Semi markov migration models for credit risk

Semi-Markov Migration Models for Credit Risk Stochastic Models for Insurance Set coordinated by Jacques Janssen Volume Semi-Markov Migration Models for Credit Risk Guglielmo D’Amico Giuseppe Di Biase Jacques Janssen Raimondo Manca First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK John Wiley & Sons, Inc 111 River Street Hoboken, NJ 07030 USA www.iste.co.uk www.wiley.com © ISTE Ltd 2017 The rights of Guglielmo D’Amico, Giuseppe Di Biase, Jacques Janssen and Raimondo Manca to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 Library of Congress Control Number: 2017931483 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-905-2 Contents Introduction ix Chapter Semi-Markov Processes Migration Credit Risk Models 1.1 Rating and migration problems 1.1.1 Ratings 1.1.2 Migration problem 1.1.3 Impact of rating on spreads for zero bonds 1.1.4 Homogeneous Markov chain model 1.1.5 Migration models 1.2 Homogeneous semi-Markov processes 1.2.1 Basic definitions 1.2.2 The Z SMP and the evolution equation system 1.2.3 Special cases of SMP 1.2.4 Sojourn times and their distributions 1.3 Homogeneous semi-Markov reliability model 1.4 Homogeneous semi-Markov migration model 1.4.1 Equivalence with the reliability problem 1.4.2 Transient results 1.4.3 Asymptotic results 1.4.4 Example 1.5 Discrete time non-homogeneous case 1.5.1 NHSMPs and evolution equations 1.5.2 The Z NHSMP 1.5.3 Sojourn times and their distributions 1.5.4 Non-homogeneous semi-Markov reliability model 1 10 10 14 16 19 21 23 23 24 26 28 33 33 34 36 37 vi Semi-Markov Migration Models for Credit Risk 1.5.5 The non-homogeneous semi-Markov migration model 1.5.6 A non-homogeneous example 38 39 Chapter Recurrence Time HSMP and NHSMP: Credit Risk Applications 51 2.1 Introduction 2.2 Recurrence times 2.3 Transition probabilities of homogeneous SMP and non-homogeneous SMP with recurrence times 2.3.1 Transition probabilities with initial backward 2.3.2 Transition probabilities with initial forward 2.3.3 Transition probabilities with final backward and forward 2.3.4 Transition probabilities with initial and final backward 2.3.5 Transition probabilities with initial and final forward 2.3.6 Transition probabilities with initial and final backward and forward 2.4 Reliability indicators of HSMP and NHSMP with recurrence times 2.4.1 Reliability indicators with initial backward 2.4.2 Reliability indicators with initial forward 2.4.3 Reliability indicators with initial and final backward 2.4.4 Reliability indicators with initial and final backward and forward 51 52 53 53 55 57 58 60 61 63 63 66 70 73 Chapter Recurrence Time Credit Risk Applications 79 3.1 S&P’s basic rating classes 3.1.1 Homogeneous case 3.1.2 Non-homogeneous case 3.2 S&P’s basic rating classes and NR state 3.2.1 Homogeneous case 3.2.2 Non-homogeneous case 3.3 S&P’s downward rating classes 3.3.1 An application 3.4 S&P’s basic rating classes & NR1 and NR2 states 3.5 Cost of capital implications 80 81 86 90 91 106 120 122 127 134 Chapter Mono-Unireducible Markov and Semi-Markov Processes 137 4.1 Introduction 4.2 Graphs and matrices 4.3 Single-unireducible non-homogeneous Markov chains 137 138 145 Contents vii 4.4 Single-unireducible semi-Markov chains 4.5 Mono-unireducible non-homogeneous backward semi-Markov chains 4.6 Real data credit risk application 158 160 Chapter Non-Homogeneous Semi-Markov Reward Processes and Credit Spread Computation 165 5.1 Introduction 5.2 The reward introduction 5.3 The DTNHSMRWP spread rating model 5.4 The algorithm description 5.5 A numerical example 5.5.1 Data 5.5.2 Results 165 166 168 170 173 173 176 Chapter NHSMP Model for the Evaluation of Credit Default Swaps 183 6.1 The price and the value of the swap: the fixed recovery rate case 6.2 The price and the value of the swap: the random recovery rate case 6.3 The determination of the n-period random recovery rate 6.4 A numerical example 184 188 196 198 Chapter Bivariate Semi-Markov Processes and Related Reward Processes for Counterparty Credit Risk and Credit Spreads 205 7.1 Introduction 7.2 Multivariate semi-Markov chains 7.3 The two-component reliability model 7.4 Counterparty credit risk in a CDS contract 7.4.1 Pricing a risky CDS and CVA evaluation 7.5 A numerical example 7.6 Bivariate semi-Markov reward chains 7.7 The estimation methodology 7.8 Credit spreads evaluation 7.9 Numerical experience 152 206 208 220 224 227 230 233 247 249 259 Chapter Semi-Markov Credit Risk Simulation Models 267 8.1 Introduction 8.2 Monte Carlo semi-Markov credit risk model for the Basel II Capital at Risk problem 267 267 viii Semi-Markov Migration Models for Credit Risk 8.2.1 The homogeneous MCSM evolution with D as absorbing state 8.3 Results of the MCSMP credit model in a homogeneous environment 273 Bibliography 279 Index 297 269 Introduction This book is a summary of several papers that the authors wrote on credit risk starting from 2003 to 2016 Credit risk problem is one of the most important contemporary problems that has been developed in the financial literature The basic idea of our approach is to consider the credit risk of a company like a reliability evaluation of the company that issues a bond to reimburse its debt Considering that semi-Markov processes (SMPs) were applied in the engineering field for the study of reliability of complex mechanical systems, we decided to apply this process and develop it for the study of credit risk evaluation Our first paper [D’AM 05] was presented at the 27th Congress AMASES held in Cagliari, 2003 The second paper [D’AM 06] was presented at IWAP 2004 Athens The third paper [D’AM 11] was presented at QMF 2004 Sidney Our remaining research articles are as follows: [D’AM 07, D’AM 08a, D’AM 08b, SIL08, D’AM 09, D’AM 10, D’AM 11a, D’AM 11b, D’AM 12, D’AM 14a, D’AM 14b, D’AM 15, D’AM 16a] and [D’AM 16b] Other credit risk studies in a semi-Markov setting were from [VAS 06, VAS 13] and [VAS 13] We should also outline that up to now, at author’s knowledge, no papers were written for outline problems or criticisms to the applications of SMPs to the migration credit risk x Semi-Markov Migration Models for Credit Risk The study of credit risk began with so-called structural form models (SFM) Merton [MER 74] proposed the first paper regarding this approach This paper was an application of the seminal papers by Black and Scholes [BLA 73] According to Merton’s paper, default can only happen at the maturity date of the debt Many criticisms were made on this approach Indeed, it was supposed that there are no transaction costs, no taxes and that the assets are perfectly divisible Furthermore, the short sales of assets are allowed Finally, it is supposed that the time evolution of the firm’s value follows a diffusion process (see [BEN 05]) In Merton’s paper [MER 74], the stochastic differential equation was the same that could be used for the pricing of a European option This problem was solved by Black and Cox [BLA 76] by extending Merton’s model, which allowed the default to occur at any time and not only at the maturity of the bond In this book, techniques useful for the pricing of American type options are discussed Many other papers generalized the Merton and Black and Cox results We recall the following papers: [DUA 94, LON 95, LEL 94, LEL 06, JON 84, OGD 87, LYD 00, EOM 03] and [GES 77] The second approach to the study of credit risk involves reduced form models (RFMs) In this case, pricing and hedging are evaluated by public data, which are fully observable by everybody In SFM, the data used for the evaluation of risk are known only within the company More precisely, [JAR 04] explains that in the case of RFM, the information set is observed by the market, and in the case of SFM, the information set is known only inside the company The first RFM was given in [JAR 92] In the late 1990s, these models developed The seminal paper [JAR 97] introduced Markov models for following the evolution of rating Starting from this paper, although many models make use of Markov chains, the problem of the poorly fitting Markov processes in the credit risk environment has been outlined Ratings change with time and a way of following their evolution their by means of Markov processes (see, for example, [JAR 97, ISR 01, HU 02] In this environment, Markov models are called migration models The problem Introduction xi of poorly fitting Markov processes in the credit risk environment has been outlined in some papers, including [ALT 98, CAR 94] and [LAN 02] These problems include the following: – the duration inside a state: actually, the probability of changing rating depends on the time that a firm remains in the same rating Under the Markov assumption, this probability depends only on the rank at the previous transition; – the dependence of the rating evaluation from the epoch of the assessment: this means that, in general, the rating evaluation depends on when it is done and, in particular, on the business cycle; – the dependence of the new rating from all history of the firm’s rank evolution, not only from the last evaluation: actually, the effect exists only in the downward cases but not in the case of upward ratings in the sense that if a firm gets a lower rating (for almost all rating classes), then there is a higher probability that the next rating will be lower than the preceding one All these problems were solved by means of models that applied the SMPs, generalizing the Markov migration models This book is self-contained and is divided into nine chapters The first part of the Chapter briefly describes the rating evolution and introduces to the meaning of migration and the importance of the evaluation of the probability of default for a company that issues bonds In the second part, Markov chains are described as a mathematical tool useful for rating migration modeling The subsequent step shows how rating migration models can be constructed by means of Markov processes Once the Markov limits in the management of migration models are defined, the chapter introduces the homogeneous semi-Markov environment The last tool that is presented is the non-homogeneous semi-Markov model Real-life examples are also presented In Chapter 2, it is shown how it is possible to take into account simultaneously recurrence times, i.e backward and forward processes at the beginning and at the end of the time in which the credit risk model is observed With such a generalization, it is possible to consider what happens Bibliography 287 [DUF 03b] DUFFIE D., SINGLETON K., Credit risk, Princeton University Press, Princeton, 2003 [DUF 04] DUFFIE D., Credit Risk Modeling with Affine Processes, Springer, Berlin, 2004 [DUF 07] DUFFIE D., SAITA L., WANG K., “Multi-period corporate default prediction with stochastic covariates”, Journal of Financial Economics, vol 83, pp 635–665, 2007 [DUF 11] DUFFIE D., Measuring Corporate Default Risk, Oxford University Press, Oxford, 2011 [ELI 03a] ELIZALDE A., “Credit risk models I: default correlation in intensity models”, Working Paper, CEMFI, 2003 [ELI 03b] ELIZALDE A., “Credit risk models III: reconciliation reduced-structural models”, Working Paper, CEMFI, 2003 [ELT 01] ELTON E.J., GRUBER M.J., AGRAWAL D et al., “Explaining the rate spread on corporate bonds”, The Journal of Finance, vol 56, pp 247–277, 2001 [EOM 03] EOM Y.H., HELWEGE J., HUANG J.Z., “Structural models of corporate bond pricing: an empirical analysis”, The Review of Financial Studies, vol 17, pp 499–544, 2003 [ERI 02] ERICSSON J., RENEBY J., “Estimating structural bond pricing models”, Working Paper, Stockholm School of Economics, 2002 [ERI 03] ERICSSON J., RENEBY J., “The valuation of corporate liabilities: theory and tests”, Working Paper, SSE/EFI No 445, 2003 [FEL 71] FELLER W., An Introduction to Probability Theory and its Applications, vol II, 2nd ed., John Wiley & Sons, New York, 1971 [FIG 11] FIGINI S., GIUDICI P., “Statistical merging of rating models”, Journal of the Operational Research Society, vol 62, pp 1067–1074, 2011 [FON 94] FONS J., “Using default rates to model the term structure of credit risk”, Financial Analysts Journal, vol 50, pp 25–32, 1994 [GAP 12] GAPKO P., ŠMÍD M., “Dynamic multi-factor credit risk model with fattailed factors”, Finance a úvěr-Czech Journal of Economics and Finance, vol 62, pp 125–140, 2012 [GAR 09] GARCIA J., GOOSSENS S., The Art of Credit Derivatives: Demystifying the Black Swan, John Wiley & Sons, London, 2009 [GES 77] GESKE R., “The valuation of corporate liabilities as compound options”, Journal of Financial and Quantitative Analysis, vol 12, pp 541–552, 1977 288 Semi-Markov Migration Models for Credit Risk [GIE 03] GIESECKE K., GOLDBERG L., “Forecasting default in the face of uncertainty”, Working Paper, Cornell University, 2003 [GIE 04] GIESECKE K., “Credit risk modeling and valuation: an introduction”, Working Paper, Cornell University, 2004 [GIE 06] GIESECKE K., “Default and Information”, Journal of Economic Dynamics and Control, vol 30, pp 2281–2303, 2006 [GRE 12] GREGORY J., Counterparty Credit Risk and Credit Value Adjustment: A Continuing Challenge for Global Financial Markets, 2nd ed., John Wiley & Sons, London, 2012 [GRE 14] GREGORY J., Central Counterparties: Mandatory Central Clearing and Initial Margin Requirements for OTC Derivatives, John Wiley & Sons, London, 2014 [GRE 15a] GREEN A., XVA: Credit, Funding and Capital Valuation Adjustments, John Wiley & Sons, London, 2015 [GRE 15b] GREGORY J., The xVA Challenge: Counterparty Credit Risk, Funding, Collateral, and Capital, 3rd ed., John Wiley & Sons, London, 2015 [GUR 11] GURGUR C.Z., “Dynamic cash management of warranty reserves”, The Engineering Economist, vol 56, pp 1–27, 2011 [HAM 04] HAMILTON D.T., CANTOR R., “Rating transitions and defaults conditional on watch list, outlook and rating history”, Moody’s Investor Service Special Comment, 2004 [HIB 10] HIBBELN M., Risk Management in Credit Portfolios, Springer, Berlin, 2010 [HOE 72] HOEM J.M., “Inhomogeneous semi-Markov processes, select actuarial tables, and duration-dependence in demography”, in GREVILLE T.N.E (ed.), Population, Dynamics, Academic Press, London, 1972 [HOW 71a] HOWARD R., Dynamic Probabilistic Systems, vol 1, John Wiley & Sons, New York, 1971 [HOW 71b] HOWARD R., Dynamic Probabilistic Systems, vol 2, John Wiley & Sons, New York, 1971 [HU 02] HU Y., KIESEL R., PERRAUDIN W., “The estimation of transition matrices for sovereign credit ratings”, Journal of Banking and Finance, vol 26, pp 1383– 1406, 2002 [HUA 03] HUANG J., HUANG M., “How much of the corporate-treasury yield spread is due to credit risk?”, Working Paper, 2003 Bibliography 289 [HUB 99] HUBNER G., “Comment on swap pricing with two-sided default risk in a rating-based model”, European Finance Review, vol 3, pp 269–272, 1999 [HUG 99] HUGE B., LANDO D., “Swap pricing with two-sided default risk in a rating-based model”, European Finance Review, vol 3, pp 239–268, 1999 [HUI 03] HUI C.H., LO C.F., TSANG S.W., “Pricing corporate bonds with dynamic default barriers”, Journal of Risk, vol 5, pp 17–39, 2003 [HUL 00] HULL J.C., White A., “Valuing credit default swap i: no counterparty default risk”, Journal of Derivatives, vol 8, pp 29–41, 2000 [HUL 03] HULL J.C., Options, Futures and Other Derivative Securities, 5th ed., Prentice-Hall, New York, 2003 [IOS 72] IOSIFESCU-MANU A., “Non-homogeneous semi-Markov processes”, Studiisi Cercetuari Matematice, vol 24, pp 529–533, 1972 [ISA 76] ISAACSON D.L., MADSEN R.W., “Markov chains theory and applications”, Series in Probability and Mathematical Statistics, John Wiley & Sons, New York, 1976 [ISR 01] ISRAEL R.B., ROSENTHAL J.S., JASON Z.W., “Finding generators for Markov chains via empirical transitions matrices, with applications to credit ratings”, Mathematical Finance, vol 11, pp 245–265, 2001 [JAF 04] JAFRY Y., SCHUERMANN T., “Measurement, estimation and comparison of credit migration matrices”, Journal of Banking and Finance, vol 28, pp 2603– 2639, 2004 [JAG 11] JAGRIC T., JAGRIC V., KRACUN D., “Does non-linearity matter in retail credit risk modeling?”, Finance a úvěr-Czech Journal of Economics and Finance, vol 61, pp 384–402, 2011 [JAN 66] JANSSEN J., “Application des processus semi-Markoviens un problème d’invalidité”, Bulletin de l’Association Royale des Actuaries Belges, vol 63, pp 35–52, 1966 [JAN 84] JANSSEN J., DE DOMINICIS R., “An algorithmic approach to nonhomogeneous semi-Markov processes”, Insurance: Mathematics and Economics, vol 3, pp 157–165, 1984 [JAN 97] JANSSEN J., MANCA R., “A realistic non-homogeneous stochastic pension funds model on scenario basis”, Scandinavian Actuarial Journal, pp 113–137, 1997 [JAN 98] JANSSEN J., MANCA R., “Non-homogeneous stochastic pension fund model: an algorithmic approach”, ICA Transactions, vol V, 1998 290 Semi-Markov Migration Models for Credit Risk [JAN 01] JANSSEN J., MANCA R., “Numerical solution of non-homogeneous semiMarkov processes in transient case”, Methodology and Computing in Applied Probability, vol 3, pp 271–293, 2001 [JAN 06] JANSSEN J., MANCA R., Applied semi-Markov Processes, Springer, New York, 2006 [JAN 07] JANSSEN J., MANCA R., Semi-Markov Risk Models for Finance, Insurance and Reliability, Springer, New York, 2007 [JAR 92] JARROW R., TURNBULL S., “Credit risk: drawing the analogy”, Risk Magazine, vol 5, no 9, pp 63–70, 1992 [JAR 95] JARROW R., TURNBULL S., “Pricing derivatives on financial securities subject to credit risk”, Journal of Finance, vol 50, pp 53–85, 1995 [JAR 97] JARROW A.J., LANDO D., TURNBULL, S.M., “A Markov model for the term structure of credit risk spreads”, The Review of Financial Studies, vol 10, pp 481–523, 1997 [JAR 01] JARROW R., YU F., “Counterparty risk and the pricing of defaultable securities”, Journal of Finance, vol 56, pp 1765–1799, 2001 [JAR 03] JARROW R., VAN DEVENTER D., WANG X., “A robust test of Merton’s structural model for credit risk”, Journal of Risk, vol 6, no 1, pp 39–58, 2003 [JAR 04] JARROW R., PROTTER P., “Structural versus reduced form models: a new information based perspective”, Journal of Investment Management, vol 2, pp 1–10, 2004 [JON 84] JONES E., MASON S., ROSENFELD E., “Contingent claims analysis of corporate capital structures: an empirical investigation”, Journal of Finance, vol 39, pp 611–627, 1984 [KAV 01] KAVVATHAS D., “Estimating credit rating transition probabilities for corporate bonds”, Working Paper, University of Chicago, 2001 [KNU 69] KNUTH D.E., The Art of Computer Programming, Addison Wesley, New York, 1969 [KOR 95] KOROLYUK V., SWISHCHUK A., Semi-Markov Random Evolutions, Kluwer Academic Publisher, New York, 1995 [KUS 99] KUSUOKA S., “A remark on default risk models”, Advances in Mathematical Economics, vol 1, pp 69–82, 1999 [LAN 98] LANDO D., “On cox processes and credit risky securities”, The Review of Derivatives Research, vol 2, pp 99–120, 1998 Bibliography 291 [LAN 02] LANDO D., SKODEBERG T.M., “Analyzing rating transitions and rating drift with continuous observations”, Journal of Banking and Finance, vol 26, pp 423–444, 2002 [LAN 04] LANDO D., Credit Risk Modeling: Theory and Applications, Princeton, Scottsdale, 2004 [LAY 14] LAYCOCK M., Risk Management at the Top: A Guide to Risk and Its Governance in Financial Institutions, John Wiley & Sons, London, 2014 [LEL 96] LELAND H.E., TOFT K.B., “Optimal capital structure, endogenous bankruptcy and the term structure of credit spreads”, Journal of Finance, vol 51, pp 987–1019, 1996 [LEV 54] LEVY P., “Processus semi-Markoviens”, Proceedings of International Congress of Mathematics, Amsterdam, 1954 [LEV 02] LEVICH R.M., MAJNONI G., REINHART C., Ratings, Rating Agencies and the Global Financial System, Kluwer, Boston, 2002 [LIA 12] LIANG G., JIANG L., “A modified structural model for credit risk”, IMA Journal of Management Mathematics, vol 23, pp 147–170, 2012 [LIM 01] LIMNIOS N., OPRIŞAN G., Semi-Markov Processes and Reliability Modeling, Birkhauser, Boston, 2001 [LÖE 10] LÖEFFLER G., POSCH P.N., Credit Risk Modeling Using Excel and VBA, 2nd ed John Wiley & Sons, London, 2010 [LON 95] LONGSTAFF F.A., SCHWARTZ E.S., “A simple approach to valuing risky fixed and floating rate debt”, Journal of Finance, vol 50, pp 789–819, 1995 [LUC 06] LUCAS A., MONTEIRO A., SMIRNOV G.V., Nonparametric Estimation for Non-Homogeneous Semi-Markov Processes: An Application to Credit Risk, Working Paper, Vrije Universiteit, 2006 [LÜT 09] LÜTKEBOHMERT E., Concentration Risk in Credit Portfolios, Springer, Berlin, 2009 [LYD 00] LYDEN S., SARANITI D., An Empirical Examination of the Classical Theory of Corporate Security Valuation, Barclays Global Investors, San Francisco, 2000 [MAD 73] MADSEN R.W., ISAACSON D.L., “Strongly ergodig behaviour for nonstationary Markov processes”, Annals of Probability, vol 1, pp 329–335, 1973 [MAD 98] MADAN D., UNAL H., “Pricing the risks of default”, Review of Derivatives Research, vol 2, pp 121–160, 1998 292 Semi-Markov Migration Models for Credit Risk [MAR 10] MARTENS D., VAN GESTEL T., DE BACKER M et al., “Credit rating prediction using ant colony optimization, Journal of the Operational Research Society, vol 61, pp 561–573, 2010 [MER 74] MERTON C.R., “On the pricing of corporate debt: the risk structure of interest rates”, Journal of Finance, vol 29, pp 449–470, 1974 [MIL 02] MILLOSSOVICH P., “An extension of the Jarrow-Lando-Turnbull model to random recovery rate”, Working Paper, University of Trieste, 2002 [MIN 70] MINE H., OSAKI S., Markovian Decision Processes, Elsevier, Amsterdam, 1970 [MOO 11] MOON T.H., KIM Y., SOHN S.Y., “Technology credit rating system for funding SMEs”, Journal of the Operational Research Society, vol 62, pp 608– 615, 2011 [MOO 12] MOODY’S, Default and Recovery rates of Corporate Bond Issuers, pp 1920–2010, Moody’s New York, 2012 [NIC 00] NICKELL P., PERRAUDIN W., VAROTTO S., “Stability of rating transitions”, Journal of Banking and Finance, vol 24, pp 203–227, 2000 [NIE 93] NIELSEN L.T., SAA-REQUEJO J., SANTA-CLARA P., “Default risk and interest rate risk: the term structure of default spreads”, Working Paper, INSEAD, 1993 [OGD 87] OGDEN J., “Determinants of the ratings and yields on corporate bonds: tests of the contingent claims model”, The Journal of Financial Research, vol 10, pp 329–339, 1987 [OSA 85] OSAKI S., Stochastic System Reliability Modeling, World Scientific Publishing Co., Singapore, 1985 [PAP 07] PAPADOPOULOU A.A., TSAKLIDIS G., “Some reward paths in semi-Markov models with stochastic selection of the transition probabilities”, Methodology and Computing in Applied Probability, vol 9, pp 399–411, 2007 [PAP 12] PAPADOPOULOU A.A., TSAKLIDIS G., MCCLEAN S et al., “On the moments and the distribution of the cost of a semi-Markov model for healthcare systems”, Methodology and Computing in Applied Probability, vol 14, pp 717– 737, 2012 [PAP 13] PAPADOPOULOU A.A., “Some results on modeling biological sequences and web navigation with a semi-Markov chain”, Communications in Statistics Theory and Methods, vol 42, pp 2153–2171, 2013 [PAZ 70] PAZ A., “Ergodic theorems for infinite probabilistic tables”, Annals of Mathematical Statistics, vol 41, pp 539–550, 1970 Bibliography 293 [PRO 04] PROTTER P., Stochastic Integration and Differential Equations, 2nd ed., Springer, New York, 2004 [PYK 61a] PYKE R., “Markov renewal processes with finitely many states”, Annals of Mathematical Statistics, vol 32, pp 1243–1259, 1961 [PYK 61b] PYKE R., “Markov renewal processes: definitions and preliminary properties”, Annals of Mathematical Statistics, vol 32, pp 1231–1242, 1961 [PYK 07] PYKHTIN M., ZHU S., “A guide to modeling counterparty credit risk”, Global Association of Risk Professionals, vol 37, pp 16–22, 2007 [QUR 94] QURESHI M.A., SANDERS H.W., “Reward model solution methods with impulse and rate rewards: an algorithmic and numerical results”, Performance Evaluation, vol 20, pp 413–436, 1994 [RIE 13] RIESZ F., Les Systèmes d’Équations linéaires une infinité d’inconnues, Gauthiers-Villars, Paris, 2013 [ROG 99] ROGERS L.C.G., “Modeling credit risk”, Working Paper, University of Bath, 1999 [RON 15] RONCORONI A., FUSAI G., CUMMINS M., Handbook of Multi-Commodity Markets and Products: Structuring, Trading and Risk Management, John Wiley & Sons, London, 2015 [ROS 13] ROSCH D., SCHEULE H., Credit Securitisations and Derivatives: Challenges for the Global Markets, John Wiley & Sons, London, 2013 [ROS 14] ROSSI C., A Risk Professional's Survival Guide: Applied Best Practices in Risk Management, John Wiley & Sons, London, 2014 [RUT 13] RUTTIENS A., Mathematics of the Financial Markets: Financial Instruments and Derivatives Modeling, Valuation and Risk Issues, John Wiley & Sons, London, 2013 [SCH 04a] SCHAEFER S.M., STREBULAEV Y.A., “Structural models of credit risk are useful: evidence from hedge ratios on corporate bonds”, Working Paper, 2004 [SCH 04b] SCHIRM A., Credit Risk Securitisation, Springer, Berlin, 2004 [SCH 04c] SCHMID B., Credit Risk Pricing Models, Springer, Berlin, 2004 [SCH 09] SCHOUTENS W., CARIBONI J., Levy Processes in Credit Risk, John Wiley & Sons, London, 2009 [SCH 11] SCHLÄFER T., Recovery Risk in Credit Default Swap Premia, Springer, Berlin, 2011 294 Semi-Markov Migration Models for Credit Risk [SEN 81] SENETA E., Non-Negative Matrices and Markov Chains, Springer, New York, 1981 [SHI 93] SHIMKO D., TEJIMA H., VAN DEVENTER D., “The pricing of risky debt when interest rates are stochastic”, Journal of Fixed Income, vol 3, pp 58–66, 1993 [SID 17] SIDDIQI N., Intelligent Credit Scoring: Building and Implementing Better Credit Risk Scorecards, 2nd ed., John Wiley & Sons, London, 2017 [SIL 04] SILVESTROV D.S., STENBERG F., “A pricing process with stochastic volatility controlled by a semi-Markov process”, Communications in Statistics, Theory and Method, vol 33, pp 591–608, 2004 [SIL 08] SILVESTROV D., MANCA R., SILVESTROVA E., “Stochastically ordered models for credit rating dynamics”, Journal of Numerical and Applied Mathematics, vol 96, pp 206–215, 2008 [SIM 15] SIMOZAR S., The Advanced Fixed Income and Derivatives Management Guide, John Wiley & Sons, London, 2015 [SKO 15] SKOGLUND J., Chen W., Financial Risk Management: Applications in Market, Credit, Asset and Liability Management and Firmwide Risk, John Wiley & Sons, London, 2015 [SMI 55] SMITH W.L., “Regenerative stochastic processes”, Proceedings of Royal Society of London Series A, vol 232, pp 6–31, 1955 [SOL 98] SOLTANI A.R., KHORSHIDIAN K., “Reward processes for semi-Markov processes: asymptotic behavior”, Journal of Applied Probability, vol 35, pp 833–842, 1998 [SON 90] SONIN I., “The asymptotic behaviour of a general finite nonhomogeneous Markov chain (the decomposition-separation theorem)”, Statistics, Probability and Game Theory, vol 30, pp 337–346, 1990 [STA 01a] STANDARD & POOR’S, Rating Performance 2000 Default, Transition, Recovery and Spreads, McGraw Hill, New York, 2001 [STA 01b] STANDARD & POOR’S CREDIT REVIEW, Corporate Default, Rating Transition Study Updated, McGraw Hill, New York, 2001 [STA 08] STANDARD & POOR’S, Annual xxxx Global Corporate Default Study and Rating Transitions, McGraw Hill, 2008 Bibliography 295 [STE 06] STENBERG F., MANCA R., SILVESTROV D., “Semi-Markov reward models for disability insurance”, Theory of Stochastic Processes, vol 12, pp 239–254, 2006 [STE 07] STENBERG F., MANCA R., SILVESTROV D., “An algorithmic approach to discrete time non-homogeneous backward semi-Markov reward processes with an application to disability insurance”, Methodology and Computing in Applied Probability, vol 9, pp 497–519, 2007 [SVI 95] SVISHCHUK A.V., “Hedging of options under mean-square criterion and semi-Markov volatility”, Ukrainian Mathematical Journal, vol 7, pp 1119– 1127, 1995 [TAK 54] TAKÁCS L., “Some investigations concerning recurrent stochastic processes of a certain type”, Magyar Tudomanyok Akadémia Matematlkal Kutato Interetenek Koezlemenyel, vol 3, pp 115–128, 1954 [TAU 99] TAUREN M., “A model of corporate bond prices with dynamic capital structure”, Working Paper, Indiana University, 1999 [TEW 73] TEWARSON R.P., Sparse Matrices, Academic Press, London, 1973 [TRU 09] TRUECK S., RACHEV S.T., Rating Based modeling of credit risk, Academic Press, London, 2009 [USS 75] US SECURITIES AND EXCHANGE COMMISSION, “Adoption of amendments to rule 15c3-1 and adoption of alternative net capital requirement for certain brokers and dealers”, Exchange Act Release No 11497, 1975 [VAS 92] VASSILIOU P.-C.G., PAPADOPOULOU A.A., “Non-homogeneous semiMarkov systems + maintainability of the state sizes”, Journal of Applied Probability, vol 29, pp 519–534, 1992 [VAS 06] VASILEIOU A., VASSILIOU P.C.G., “An inhomogeneous semi-Markov model for the term structure of credit risk spreads”, Advances in Applied Probability, vol 38, pp 171–198, 2006 [VAS 13a] VASSILIOU P.-C.G., “Fuzzy semi-Markov migration process in credit risk”, Fuzzy Sets and Systems, vol 223, pp 39–58, 2013 [VAS 13b] VASSILIOU P.-C.G., VASILEIOU A., “Asymptotic behaviour of the survival probabilities in an inhomogeneous Semi-Markov model for the migration process in credit risk”, Linear Algebra and its Application, vol 438, pp 2880–2903, 2013 [WAJ 92] WAJDA W., “Uniformly strong ergodicity for non-homogeneous semiMarkov processes”, Demonstratio Mathematica, vol 25, pp 755–764, 1992 296 Semi-Markov Migration Models for Credit Risk [WIT 17] WITZANY J., Credit Risk Management, Springer, 2017 [WU 07] WU Y., FYFE C., “The on-line cross entropy method for unsupervised data exploration”, Wseas Transactions on Mathematics, vol 6, pp.865–877, 2007 [YAC 66] YACKEL J., “Limit theorems for semi-Markov processes”, Transactions of the American Mathematical Society, vol 123, pp 402–424, 1966 [YU 08] YU L., WANG S., LAI K.K et al., Bio-Inspired Credit Risk Analysis, Springer, Berlin, 2008 [ZHO 01] ZHOU C., “The term structure of credit spreads with jump risk”, Journal of Banking & Finance, vol 25, pp 2015–2040, 2001 Index A, B, C algorithm, 166, 170, 171 asymptotic behavior, 137, 138, 158 backward and forward processes, 51– 53, 57 bivariate semi-Markov chain, 207 counterparty credit risk, 205, 206, 208, 224, 228 credit rating, 3, 4, 10, 167 risk, 267, 269 spread, 205, 249–252, 261 D, F, G, H default probability, 2, 5, 83, 84, 121 swap, 183, 200, 201 finance, 80, 134, 135, 145, 161 graph, 138, 150 homogeneous, 1, 7, 8, 10, 12, 16, 21, 23, 28, 33–37 M, N migration models, 267–278 mono-unireducibility, 158, 159 Monte Carlo methods, 267, 268 non-homogeneity, 162, 183 non-homogeneous, 1, 10, 28, 33, 36–41 P, R point wise availability function, 22, 37 rating, 1–5, 8–10, 13, 15, 16, 24, 25, 29, 34, 35, 38, 39, 41, 46, 47, 49 reliability, 63-76, 85, 88–90, 96, 105, 113, 119, 121, 124–130, 132–134, 162, 183, 186, 198 function, 22, 24, 37 reward, 167–169, 171, 172 process, 205, 207, 208, 235, 236, 239, 246, 249, 254, 256, 257 risk, 79, 126, 128, 134–137, 148, 154, 160 S semi-Markov models, 88, 124, 125, 131 processes, 51–55, 59, 184, 190, 198, 267–269 sojourn times, 19, 20, 36 spread, 3, 5–7, 25 stochastic processes, 160 recovery rate, 183, 184, 188, 194, 202 Semi-Markov Migration Models for Credit Risk, First Edition Guglielmo D’Amico, Giuseppe Di Biase, Jacques Janssen and Raimondo Manca © ISTE Ltd 2017 Published by ISTE Ltd and John Wiley & Sons, Inc Other titles from in Mathematics and Statistics 2017 CELANT Giorgio, BRONIATOWSKI Michel Interpolation and Extrapolation Optimal Designs 2: Finite Dimensional General Models HARLAMOV Boris Stochastic Analysis of Risk and Management (Stochastic Models in Survival Analysis and Reliability Set – Volume 2) 2016 CELANT Giorgio, BRONIATOWSKI Michel Interpolation and Extrapolation Optimal Designs CHIASSERINI Carla Fabiana, GRIBAUDO Marco, MANINI Daniele Analytical Modeling of Wireless Communication Systems (Stochastic Models in Computer Science and Telecommunication Networks Set – Volume 1) GOUDON Thierry Mathematics for Modeling and Scientific Computing KAHLE Waltraud, MERCIER Sophie, PAROISSIN Christian Degradation Processes in Reliability (Mathematial Models and Methods in Reliability Set – Volume 3) KERN Michel Numerical Methods for Inverse Problems RYKOV Vladimir Reliability of Engineering Systems and Technological Risks (Stochastic Models in Survival Analysis and Reliability Set – Volume 1) 2015 DE SAPORTA Benoợte, DUFOUR Franỗois, ZHANG Huilong Numerical Methods for Simulation and Optimization of Piecewise Deterministic Markov Processes DEVOLDER Pierre, JANSSEN Jacques, MANCA Raimondo Basic Stochastic Processes LE GAT Yves Recurrent Event Modeling Based on the Yule Process (Mathematical Models and Methods in Reliability Set – Volume 2) 2014 COOKE Roger M., NIEBOER Daan, MISIEWICZ Jolanta Fat-tailed Distributions: Data, Diagnostics and Dependence (Mathematical Models and Methods in Reliability Set – Volume 1) MACKEVIČIUS Vigirdas Integral and Measure: From Rather Simple to Rather Complex PASCHOS Vangelis Th Combinatorial Optimization – 3-volume series – 2nd edition Concepts of Combinatorial Optimization / Concepts and Fundamentals – volume Paradigms of Combinatorial Optimization – volume Applications of Combinatorial Optimization – volume 2013 COUALLIER Vincent, GERVILLE-RÉACHE Léo, HUBER Catherine, LIMNIOS Nikolaos, MESBAH Mounir Statistical Models and Methods for Reliability and Survival Analysis JANSSEN Jacques, MANCA Oronzio, MANCA Raimondo Applied Diffusion Processes from Engineering to Finance SERICOLA Bruno Markov Chains: Theory, Algorithms and Applications 2012 BOSQ Denis Mathematical Statistics and Stochastic Processes CHRISTENSEN Karl Bang, KREINER Svend, MESBAH Mounir Rasch Models in Health DEVOLDER Pierre, JANSSEN Jacques, MANCA Raimondo Stochastic Methods for Pension Funds 2011 MACKEVIČIUS Vigirdas Introduction to Stochastic Analysis: Integrals and Differential Equations MAHJOUB Ridha Recent Progress in Combinatorial Optimization – ISCO2010 RAYNAUD Hervé, ARROW Kenneth Managerial Logic 2010 BAGDONAVIČIUS Vilijandas, KRUOPIS Julius, NIKULIN Mikhail Nonparametric Tests for Censored Data BAGDONAVIČIUS Vilijandas, KRUOPIS Julius, NIKULIN Mikhail Nonparametric Tests for Complete Data IOSIFESCU Marius et al Introduction to Stochastic Models VASSILIOU PCG Discrete-time Asset Pricing Models in Applied Stochastic Finance 2008 ANISIMOV Vladimir Switching Processes in Queuing Models FICHE Georges, HÉBUTERNE Gérard Mathematics for Engineers HUBER Catherine, LIMNIOS Nikolaos et al Mathematical Methods in Survival Analysis, Reliability and Quality of Life JANSSEN Jacques, MANCA Raimondo, VOLPE Ernesto Mathematical Finance 2007 HARLAMOV Boris Continuous Semi-Markov Processes 2006 CLERC Maurice Particle Swarm Optimization ... the applications of SMPs to the migration credit risk x Semi- Markov Migration Models for Credit Risk The study of credit risk began with so-called structural form models (SFM) Merton [MER 74] proposed... as it is xiv Semi- Markov Migration Models for Credit Risk well known, in the field of reliability For this reason, it is quite natural to construct semi- Markov credit risk migration models This... (0.77% – 0.34%)/(1 – 0.34%) = 0.43% 8 Semi- Markov Migration Models for Credit Risk 1.1.5 Migration models 1) Credit risk and reliability problems Homogeneous semi- Markov processes (HSMPs) were defined