Marine Habart-Corlosquet & Jacques Janssen 
VaR Methodology for Non-Gaussian Finance 

With the impact of the recent financial crises, more attention must be given to new models in finance rejecting ‘Black-Scholes-Samuelson’ assumptions leading to what is called non-Gaussian finance. With the growing importance of Solvency II, Basel II and III regulatory rules for insurance companies and banks, value at risk (Va R) – one of the most popular risk indicator techniques plays a fundamental role in defining appropriate levels of equities. The aim of this book is to show how new Va R techniques can be built more appropriately for a crisis situation.
Va R methodology for non-Gaussian finance looks at the importance of Va R in standard international rules for banks and insurance companies; gives the first non-Gaussian extensions of Va R and applies several basic statistical theories to extend classical results of Va R techniques such as the NP approximation, the Cornish-Fisher approximation, extreme and a Pareto distribution. Several non-Gaussian models using Copula methodology, Lévy processes along with particular attention to models with jumps such as the Merton model are presented; as are the consideration of time homogeneous and non-homogeneous Markov and semi-Markov processes and for each of these models.

Contents

1. Use of Value-at-Risk (Va R) Techniques for Solvency II, Basel II and III.
2. Classical Value-at-Risk (Va R) Methods.
3. Va R Extensions from Gaussian Finance to Non-Gaussian Finance.
4. New Va R Methods of Non-Gaussian Finance.
5. Non-Gaussian Finance: Semi-Markov Models.

Table of Content

INTRODUCTION ix

CHAPTER 1. USE OF VALUE-AT-RISK (VAR) TECHNIQUES FOR SOLVENCY
II, BASEL II AND III 1

1.1. Basic notions of Va R 1

1.2. The use of Va R for insurance companies 6

1.3. The use of Va R for banks 13

1.4. Conclusion 16

CHAPTER 2. CLASSICAL VALUE-AT-RISK (VAR) METHODS 17

2.1. Introduction 17

2.2. Risk measures 18

2.3. General form of the Va R 19

2.4. Va R extensions: tail Va R and conditional Va R 25

2.5. Va R of an asset portfolio 28

2.6. A simulation example: the rates of investment of assets
32

CHAPTER 3. VAR EXTENSIONS FROM GAUSSIAN FINANCE TO
NON-GAUSSIAN FINANCE 35

3.1. Motivation 35

3.2. The normal power approximation 37

3.3. Va R computation with extreme values 40

3.4. Va R value for a risk with Pareto distribution 56

3.5. Conclusion 62

CHAPTER 4. NEW VAR METHODS OF NON-GAUSSIAN FINANCE 63

4.1. Lévy processes 63 model with jumps 76

4.2. Copula models and Va R techniques 90

4.3. Va R for insurance 109

CHAPTER 5. NON-GAUSSIAN FINANCE: SEMI-MARKOV MODELS
115

5.1. Introduction 115

5.2. Homogeneous semi-Markov process 116

5.3. Semi-Markov option model 139

5.4. Semi-Markov Va R models 143

5.5. The Semi-Markov Monte Carlo Model in a homogeneous
environment 147

CONCLUSION 159

BIBLIOGRAPHY 161

INDEX 165

About the author

Marine Habart-Corlosquet is a Qualified and Certified Actuary at BNP Paribas Cardif, Paris, France. She is co-director of EURIA (Euro-Institut d’Actuariat, University of West Brittany, Brest, France), and associate researcher at Telecom Bretagne (Brest, France) as well as a board member of the French Institute of Actuaries. She teaches at EURIA, Telecom Bretagne and Ecole Centrale Paris (France). Her main research interests are pandemics, Solvency II internal models and ALM issues for insurance companies.

Jacques Janssen is now Honorary Professor at the Solvay Business School (ULB) in Brussels, Belgium, having previously taught at EURIA (Euro-Institut d’Actuariat, University of West Brittany, Brest, France) and Telecom Bretagne (Brest, France) as well as being a director of Jacan Insurance and Finance Services, a consultancy and training company.

Raimondo Manca is Professor of mathematical methods applied to economics, finance and actuarial science at University of Roma ‘La Sapienza’ in Italy. He is associate editor for the journal Methodology and Computing in Applied Probability. His main research interests are multidimensional linear algebra, computational probability, application of stochastic processes to economics, finance and insurance and simulation models.