CreditGrades Framework within Stochastic Covariance Models

Affiliation(s)

Department of Mathematics, Ryerson University, Toronto, Canada.

Royal Bank of Canada, Toronto, Canada.

Department of Mathematics, University of Toronto, Toronto, Canada.

Department of Mathematics, Ryerson University, Toronto, Canada.

Royal Bank of Canada, Toronto, Canada.

Department of Mathematics, University of Toronto, Toronto, Canada.

ABSTRACT

In this paper we study a multivariate extension of a structural credit risk model, the CreditGrades model, under the assumption of stochastic volatility and correlation between the assets of the companies. The covariance of the assets follows two popular models which are non-overlapping extensions of the CIR model to dimensions greater than one, the Wishart process and the Principal component process. Under CreditGrades, we find quasi closed-form solutions for equity options, marginal probabilities of defaults, and some other major financial derivatives.

In this paper we study a multivariate extension of a structural credit risk model, the CreditGrades model, under the assumption of stochastic volatility and correlation between the assets of the companies. The covariance of the assets follows two popular models which are non-overlapping extensions of the CIR model to dimensions greater than one, the Wishart process and the Principal component process. Under CreditGrades, we find quasi closed-form solutions for equity options, marginal probabilities of defaults, and some other major financial derivatives.

Cite this paper

M. Escobar, H. Arian and L. Seco, "CreditGrades Framework within Stochastic Covariance Models,"*Journal of Mathematical Finance*, Vol. 2 No. 4, 2012, pp. 303-313. doi: 10.4236/jmf.2012.24033.

M. Escobar, H. Arian and L. Seco, "CreditGrades Framework within Stochastic Covariance Models,"

References

[1] M.-F. Bru, “Wishart Processes,” Journal of Theoretical Probability, Vol. 4, No. 4, 1991, pp. 725-751.

[2] M. Escobar, B. Gotz, L. Seco and R. Zagst, “Pricing of a CDO on Stochastically Correlated Underlyings,” Quantitative Finance, Vol. 10, No. 3, 2007, pp. 265-277.

[3] C. Gourieroux, J. Jasiak and R. Sufana, “Derivative Pricing with Multivariate Stochastic Volatility: Application to Credit Risk,” Working Paper, 2004.

[4] C. Gourieroux and R. Sufana, “The Wishart Autoregressive Process of Multivariate Stochastic Volatility,” Econometrics, Vol. 150, No. 2, 2009, pp. 167-181. doi:10.1016/j.jeconom.2008.12.016

[5] J. DaFonseca, M. Grasselli and F. Ielpo, “Estimating the Wishart Affine Stochastic Correlation Model Using the Empirical Characteristic Functionk,” Working paper ESILV, RR-35, 2007.

[6] J. DaFonseca, M. Grasselli and C. Tebaldi, “A Multifactor Volatility Heston Model,” Quantitative Finance, Vol. 8, No. 6, 2006, pp. 591-604.

[7] J. DaFonseca, M. Grasselli and C. Tebaldi, “Option Pricing When Correlations Are Stochastic: An Analytical Framework,” Review of Derivatives Research, Vol. 10, No. 2, 2007, pp. 151-180.

[8] A. Lipton, “Mathematical Methods for Foreign Exchange: A Financial Engineers Approach,” World Scientific, Singapore, 2001.

[9] A. Sepp, “Extended Creditgrades Model with Stochastic Volatility and Jumps,” Wilmott Magazine, 2006, pp. 50-62.

[10] S. L. Heston, “A Closed-Form Solution for Options with Stochastic Volatility, with Applications to Bond and Currency Options,” Review of Financial Studies, Vol. 6, No. 2, 1993, pp. 327-343. doi:10.1093/rfs/6.2.327

[11] R. C. Merton, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” Journal of Finance, Vol. 29, No. 2, 1974, pp. 449-470.

[12] R. Stamicar and C Finger, “Incorporating Equity Derivatives into the Creditgrades Model,” RiskMetrics Group, Tampa, 2005.

[13] H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, “Matrix Riccati Equations in Control and Systems Theory,” Springer, Berlin, 2003. doi:10.1007/978-3-0348-8081-7

[14] M. Grasselli and C. Tebaldi, “Solvable Affine Term Structure Models,” Mathematical Finance, Vol. 18, No. 1, 2004, pp. 135-153.

[1] M.-F. Bru, “Wishart Processes,” Journal of Theoretical Probability, Vol. 4, No. 4, 1991, pp. 725-751.

[2] M. Escobar, B. Gotz, L. Seco and R. Zagst, “Pricing of a CDO on Stochastically Correlated Underlyings,” Quantitative Finance, Vol. 10, No. 3, 2007, pp. 265-277.

[3] C. Gourieroux, J. Jasiak and R. Sufana, “Derivative Pricing with Multivariate Stochastic Volatility: Application to Credit Risk,” Working Paper, 2004.

[4] C. Gourieroux and R. Sufana, “The Wishart Autoregressive Process of Multivariate Stochastic Volatility,” Econometrics, Vol. 150, No. 2, 2009, pp. 167-181. doi:10.1016/j.jeconom.2008.12.016

[5] J. DaFonseca, M. Grasselli and F. Ielpo, “Estimating the Wishart Affine Stochastic Correlation Model Using the Empirical Characteristic Functionk,” Working paper ESILV, RR-35, 2007.

[6] J. DaFonseca, M. Grasselli and C. Tebaldi, “A Multifactor Volatility Heston Model,” Quantitative Finance, Vol. 8, No. 6, 2006, pp. 591-604.

[7] J. DaFonseca, M. Grasselli and C. Tebaldi, “Option Pricing When Correlations Are Stochastic: An Analytical Framework,” Review of Derivatives Research, Vol. 10, No. 2, 2007, pp. 151-180.

[8] A. Lipton, “Mathematical Methods for Foreign Exchange: A Financial Engineers Approach,” World Scientific, Singapore, 2001.

[9] A. Sepp, “Extended Creditgrades Model with Stochastic Volatility and Jumps,” Wilmott Magazine, 2006, pp. 50-62.

[10] S. L. Heston, “A Closed-Form Solution for Options with Stochastic Volatility, with Applications to Bond and Currency Options,” Review of Financial Studies, Vol. 6, No. 2, 1993, pp. 327-343. doi:10.1093/rfs/6.2.327

[11] R. C. Merton, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” Journal of Finance, Vol. 29, No. 2, 1974, pp. 449-470.

[12] R. Stamicar and C Finger, “Incorporating Equity Derivatives into the Creditgrades Model,” RiskMetrics Group, Tampa, 2005.

[13] H. Abou-Kandil, G. Freiling, V. Ionescu and G. Jank, “Matrix Riccati Equations in Control and Systems Theory,” Springer, Berlin, 2003. doi:10.1007/978-3-0348-8081-7

[14] M. Grasselli and C. Tebaldi, “Solvable Affine Term Structure Models,” Mathematical Finance, Vol. 18, No. 1, 2004, pp. 135-153.