×

Asymptotic behaviour of the survival probabilities in an inhomogeneous semi-Markov model for the migration process in credit risk. (English) Zbl 1282.91363

Summary: We start with the stochastic foundation of the general discrete-time market of defaultable bonds. We prove that the above market is viable, if and only if there exists an equivalent martingale measure, from which we construct the forward probability measure and under which the discounted default free bond price is a martingale. Assuming that the migration process of defaultable bonds evolves as an inhomogeneous semi-Markov process, we study the asymptotic behaviour of survival probabilities. We provide a method of estimating real-world transition probability sequences for the semi-Markov process, and statistics for testing their homogeneity over time.

MSC:

91G40 Credit risk
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
60K15 Markov renewal processes, semi-Markov processes
62M05 Markov processes: estimation; hidden Markov models
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anderson, T. W.; Goodman, L. A., Statistical inference about Markov chains, Ann. Math. Statist., 28, 89-110 (1957) · Zbl 0087.14905
[2] Barbu, V.; Boussement, M.; Limnios, N., Discrete time semi-Markov model for reliability and survival analysis, Comm. Statist. Theory Methods, 33, 11, 2833-2868 (2004) · Zbl 1089.60525
[3] Barbu, V.; Limnios, N., Empirical estimation for discrete-time semi-Markov processes with application in reliability, J. Nonparametr. Statist., 18, 7-8, 483-498 (2006) · Zbl 1117.62081
[4] Belman, A.; Plemons, R. J., Non-negative Matrices in the Mathematical Sciences (1979), Academic Press · Zbl 0484.15016
[5] Bingham, N. H.; Kiesel, R., Risk-Neutral Valuation. Risk-Neutral Valuation, Pricing and Hedging Financial Derivatives (1998), Springer-Verlag: Springer-Verlag London · Zbl 0922.90009
[6] Bielecki, T. R.; Rutkowski, M., Credit Risk: Modeling, Valuation and Hedging (2002), Springer: Springer Berlin · Zbl 0979.91050
[7] Campbell, S. L.; Meyer, C. D., Generalized Inverses of Linear Transformations (1979), Pitman: Pitman London · Zbl 0417.15002
[8] Carty, L. V.; Fons, J. S., Measuring changes in corporate credit quality, J. Fixed Income, 4, 27-41 (1994)
[9] Coolen-Schrijner, P.; Van Doorn, E. A., Quasi-stationary distributions for a class of discrete-time Markov Chains, Methodol. Comput. Appl. Probab., 8, 449-465 (2006) · Zbl 1106.60064
[10] D’amico, G.; Janssen, J.; Manca, R., Homogeneous semi-Markov reliability models for credit risk management, Decis. Econom. Finance, 28, 2, 79-93 (2005) · Zbl 1125.91341
[11] D’amico, G.; Janssen, J.; Manca, R., Valuing credit default swap in a non-homogeneous semi-Markov rating based model, Comput. Econom., 29, 119-138 (2007) · Zbl 1161.91386
[12] D’amico, G., A semi-Markov maintenance model with credit rating application, IMA J. Manage. Math., 20, 51-58 (2009) · Zbl 1154.91498
[13] D’amico, G.; Janssen, J.; Manca, R., Initial and final backward and forward discrete time non-homogeneous semi-Markov credit risk models, Methodol. Comput. Appl. Probab., 12, 215-225 (2010) · Zbl 1194.60054
[14] D’amico, G.; Janssen, J.; Manca, R., Discrete time non-homogeneous semi-Markov reliability transition credit risk models and the default distribution functions, Comput. Econom., 38, 465-481 (2011) · Zbl 1247.91194
[15] Darroch, J. N.; Seneta, E., On quasi-stationary distributions in absorbing discrete time finite Markov chains, J. Appl. Probab., 2, 88-100 (1965) · Zbl 0134.34704
[16] Das, S. R.; Tufano, P., Pricing credit-sensitive debt when interest rates, credit ratings, and credit spreads are stochastic, J. Financial Eng., 5, 161-198 (1996)
[17] Douady, R.; Et Jeanblanc, M., A rating based model for credit derivatives, Eur. Investment Rev., 1, 17-29 (2002)
[18] Duffie, D.; Singleton, K., Modeling term structures of defaultable bonds, Rev. Financial Stud., 12, 687-720 (1999)
[19] Duffie, D.; Singleton, K., Credit Risk (2003), Princeton University Press
[20] Elliot, R. J.; Kopp, P. E., Mathematics of Financial Markets (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0943.91035
[21] Elliot, R. J.; Jeanblanc, M.; Yor, M., On models of default risk, Math. Finance, 10, 2, 179-195 (2000) · Zbl 1042.91038
[22] J.S. Fons, An approach to forecasting default rates, Working paper, Moody’s Investors Service, 1991.
[23] Gantmacher, F. R., Applications of the Theory of Matrices (1959), Interscience: Interscience New York · Zbl 0085.01001
[24] Guo, G.; Zhao, W., Schwarz methods for quasi stationary distributions of Markov chains, Calcolo, 49, 21-39 (2012) · Zbl 1252.60071
[25] D.T. Hamilton, Default and recovery rates of corporate bond issuers: 2000s, Special comment, Moody’s Investors Service, 2001.
[26] Horn, R. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge University Press · Zbl 0576.15001
[27] Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge University Press · Zbl 0729.15001
[28] Howard, R. A., Dynamic Probabilistic Systems, vol. I and II (1971), John Wiley: John Wiley Chichester · Zbl 0227.90031
[29] Huang, C. C.; Isaacson, D.; Vinograde, B., The rate of convergence of certain nonhomogeneous Markov chains, Z. Warscheinlich-keitstheorie. Gebiete, 35, 141-146 (1976) · Zbl 0356.60002
[30] Hunter, J. J., Mathematical Techniques of Applied Probability (1983), Academic Press · Zbl 0539.60065
[31] Jarrow, R.; Turnbull, S., Predicting derivatives on financial securities subject to credit risk, J. Finance, 50, 53-86 (1995)
[32] Jarrow, R.; Lando, D.; Turnbull, S., A Markov model for the term structure of credit risk spreads, Rev. Financial Stud., 10, 481-523 (1997)
[33] Kijima, M., On the existence of quasi-stationary distributions in denumerable R-transient Markov chains, J. Appl. Probab., 29, 21-36 (1992) · Zbl 0758.60062
[34] Kijima, M., Markov Processes for Stochastic Modeling (1997), Chapman and Hall · Zbl 0866.60056
[35] Kijima, M., Monotonicity in a Markov chain model for valuing coupon bond subject to credit risk, Math. Finance, 8, 229-247 (1998) · Zbl 1020.91033
[36] Kijima, M.; Komoribayashi, K., A Markov chain model for valuing credit risk derivatives, J. Derivatives, 6, 97-108 (1998)
[37] Kirkland, S., The group inverse associated with an irreducible periodic nonnegative matrix, SIAM J. Matrix Anal. Appl., 16, 4, 1127-1134 (1995) · Zbl 0838.15003
[38] Lando, D., On Cox process and credit risk securities, Rev. Derivatives Res., 2, 99-120 (1998) · Zbl 1274.91459
[39] Lando, D.; Skotemberg, T., Analysis rating transitions and rating drift with continuous observations, J. Banking Finance, 26, 423-444 (2001)
[40] Lando, D., Credit Risk Modeling (2004), Princeton University Press
[41] Lee, E. T., Statistical Methods for Survival Data Analysis (1992), John Wiley: John Wiley New York
[42] McClean, S. I.; Gribbin, J. O., Estimation for incomplete manpower data, Appl. Stoch. Models Data Anal., 3, 13-25 (1987) · Zbl 0613.62147
[43] McClean, S. I.; Gribbin, J. O., A non-parametric competing risks model for manpower planning, Appl. Stoch. Models Data Anal., 7, 327-341 (1991)
[44] McClean, S. I.; Montgomery, E.; Ugwuowo, F., Inhomogeneous continuous-time Markov and semi-Markov manpower models, Appl. Stoch. Models Data Anal., 13, 191-198 (1997) · Zbl 0913.60056
[45] McClean, S. I.; Montgomery, E., Estimation for semi-Markov manpower models in a stochastic environment, (Janssen, J.; Limnios, N., Semi-Markov Models and Applications (1999), Kluwer: Kluwer Dordrecht), 219-227 · Zbl 0960.62089
[46] Meyer, C. D., The role of the group generalized inverse in the theory of finite Markov chains, SIAM Rev., 17, 443-464 (1975) · Zbl 0313.60044
[47] Meyer, C. D., The condition of a finite Markov chain and perturbation bounds for the limiting probabilities, SIAM J. Alg. Disc. Meth., 1, 273-283 (1980) · Zbl 0498.60071
[48] Meyer, C. D., Updating finite Markov chains by using techniques of group matrix inversion, J. Statist. Comput. Simul., 11, 163-181 (1980) · Zbl 0464.60075
[49] Meyer, C. D.; Shoaf, J. M., Updating finite Markov chains by using techniques for group matrix inversion, J. Stat. Comput. Simulation, 11, 163-181 (1980) · Zbl 0464.60075
[50] A. Mondeiro, G.V. Smirnov, A. Lucas, Non-parametric estimation for non-homogeneous semi-Markov processes: an application to credit risk, Tinbergen Institute Discussion Paper, TI2006-024/2, 2006.
[51] Musiela, M.; Rutkowski, M., Martingale Methods in Financial Modeling (2000), Springer: Springer Berlin
[52] Oubi, B.; Limnios, N., The rate of occurrence of failures for semi-Markov processes and estimation, Stat. Probab. Lett., 59, 3, 245-255 (2002) · Zbl 1017.62106
[53] Papadopoulou, A. A.; Vassiliou, P.-C. G., Asymptotic behavior of non-homogeneous semi-Markov systems, Linear Algebra Appl., 210, 153-198 (1994) · Zbl 0813.60084
[54] Pliska, S. R., Introduction to Mathematical Finance. Introduction to Mathematical Finance, Discrete Time Models (1997), Blackwell Publishing
[55] Pease, M. C., Methods of Matrix Algebra (1965), Academic Press · Zbl 0145.03701
[56] Seneta, E., Non-negative Matrices and Markov chains (1981), Springer · Zbl 0471.60001
[57] Seneta, E.; Vere-Jones, D., On quasi-stationary distributions in discrete-time Markov chains with denumerable infinity of states, J. Appl. Probab., 3, 403-434 (1966) · Zbl 0147.36603
[58] Seneta, E.; Tweedie, R. L., Moments for stationary and quasi-stationary distributions of Markov chains, J. Appl. Probab., 22, 148-155 (1985) · Zbl 0563.60079
[59] Vasileiou, A.; Vassiliou, P.-C. G., An inhomogeneous semi-Markov model for the term structure of credit risk spreads, Adv. Appl. Prob., 38, 171-198 (2006) · Zbl 1100.60047
[60] Vassiliou, P.-C. G., A Markov model for wastage in manpower systems, Oper. Res. Quart., 27, 57-70 (1976)
[61] Vassiliou, P.-C. G.; Papadopoulou, A. A., Nonhomogeneous semi-Markov systems and maintainability of the state sizes, J. Appl. Probab., 29, 519-534 (1992) · Zbl 0766.90051
[62] Vassiliou, P.-C. G.; Symeonaki, M. A., The perturbed non-homogeneous semi-Markov system, Linear Algebra Appl., 289, 319-332 (1999) · Zbl 0947.60076
[63] Vassiliou, P-C. G.; Tsaklidis, G., The rate of convergence of the vector of variances and covariances in non-homogeneous Markov systems, J. Appl. Probab., 26, 776-783 (1989) · Zbl 0688.60058
[64] Vassiliou, P-C. G.; Tsakiridou, H., The evolution of the population structure of the perturbed non-homogeneous semi-Markov system, (Charalambidis, Ch. A.; Koutras, M. V.; Balakrishnan, N., Probability and Statistical Models with Applications (2001), Chapman and Hall), 185-205
[65] Vassiliou, P-C. G.; Tsakiridou, H., Asymptotic behaviour of first passage probabilities in the perturbed non-homogeneous semi-Markov systems, Comm. Statist. Theory Methods, 33, 651-679 (2004) · Zbl 1114.60326
[66] Vassiliou, P-C. G.; Tsaklidis, G., Applied Matrix Theory (2003), Ziti Pub. Comp. Thessaloniki: Ziti Pub. Comp. Thessaloniki Greece
[67] Vassiliou, P.-C. G., Discrete-time Asset Pricing Models in Applied Stochastic Finance, ISTE (2010), John Wiley · Zbl 1229.91005
[68] P.-C.G. Vassiliou, Fuzzy semi-Markov migration process in credit risk, Fuzzy Sets and Systems, in press. · Zbl 1284.91563
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.