Semi-Markov migration models for credit risk

309 Pages·2017·7.005 MB·English

by  Biase, Giuseppe Di, D'Amico, Guglielmo, Janssen, Jacques, Manca, Raimondo

Semi-Markov Migration Models for Credit Risk
Stochastic Models for Insurance Set
coordinated by
Jacques Janssen
Volume 1
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 John Wiley & Sons, Inc.
27-37 St George’s Road 111 River Street
London SW19 4EU Hoboken, NJ 07030
UK 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 1. Semi-Markov Processes
Migration Credit Risk Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Rating and migration problems . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1. Ratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2. Migration problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3. Impact of rating on spreads for zero bonds . . . . . . . . . . . . . . . 5
1.1.4. Homogeneous Markov chain model . . . . . . . . . . . . . . . . . . . 7
1.1.5. Migration models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2. Homogeneous semi-Markov processes . . . . . . . . . . . . . . . . . . . 10
1.2.1. Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.2. The Z SMP and the evolution equation system . . . . . . . . . . . . 14
1.2.3. Special cases of SMP . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.4. Sojourn times and their distributions . . . . . . . . . . . . . . . . . . 19
1.3. Homogeneous semi-Markov reliability model . . . . . . . . . . . . . . . 21
1.4. Homogeneous semi-Markov migration model . . . . . . . . . . . . . . . 23
1.4.1. Equivalence with the reliability problem . . . . . . . . . . . . . . . . 23
1.4.2. Transient results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4.3. Asymptotic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4.4. Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5. Discrete time non-homogeneous case . . . . . . . . . . . . . . . . . . . . 33
1.5.1. NHSMPs and evolution equations . . . . . . . . . . . . . . . . . . . . 33
1.5.2. The Z NHSMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.5.3. Sojourn times and their distributions . . . . . . . . . . . . . . . . . . 36
1.5.4. Non-homogeneous semi-Markov reliability model . . . . . . . . . . 37
vi Semi-Markov Migration Models for Credit Risk
1.5.5. The non-homogeneous semi-Markov migration model . . . . . . . 38
1.5.6. A non-homogeneous example . . . . . . . . . . . . . . . . . . . . . . 39
Chapter 2. Recurrence Time HSMP and NHSMP:
Credit Risk Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2. Recurrence times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3. Transition probabilities of homogeneous SMP
and non-homogeneous SMP with recurrence times . . . . . . . . . . . . . . . 53
2.3.1. Transition probabilities with initial backward . . . . . . . . . . . . . 53
2.3.2. Transition probabilities with initial forward . . . . . . . . . . . . . . 55
2.3.3. Transition probabilities with final backward and forward . . . . . . 57
2.3.4. Transition probabilities with initial and final backward . . . . . . . 58
2.3.5. Transition probabilities with initial and final forward . . . . . . . . 60
2.3.6. Transition probabilities with initial and final
backward and forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.4. Reliability indicators of HSMP and NHSMP with
recurrence times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.4.1. Reliability indicators with initial backward . . . . . . . . . . . . . . 63
2.4.2. Reliability indicators with initial forward . . . . . . . . . . . . . . . . 66
2.4.3. Reliability indicators with initial and final backward . . . . . . . . . 70
2.4.4. Reliability indicators with initial and final
backward and forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Chapter 3. Recurrence Time Credit Risk Applications . . . . . . . . . . 79
3.1. S&P’s basic rating classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.1.1. Homogeneous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.1.2. Non-homogeneous case . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.2. S&P’s basic rating classes and NR state . . . . . . . . . . . . . . . . . . . 90
3.2.1. Homogeneous case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.2.2. Non-homogeneous case . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.3. S&P’s downward rating classes . . . . . . . . . . . . . . . . . . . . . . . . 120
3.3.1. An application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.4. S&P’s basic rating classes & NR1 and NR2 states . . . . . . . . . . . . . 127
3.5. Cost of capital implications . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Chapter 4. Mono-Unireducible Markov and
Semi-Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.2. Graphs and matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.3. Single-unireducible non-homogeneous Markov chains . . . . . . . . . . 145
Contents vii
4.4. Single-unireducible semi-Markov chains . . . . . . . . . . . . . . . . . . 152
4.5. Mono-unireducible non-homogeneous backward
semi-Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
4.6. Real data credit risk application . . . . . . . . . . . . . . . . . . . . . . . . 160
Chapter 5. Non-Homogeneous Semi-Markov Reward
Processes and Credit Spread Computation . . . . . . . . . . . . . . . . . 165
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.2. The reward introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.3. The DTNHSMRWP spread rating model . . . . . . . . . . . . . . . . . . 168
5.4. The algorithm description . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.5. A numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.5.1. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.5.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Chapter 6. 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 . . . . 184
6.2. The price and the value of the swap: the random recovery rate case . . 188
6.3. The determination of the n-period random recovery rate . . . . . . . . . 196
6.4. A numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Chapter 7. Bivariate Semi-Markov Processes
and Related Reward Processes for Counterparty
Credit Risk and Credit Spreads . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.2. Multivariate semi-Markov chains . . . . . . . . . . . . . . . . . . . . . . . 208
7.3. The two-component reliability model . . . . . . . . . . . . . . . . . . . . 220
7.4. Counterparty credit risk in a CDS contract . . . . . . . . . . . . . . . . . 224
7.4.1. Pricing a risky CDS and CVA evaluation . . . . . . . . . . . . . . . 227
7.5. A numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.6. Bivariate semi-Markov reward chains . . . . . . . . . . . . . . . . . . . . 233
7.7. The estimation methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 247
7.8. Credit spreads evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
7.9. Numerical experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Chapter 8. Semi-Markov Credit Risk Simulation Models . . . . . . . . 267
8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
8.2. Monte Carlo semi-Markov credit risk model
for the Basel II Capital at Risk problem . . . . . . . . . . . . . . . . . . . . . . 267
viii Semi-Markov Migration Models for Credit Risk
8.2.1. The homogeneous MCSM evolution
with D as absorbing state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
8.3. Results of the MCSMP credit model in a
homogeneous environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
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 1 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