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Valuation of a Credit Spread Put Option: The Stable Paretian model with Copulas

  • Dylan D’Souza
  • Key van Amir-Atefi
  • Borjana Racheva-Jotova

Abstract

Financial institutions are making a concerted effort to measure and manage credit risk inherent in their large defaultable portfolios. This is partly in response to regulatory requirements to have adequate capital to meet credit event contingencies, but risk managers are also concerned about the sensitivity of the value of their portfolios to potential deteriorating credit quality of issuers. These changes in portfolio value can be quite significant for financial institutions such as commercial banks, insurance companies and investment banks, exposed to credit risk inherent in their large bond and loan portfolios. Credit derivatives are instruments used to manage financial losses due to credit risk, but unlike derivatives to manage market risk they are relatively less liquid and are more complicated to price because of the relative illiquidity of the underlying reference assets.

Keywords

Credit Risk Tail Dependence Spot Rate Credit Spread Stochastic Volatility Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Alessandrini, F. (1999). Credit risk, interest rate risk and the Business cycle. The Journal of Fixed Income, 1(9):42–53.CrossRefGoogle Scholar
  2. [2]
    Altaian, E.I. (1983/1990). Corporate Financial Distress. Wiley, New York.Google Scholar
  3. [3]
    Altman, E.I., Brady, B., Resti, A., and Sironi, A. (2002). The link between default and recovery rates: Implications for credit risk models and procylicality.Google Scholar
  4. [4]
    Ammann, M. (1998). Pricing Derivative Credit Risk. Springer-Verlag, New York.Google Scholar
  5. [5]
    Anson, M. J.P. (1999). Credit Derivatives. Frank J. Fabozzi Associates, New Hope, PA.Google Scholar
  6. [6]
    Artzner, P., and Delbaen, F. (1992). Credit risk and prepayment option. 22: 81–96.Google Scholar
  7. [7]
    Bakshi, G., Madan, D., and Zhang, F. (2001). Recovery in default risk modeling: Theoretical foundations and empirical applications. University of Maryland and the Federal Reserve Board.Google Scholar
  8. [8]
    Black, F., and Cox., J. (1976). Valuing corporate securities: Some effects of bond indenture provisions. Journal of Finance, 31(2):351–367.CrossRefGoogle Scholar
  9. [9]
    Black, F., and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81: 637–654.CrossRefGoogle Scholar
  10. [10]
    Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31: 307–327.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Bollerslev, T. (1987). A conditional heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics, 69:542–547.CrossRefGoogle Scholar
  12. [12]
    Brèmaud, P. (1981). Point Processes and Queues. Springer-Verlag, Berlin, Heidelberg, New York.MATHCrossRefGoogle Scholar
  13. [13]
    Campbell, J.Y., Lo, A.W., and MacKinlay, A.C. (1997). The Econometrics of Financial Markets. Princeton University Press, Princeton, NJ.MATHGoogle Scholar
  14. [14]
    Clewlow, L., and Strickland, C. (1998). Implementing Derivatives Models. Wiley Series in Financial Engineering.Google Scholar
  15. [15]
    Cooper, I., and Martin, M. (1996). Default risk and derivative securities. Applied Mathematical Finance, 3(1): 109–126.CrossRefGoogle Scholar
  16. [16]
    Cossin, D. (1997). Credit risk pricing: A literature survey. Finanzmarkt und Portfolio Management, 11(4): 398–412.Google Scholar
  17. [17]
    Cossin, P., and Pirotte, H. (2001). Advanced Credit Risk Analysis: Financial approaches and mathematical models to assess, price and manage credit risk. John Wiley and Sons Ltd.Google Scholar
  18. [18]
    Crouhy, M., Galai, D., and Mark, R. (2000). A comparative analysis of current credit risk models. Journal of Banking and Finance, 24(1-2): 59–117.CrossRefGoogle Scholar
  19. [19]
    Das, S.R. (1995). Credit risk derivatives. Journal of Derivatives, 2(3): 7–23.CrossRefGoogle Scholar
  20. [20]
    Das, S.R. (1999). Pricing credit derivatives. Handbook of Credit Derivatives. McGraw-Hill.Google Scholar
  21. [21]
    Das, S.R., and Tufano, P. (1996). Pricing credit — sensitive debt when interest rates, credit ratings and credit spreads are stochastic. Journal of Financial Engineering, 5(2): 161–198.Google Scholar
  22. [22]
    D’Souza, D.M., Amir-Atefi, K., and Racheva-Jotova, B. (2002). Valuation of a credit default swap: The stable non-Gaussian versus the Gaussian approach, to appear in the Proceedings of the 8th Econometrics Conference on Credit Risk Management.Google Scholar
  23. [23]
    Duffle, D. (1998). Defaultable term structure models with fractional recovery of par. Working paper, Graduate School of Business, Stanford University.Google Scholar
  24. [24]
    Duffie, D. (1994). Forward rate curves with default risk. Working paper, Graduate School of Business, Stanford University.Google Scholar
  25. [25]
    Duffie, D. (1999). Credit swap valuation. Working paper, Graduate School of Business, Stanford University.Google Scholar
  26. [26]
    Duffie, D. (1999). Credit swap valuation. Association for Investment Management and Research, p. 73–86.Google Scholar
  27. [27]
    Duffie, D., and Huang, M. (1996). Swap rates and credit quality. Journal of Finance, 51: 921–949.CrossRefGoogle Scholar
  28. [28]
    Duffie, D., Schroder, M., and Skiadas, C. (1996). Recursive valuation of defaultable securities and the timing of resolution of uncertainty. Annals of Probability, 6:1075–1090.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    Duffie, D., and Singleton. K. (1997). An econometric model of the term structure of interest rate swap yields. Journal of Finance, 52(4): 1287–1322.CrossRefGoogle Scholar
  30. [30]
    Duffie, D., and K. Singleton. (1999). Modeling term structures of defaultable bonds. The Review of Financial Studies, 12(4): 687–720.CrossRefGoogle Scholar
  31. [31]
    DuMouchel, W. (1973a). Stable distributions in statistical inference: Symmetric stable distribution compared to other symmetric long-tailed distributions. Journal of the American Statistical Association, 68:469–477.MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    DuMouchel, W. (1973b). On the asymptotic normality of the maximum-likelihood estimate when sampling from a stable distribution. Annals of Statistics, 3:948–957.MathSciNetCrossRefGoogle Scholar
  33. [33]
    Embrechts, P., Klüppelberg, C, and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer-Verlag, Berlin.MATHCrossRefGoogle Scholar
  34. [34]
    Embrechts, P., McNeil, A., and Straumann, D. (1999). Correlation: Pitfalls and Alternatives. Department of Mathematics, ETH Zentrum, CH-8092 Zürich.Google Scholar
  35. [35]
    Embrechts, P., Lindskog, F., and McNeil, A. (2001). Modeling dependence with copulas and applications to risk management.Google Scholar
  36. [36]
    Fama, E. (1965a). The behavior of stock market prices. Journal of Business, 38:34–105.CrossRefGoogle Scholar
  37. [37]
    Fama, E. (1965b). Portfolio analysis in a stable Paretian market. Management Science, 11:404–19.MATHCrossRefGoogle Scholar
  38. [38]
    Flesaker, B., Houghston, L., Schreiber, L., and Sprung, L. (1994). Taking all the credit. Risk Magazine, 7:105–108.Google Scholar
  39. [39]
    Fons, J. (1994). Using default rates to model the term structures of defaultable bonds. Financial Analysts Journal, September-October, 25–32.Google Scholar
  40. [40]
    Franks, J.R. and Torous, W.N. (1994). A comparison of financial re-contracting in distressed exchanges and Chapter 11 reorganizations. Journal of Financial Economics, 35:349–370.CrossRefGoogle Scholar
  41. [41]
    Geske, R. (1977). The valuation of corporate liabilities as compound options. Journal of Financial and Quantitative Analysis, 12:541–552.CrossRefGoogle Scholar
  42. [42]
    Gupton, G.M. and Stein, R.M. (2002). LossCalc™ Moody’s Model for predicting loss given default (LGD), Global Credit Research, Moody’s Investor Services.Google Scholar
  43. [43]
    Heath, D., Jarrow, R., and Morton, A. (1992). Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica, 60:77–105.MATHCrossRefGoogle Scholar
  44. [44]
    Hull, J.C. (2000). Options, Futures, and Other Derivatives. Prentice-Hall International.Google Scholar
  45. [45]
    Hull, J. and A. White. (1990). Pricing Interest-Rate-Derivative Securities. Review of Financial Studies, 3:573–592.CrossRefGoogle Scholar
  46. [46]
    Hull, J. and A. White. (1994a). Numerical procedures for implementing term structure models I: Single-Factor models. Journal of Derivatives, 2:7–16.CrossRefGoogle Scholar
  47. [47]
    Hull, J. and A. White. (1994b). Numerical procedures for implementing term structure models II: Two-Factor models. Journal of Derivatives, 2:37–48.CrossRefGoogle Scholar
  48. [48]
    Jarrow, R.A., Lando, D., and Turnbull, S.M. (1997). A Markov model of the term structure of credit risk spreads. Review of Financial Studies, 10(2), Summer.Google Scholar
  49. [49]
    Jarrow, R.A., and S.M. Turnbull. (1995). Pricing derivatives on financial securities subject to credit risk. Journal of Finance, 50:53–85.CrossRefGoogle Scholar
  50. [50]
    Jarrow, R.A. and Turnbull, S.M. (2000). The intersection of market and credit risk. Journal of Banking and Finance, 24:271–299.CrossRefGoogle Scholar
  51. [51]
    Jarrow, R.A. and Yildirim, Y. (2002). Valuing default swaps under market and credit risk correlation. The Journal of Fixed Income, 7–19.Google Scholar
  52. [52]
    Jones, E.P., Mason, S.P., and Rosenfeld, E. (1984). Contingent claim analysis of corporate capital structures: An empirical investigation. Journal of Finance, 39(3).Google Scholar
  53. [53]
    Kou, S.G. (2002). A jump-diffusion model for option pricing. Management Science, 48(8):1086–1101.MATHCrossRefGoogle Scholar
  54. [54]
    Lando, D. (1994). Three essays on contingent claims pricing. Ph.D. thesis, Graduate School of Management, Cornell University.Google Scholar
  55. [55]
    Lando, D. (1998). On Cox processes and credit risky securities. Review of Derivatives Research, 2(2/3):99–120.CrossRefGoogle Scholar
  56. [56]
    Longstaff, F.A., and Schwartz, E. (1995a). A simple approach to valuing risky fixed and floating rate debt. Journal of Finance 50(3): 789–819.CrossRefGoogle Scholar
  57. [57]
    Longstaff, F.A., and Schwartz, E. (1995b). The pricing of credit derivatives. Journal of Fixed Income, 5(1): 6–14.CrossRefGoogle Scholar
  58. [58]
    Madan, D and H. Unal. (1998). Pricing the risks of default. Review of Derivatives Research, 2(2/3): 121–160.CrossRefGoogle Scholar
  59. [59]
    Mandelbrot, B.B., (1963). The variation of certain speculative prices. Journal of Business, 26:394–419.Google Scholar
  60. [60]
    Mandelbrot, B.B., (1967). The variation of some other speculative prices. Journal of Business, 40:393–413.CrossRefGoogle Scholar
  61. [61]
    Marinelli, C, Rachev, S.T., Roll, R., and Göppl, H. (1999). Subordinated stock price models: Heavy tails and long-range dependence in the high-frequency Deutsche bank price record. Technical Report, Department of Statistics and Mathematical Economics, University of Karlsruhe.Google Scholar
  62. [62]
    Martin, B., Rachev, S.T., Schwartz, E.S. (2000). Stable non-Gaussian models for credit risk management. Working paper, University of Karlsruhe, Germany.Google Scholar
  63. [63]
    Mashal, R., and Zeevi, A. (2002). Beyond correlation: Extreme co-movements between financial assets. Columbia, Graduate School of Business, Columbia University.Google Scholar
  64. [64]
    Merton, R.C. (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29:449–470.Google Scholar
  65. [65]
    Mittnik, S., Rachev, S.T., Doganoglu, T. and Chenyao, D. (1996) Maximum likelihood estimation of stable Paretian models. Working paper, Christian Albrechts University, Kiel.Google Scholar
  66. [66]
    Mittnik, S., and Rachev, S.T., (2000). Diagnosing and treating the fat tails in financial returns data. Journal of Empirical Finance 7:389–416.CrossRefGoogle Scholar
  67. [67]
    Nelsen, R.B. (1999). An Introduction to Copulas. Springer-Verlag, New York.MATHCrossRefGoogle Scholar
  68. [68]
    Nielsen, L. (1999). Pricing and Hedging of Derivative Securities. Oxford University Press.Google Scholar
  69. [69]
    Paulauskas, V. and Rachev, S.T. (1999). Maximum likelihood estimators in regression models with infinite variance innovation. Working paper, Vilnius University, LithuaniaGoogle Scholar
  70. [70]
    Pedrosa, M., and Roll, R. (1998). Systematic risk in corporate bond credit spreads. The Journal of Fixed Income, 7–26.Google Scholar
  71. [71]
    Prigent, J.L., Renault, O., Scaillet, O. (2001). An empirical investigation in credit spread indices.Google Scholar
  72. [72]
    Rachev, S.T., (1991). Probability Metrics and the Stability of Stochastic Models. John Wiley&Sons, Chichester, New York.MATHGoogle Scholar
  73. [73]
    Rachev, S.T., and Mittnik, S. (2000). Stable Paretian Models in Finance. Wiley&Sons, New York.MATHGoogle Scholar
  74. [74]
    Rachev, S.T, Racheva-Jotova, B., Hristov, B., I. Mandev (1999). Software Package for Market Risk (VaR) Modeling for Stable Distributed Financial Distributed Returns Mercury 1.0 Documentation.Google Scholar
  75. [75]
    Rachev, S.T., Schwartz, E., and Khindanova, I., (2000). Stable modeling of credit risk. Working paper UCLA.Google Scholar
  76. [76]
    Rachev, S.T., and Tokat, Y. (2000) Asset and liability management: recent advances. CRC Handbook on Analytic Computational Methods in Applied Mathematics.Google Scholar
  77. [77]
    Samorodnitsky, G., Taqqu, M.S. (1994). Stable Non-Gaussian Random Variables. Chapman and Hall, New York.Google Scholar
  78. [78]
    Sarig, O., and Warga, A. (1989). Some empirical estimates of the risk structure of interest rates. Journal of Finance, 44(5), 1351–1360.CrossRefGoogle Scholar
  79. [79]
    Schönbucher, P. (August 1996). The term structure of defaultable bond prices. Discussion Paper B-384, University of Bonn, SFB 303.Google Scholar
  80. [80]
    Schönbucher, P. (1998). Term structure modelling of defaultable bonds. Discussion Paper B-384,University of Bonn, SFB 303. Review of Derivatives Studies, Special Issue: Credit Risk 2(2/3): 161–192.Google Scholar
  81. [81]
    Schönbucher, P. (1999). Credit Risk Modelling and Credit Derivatives. Ph.D.-thesis, Faculty of Economics, Bonn University.Google Scholar
  82. [82]
    Schönbucher, P. (June 1999). A tree implementation of a credit spread model for credit derivatives. Department of Statistics, Bonn University.Google Scholar
  83. [83]
    Schönbucher, P. (2002). A tree implementation of a credit spread model for credit derivatives. The Journal of Computational Finance, 6(2), Winter 2002/3.Google Scholar
  84. [84]
    Schweizer, B. (1991). Thirty Years of Copulas. G. Dall’Aglio, S. Kotz, G. Salinetti, eds. Advances in Probability Distributions with Given Marginals, Kluwer, Dordrecht, Netherlands. 13–50CrossRefGoogle Scholar
  85. [85]
    Shane, H. (1994). Comovements of low grade debt and equity returns of highly levered firms. Journal of Fixed Income 3(4):79–89.CrossRefGoogle Scholar
  86. [86]
    Sklar, A. (1959). Fonctions de Répartition à n Dimensions et Leurs Marges. Publications de l’Institut Statistique de l’Université de Paris, 8:229–231.MathSciNetGoogle Scholar
  87. [87]
    Tavakoli, Janet M. (1998). Credit Derivatives: A guide to instruments and applications. John Wiley&Sons.Google Scholar
  88. [88]
    Tokat, Y, Rachev, S.T., Schwartz, E.S. (2001). Asset Liability Management: A review and some new results in the presence of heavy tails. Ziemba, W.T. (Ed) Handbook of Heavy Tailed Distributions in Finance, Handbook of Finance.Google Scholar
  89. [89]
    Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5: 177–188.CrossRefGoogle Scholar
  90. [90]
    Wilson, T. (1997a). Portfolio credit risk (1). Risk 10(9): 111–116.Google Scholar
  91. [91]
    Wilson, T. (1997b). Portfolio credit risk (2). Risk 10(10): 56–61.Google Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Dylan D’Souza
    • 1
  • Key van Amir-Atefi
    • 1
  • Borjana Racheva-Jotova
    • 2
  1. 1.Credit Risk ManagementHSBC Bank USANew York, NYUSA
  2. 2.Faculty of Economics and BusinessSofia University, BulgariaSofiaBulgaria

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