Pricing such that Pricing C (F1 (x1 ), . . . , Fn (xn )) := F (x1 , . . . , xn ). model: Credit- Metrics Critical View Conclusion Nelson, R. B. , An Introduction to Copulas. In: Journal of Finance 28, New York: Springer, 1999.
  85. 85. Modeling Correla- tion in Credit Copulas Risk Robbin Definition (Copula) Tops If F is a n-dimensional joint distribution function with marginals “The F1 , F2 , . . . , Fn then a copula C is a function Crisis” Credit C : [0, 1]n −→ [0, 1] Derivative Products Pricing such that Pricing C (F1 (x1 ), . . . , Fn (xn )) := F (x1 , . . . , xn ). model: Credit- Metrics Critical Theorem (Sklar’s Theorem (modified)) View Let F be a n-dimensional distribution function with continuous Conclusion marginals F1 , . . . , Fn . Then there exists a unique n-dimensional copula C such that for all x ∈ Rn , ¯ F (x1 , . . . , xn ) = C (F1 (x1 ), . . . , Fn (xn )). (1) Conversely, if C is a copula and F1 , . . . , Fn are univariate continuous distribution functions, then the function F defined in (1) is a multivariate distribution function with marginals F1 , . . . , Fn . Nelson, R. B. , An Introduction to Copulas. In: Journal of Finance 28, New York: Springer, 1999.
  86. 86. Modeling Correla- tion in Credit Gaussian Copula Risk Robbin Tops “The Bivariate Gaussian Copula Crisis” Let F1 and F2 be standard normal distribution functions, then Credit Derivative z1 z2 Ga Products Cρ (F1 (z1 ), F2 (z2 )) = φ2 (x, y |ρ)dxdy = Φ2 (z1 , z2 , ρ) Pricing −∞ −∞ Pricing model: Credit- with ρ the correlation parameter between F1 and F2 . Metrics Critical View Conclusion
  87. 87. Modeling Correla- tion in Credit Gaussian Copula Risk Robbin Tops “The Bivariate Gaussian Copula Crisis” Let F1 and F2 be standard normal distribution functions, then Credit Derivative z1 z2 Ga Products Cρ (F1 (z1 ), F2 (z2 )) = φ2 (x, y |ρ)dxdy = Φ2 (z1 , z2 , ρ) Pricing −∞ −∞ Pricing model: Credit- with ρ the correlation parameter between F1 and F2 . Metrics Critical Thus: View The joint default probability is assumed to be Conclusion P (Vti ≤ vdef , Vti ≤ vdef ) = Cρasset (Φ(ri ), Φ(rj )) = Φ2 (ri , rj , ρasset ) i i Ga vk ln def −(µk − 1 σk )t 2 Vk 2 where k vdef are default thresholds and rk := 0 √ for σk t k = i, j are normalized thresholds.
  88. 88. Modeling Correla- tion in Credit Rescale Marginals to Survival Time and Risk Default correlation Robbin Tops Joint Survival Function “The If Sk (t ) = 1 − Gk (t ) with k = i, j are the survival functions for Ti and Crisis” Tj , then the joint survival function can be defined as Credit Derivative Products P (Ti ≤ ti , Tj ≤ tj ) Ga = Cρasset (Gi (ti ), Gj (tj )) Pricing (2) Pricing model: Credit- = Φ2 (Φ−1 (Gi (ti )), Φ−1 (Gj (tj )), ρasset ). Metrics Critical View Conclusion Li, D. X. , On Default Correlation: A Copula Function Approach. In: Journal of Fixed Income 9(4), pp. 43–54, 2000.
  89. 89. Modeling Correla- tion in Credit Rescale Marginals to Survival Time and Risk Default correlation Robbin Tops Joint Survival Function “The If Sk (t ) = 1 − Gk (t ) with k = i, j are the survival functions for Ti and Crisis” Tj , then the joint survival function can be defined as Credit Derivative Products P (Ti ≤ ti , Tj ≤ tj ) Ga = Cρasset (Gi (ti ), Gj (tj )) Pricing (2) Pricing model: Credit- = Φ2 (Φ−1 (Gi (ti )), Φ−1 (Gj (tj )), ρasset ). Metrics Critical View Default correlation Conclusion We generate from Equation (2) to calculate (linear) default correlation as follows, E (Ti Tj ) − E (Ti )E (Tj ) ρdef = ρ(Ti , Tj ) = . Var (Ti )Var (Tj ) Li, D. X. , On Default Correlation: A Copula Function Approach. In: Journal of Fixed Income 9(4), pp. 43–54, 2000.
  90. 90. Modeling Correla- tion in Credit Asset values Risk Robbin Tops Theorem “The Crisis” Let Di (t ) and Dj (t ) be the comprehensive default events of the Credit Derivative financial entities i and j, respectively. If asset value processes Vti and Products j Vt for financial entities i and j, respectively, then Pricing Pricing j j model: Vti < vti , Vt < vt ⊂ Di ( t ) ∩ Dj ( t ) . Credit- Metrics Critical View Conclusion
  91. 91. Modeling Correla- tion in Credit Asset values Risk Robbin Tops Theorem “The Crisis” Let Di (t ) and Dj (t ) be the comprehensive default events of the Credit Derivative financial entities i and j, respectively. If asset value processes Vti and Products j Vt for financial entities i and j, respectively, then Pricing Pricing j j model: Vti < vti , Vt < vt ⊂ Di ( t ) ∩ Dj ( t ) . Credit- Metrics Critical View Conclusion Example Insurance Company XYZ insurance insurance Zero asset correlation Financial Financial Entity i ←→ Entity j
  92. 92. Modeling Correla- tion in Credit Tail dependence of Gaussian Copula Risk Robbin Tops “The Definition (Tail Dependence) Crisis” The upper and lower tail dependence coefficients are defined by Credit Derivative − − Products λu = lim P X2 > F2 1 (q )|X1 > F1 1 (q ) Pricing q ↑1 Pricing − − model: Credit- λl = lim P X2 < F2 1 (q )|X1 < F1 1 (q ) , Metrics q ↑1 Critical View respectively, and measure the probability of joint extreme events. Conclusion
  93. 93. Modeling Correla- tion in Credit Tail dependence of Gaussian Copula Risk Robbin Tops “The Definition (Tail Dependence) Crisis” The upper and lower tail dependence coefficients are defined by Credit Derivative − − Products λu = lim P X2 > F2 1 (q )|X1 > F1 1 (q ) Pricing q ↑1 Pricing − − model: Credit- λl = lim P X2 < F2 1 (q )|X1 < F1 1 (q ) , Metrics q ↑1 Critical View respectively, and measure the probability of joint extreme events. Conclusion Theorem The upper and lower tail dependence coefficients of the Gaussian Ga copula Cρ are zero, that is, λu = λl = 0 for ρ < 1.
  94. 94. Modeling Correla- tion in Credit Tail dependence of Gaussian Copula Risk Robbin Tops “The Crisis” Example Credit Derivative Tails of bivariate Gaussian versus bivariate t-distribution. Products Pricing Pricing model: Credit- Metrics Critical View Conclusion
  95. 95. Modeling Correla- tion in Credit Dependence Structure in Rescaling Risk Marginals Robbin Tops Theorem “The Let H1 and H2 be strictly monotonic continuous functions defined on Crisis” the range of random variables X1 and X2 , respectively, then Credit Derivative Products |ρ(X1 , X2 )| > |ρ (H1 (X1 ), H2 (X2 )) |, Pricing where ρ is the linear correlation coefficient. Pricing model: ⇒ Dependence structure always reduces if marginals are Credit- Metrics transformed! Critical View Conclusion
  96. 96. Modeling Correla- tion in Credit Dependence Structure in Rescaling Risk Marginals Robbin Tops Theorem “The Let H1 and H2 be strictly monotonic continuous functions defined on Crisis” the range of random variables X1 and X2 , respectively, then Credit Derivative Products |ρ(X1 , X2 )| > |ρ (H1 (X1 ), H2 (X2 )) |, Pricing where ρ is the linear correlation coefficient. Pricing model: ⇒ Dependence structure always reduces if marginals are Credit- Metrics transformed! Critical View Example Conclusion
  97. 97. Modeling Correla- tion in Credit Summary Risk Robbin Tops “The Crisis” Credit Derivative Products Pricing Which flaws have we seen Pricing Defaultable bond prices or asset swap spreads to estimate survival model: Credit- functions, Metrics Asset value as underlying information for dependence structure, Critical View Gaussian copula and simultaneous extreme events, Conclusion Rescaling results in weaker correlation structure, Linear correlation has many undesirable properties.
  98. 98. Modeling Correla- tion in Credit Conclusion Risk Robbin Tops “The Crisis” Credit Derivative Products CreditMetric Approach Pricing Pricing of CDOs is complicated, due to complexity, Pricing model: Assumptions should not be indiscriminately excepted. Credit- Metrics Critical View Conclusion
  99. 99. Modeling Correla- tion in Credit Conclusion Risk Robbin Tops “The Crisis” Credit Derivative Products CreditMetric Approach Pricing Pricing of CDOs is complicated, due to complexity, Pricing model: Assumptions should not be indiscriminately excepted. Credit- Metrics Critical Possible Solutions View Jump Levy component in asset value process, Conclusion Factor models for default, t-copula model, Non-parametric model.

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