Thierry Roncalli

Lecture Notes on RiskManagement & FinancialRegulation

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Structure of the book

1 Introduction

I Risk Management in the Banking Sector

2 Market Risk3 Credit Risk4 Counterparty Credit Risk and Collateral Risk5 Operational Risk6 Liquidity Risk7 Asset/Liability Management Risk

II Risk Management in Other Financial Sectors

8 The Insurance Regulation and Solvency II9 Asset Managers10 Asset Owners11 Market Intermediaries and Infrastructure12 Systemic Risk and Shadow Banking System

III Mathematical and Statistical Tools of Risk Management

13 Model Risk of Derivative Instruments14 Statistical Inference and Model Estimation15 Copulas and Dependence16 Extreme Value Theory17 Monte Carlo Simulation Methods18 Stress Testing and Scenario Analysis19 Credit Scoring Models

20 Conclusion

Appendix

A Technical Appendix

Preface

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Warning

These lectures notes are currently being written and far from com-plete. My objective is first to provide a preliminary draft for eachchapter. In a second time, I will improve the English text of themanuscript and illustrate with more examples and data. Correc-tions and comments are then welcome!

Three decades of financial regulation

Two decades of risk management

One decade of financial instability

About these lecture notesThese lecture notes are divided into three parts. After an introductory

chapter presenting the main concepts of risk management and an overviewof the financial regulation, the first part is dedicated to the risk managementin the banking sector and consists of six chapters: market risk, credit risk,counterparty credit risk and collateral risk, operational risk, liquidity risk andasset/liability management risk. We begin with the market risk, because itpermits to introduce naturally the concepts of risk factor and risk measureand to define the risk allocation approach. For each chapter, we present thecorresponding regulation framework and the risk management tools. The sec-ond part is dedicated to non-banking financial sectors with four chapters ded-icated to insurance, asset management, investors and market infrastructure

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(including central counterparties). This second part ends with a fifth chapteron systemic risk and shadow banking system. The third part of these lecturenotes develops the mathematical and statistical tools used in risk management.It contains seven chapters: risk model and derivatives hedging, statistical in-ference and model estimation, copula functions, extreme value theory, MonteCarlo simulation, stress testing methods and scoring models. Each chapterof these lectures notes are extensively illustrated by numerical examples andcontains also tutorial exercises. Finally, a technical appendix completes thelecture notes and contains some important elements on numerical analysis.

The writing of these lectures notes started in April 2015 and is the resultof fifteen years of academic courses. When I began to teach risk management,a large part of my course was dedicated to statistical tools. Over the years,financial regulation became however increasingly important. This is why riskmanagement is now mainly driven by the regulation, not by the progress withthe mathematical models. The preparation of this book has benefited fromthe existing materials of my French book called La Gestion des Risques Fi-nanciers. Nevertheless, the structure of the two books is different, becausemy previous book only concerned risk management in the banking sector andbefore Basel III. Three years ago, I decided to extend the course to otherfinancial sectors, especially insurance, asset management and market infras-tructure. It appears that even if the quantitative tools of risk managementare the same across the different financial areas, each sector presents someparticular aspects. The knowledge of the different regulations is especially notan easy task for students. However, it is necessary if one would like to un-derstand what is the role of risk management in financial institutions in thepresent-day world. Moreover, reducing the practice of risk management to theassimilation of the regulation rules is not sufficient. The sound understandingof the financial products and the mathematical models are essential to knowwhere the risks are. This is why some parts of this book can be particularlydifficult because risk management is today complex in finance. A companionbook is available in order to facilitate learning and knowledge assimilation atthe following internet web page:

http://www.thierry-roncalli.com/RiskManagement.html

It contains additional information like some detailed calculation and the so-lution of the tutorial exercises.

Contents

Preface i

List of Figures xvii

List of Tables xxiii

List of Symbols and Notations xxvii

1 Introduction 11.1 The need for risk management . . . . . . . . . . . . . . . . . 1

1.1.1 Risk management and the financial system . . . . . . 11.1.2 The development of financial markets . . . . . . . . . 31.1.3 Financial crises and systemic risk . . . . . . . . . . . 9

1.2 Financial regulation . . . . . . . . . . . . . . . . . . . . . . . 131.2.1 Banking regulation . . . . . . . . . . . . . . . . . . . 141.2.2 Insurance regulation . . . . . . . . . . . . . . . . . . . 261.2.3 Market regulation . . . . . . . . . . . . . . . . . . . . 291.2.4 Systemic risk . . . . . . . . . . . . . . . . . . . . . . . 31

1.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.3.1 List of supervisory authorities . . . . . . . . . . . . . 331.3.2 Timeline of financial regulation . . . . . . . . . . . . . 35

I Risk Management in the Banking Sector 41

2 Market Risk 432.1 Regulatory framework . . . . . . . . . . . . . . . . . . . . . . 43

2.1.1 Standardized measurement method . . . . . . . . . . 452.1.1.1 Interest rate risk . . . . . . . . . . . . . . . 452.1.1.2 Equity risk . . . . . . . . . . . . . . . . . . . 482.1.1.3 Foreign exchange risk . . . . . . . . . . . . . 492.1.1.4 Commodity risk . . . . . . . . . . . . . . . . 502.1.1.5 Options market risk . . . . . . . . . . . . . 522.1.1.6 Securitization instruments . . . . . . . . . . 55

2.1.2 Internal model-based approach . . . . . . . . . . . . . 552.1.2.1 Qualitative criteria . . . . . . . . . . . . . . 562.1.2.2 Quantitative criteria . . . . . . . . . . . . . 572.1.2.3 Stress testing . . . . . . . . . . . . . . . . . 59

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2.1.2.4 Specific risk and other risk charges . . . . . 602.1.2.5 Backtesting and the ex-post evaluation of the

internal model . . . . . . . . . . . . . . . . . 612.2 Value-at-risk method . . . . . . . . . . . . . . . . . . . . . . 65

2.2.1 Historical value-at-risk . . . . . . . . . . . . . . . . . 672.2.1.1 The order statistics approach . . . . . . . . 672.2.1.2 The kernel density approach . . . . . . . . . 71

2.2.2 Analytical value-at-risk . . . . . . . . . . . . . . . . . 732.2.2.1 Derivation of the closed-form formula . . . . 732.2.2.2 Gaussian VaR and Linear factor models . . 752.2.2.3 Volatility forecasting . . . . . . . . . . . . . 822.2.2.4 Extension to other probability distributions 86

2.2.3 Monte Carlo value-at-risk . . . . . . . . . . . . . . . . 942.2.4 The case of options and derivatives . . . . . . . . . . 96

2.2.4.1 Identification of risk factors . . . . . . . . . 972.2.4.2 Methods to calculate the value-at-risk . . . 992.2.4.3 Backtesting . . . . . . . . . . . . . . . . . . 1072.2.4.4 Model risk . . . . . . . . . . . . . . . . . . . 108

2.3 Risk allocation . . . . . . . . . . . . . . . . . . . . . . . . . . 1112.3.1 Properties of a risk measure . . . . . . . . . . . . . . 111

2.3.1.1 Coherency and convexity of risk measures . 1112.3.1.2 Euler allocation principle . . . . . . . . . . . 115

2.3.2 Application to non-normal risk measures . . . . . . . 1182.3.2.1 Main results . . . . . . . . . . . . . . . . . . 1182.3.2.2 Calculating risk contributions with historical

and simulated scenarios . . . . . . . . . . . 1202.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

2.4.1 Calculating regulatory capital with the standardizedmeasurement method . . . . . . . . . . . . . . . . . . 127

2.4.2 Covariance matrix . . . . . . . . . . . . . . . . . . . . 1292.4.3 Risk measure . . . . . . . . . . . . . . . . . . . . . . . 1302.4.4 Value-at-risk of a long/short portfolio . . . . . . . . . 1312.4.5 Value-at-risk of an equity portfolio hedged with put

options . . . . . . . . . . . . . . . . . . . . . . . . . . 1322.4.6 Risk management of exotic options . . . . . . . . . . 1332.4.7 P&L approximation with Greek sensitivities . . . . . 1332.4.8 Calculating the non-linear quadratic value-at-risk . . 1342.4.9 Risk decomposition of the expected shortfall . . . . . 1362.4.10 Expected shortfall of an equity portfolio . . . . . . . 1362.4.11 Risk measure of a long/short portfolio . . . . . . . . . 136

3 Credit Risk 1393.1 The market of credit risk . . . . . . . . . . . . . . . . . . . . 139

3.1.1 The loan market . . . . . . . . . . . . . . . . . . . . . 1393.1.2 The bond market . . . . . . . . . . . . . . . . . . . . 141

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3.1.2.1 Statistics of the bond market . . . . . . . . 1423.1.2.2 Pricing of bonds . . . . . . . . . . . . . . . . 145

3.1.3 Securitization and credit derivatives . . . . . . . . . . 1543.1.3.1 Credit securitization . . . . . . . . . . . . . 1543.1.3.2 Credit default swap . . . . . . . . . . . . . . 1583.1.3.3 Basket default swap . . . . . . . . . . . . . . 1703.1.3.4 Collateralized debt obligations . . . . . . . . 175

3.2 Capital requirements . . . . . . . . . . . . . . . . . . . . . . 1793.2.1 The Basel I framework . . . . . . . . . . . . . . . . . 1803.2.2 The Basel II standardized approach . . . . . . . . . . 183

3.2.2.1 Standardized risk weights . . . . . . . . . . 1833.2.2.2 Credit risk mitigation . . . . . . . . . . . . . 186

3.2.3 The Basel II internal ratings-based approach . . . . . 1893.2.3.1 The general framework . . . . . . . . . . . . 1903.2.3.2 The credit risk model of Basel II . . . . . . 1913.2.3.3 The IRB formulas . . . . . . . . . . . . . . . 200

3.2.4 The securitization framework . . . . . . . . . . . . . . 2063.2.5 Basel IV proposals . . . . . . . . . . . . . . . . . . . . 206

3.3 Credit risk modeling . . . . . . . . . . . . . . . . . . . . . . . 2063.3.1 Exposure at default . . . . . . . . . . . . . . . . . . . 2063.3.2 Loss given default . . . . . . . . . . . . . . . . . . . . 206

3.3.2.1 Economic modeling . . . . . . . . . . . . . . 2063.3.2.2 Stochastic modeling . . . . . . . . . . . . . . 206

3.3.3 Probability of default . . . . . . . . . . . . . . . . . . 2063.3.3.1 Survival function . . . . . . . . . . . . . . . 2063.3.3.2 Transition probability matrix . . . . . . . . 2123.3.3.3 Structural models . . . . . . . . . . . . . . . 223

3.3.4 Default correlation . . . . . . . . . . . . . . . . . . . . 2233.3.4.1 Factor models . . . . . . . . . . . . . . . . . 2233.3.4.2 Copula models . . . . . . . . . . . . . . . . . 2233.3.4.3 Estimation methods . . . . . . . . . . . . . 223

3.3.5 Granularity and concentration . . . . . . . . . . . . . 2233.3.5.1 Difference between fine-grained and concen-

trated portfolios . . . . . . . . . . . . . . . . 2233.3.5.2 Granularity adjustment . . . . . . . . . . . . 223

3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2243.4.1 Single and multi-name credit default swaps . . . . . . 2243.4.2 Risk contribution in the Basel II model . . . . . . . . 2253.4.3 Calibration of the piecewise exponential model . . . . 2273.4.4 Modeling loss given default . . . . . . . . . . . . . . . 2283.4.5 Modeling default times with a Markov chain . . . . . 2293.4.6 Calculating the credit value-at-risk with non granular

portfolios . . . . . . . . . . . . . . . . . . . . . . . . . 2303.4.7 Understanding CDO cash flows . . . . . . . . . . . . 2303.4.8 Modeling default correlations . . . . . . . . . . . . . . 230

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3.4.9 Continuous-time modeling of default risk . . . . . . . 231

4 Counterparty Credit Risk and Collateral Risk 2334.1 Counterparty credit risk . . . . . . . . . . . . . . . . . . . . . 233

4.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 2334.1.2 Modeling the exposure at default . . . . . . . . . . . 236

4.1.2.1 An illustrative example . . . . . . . . . . . . 2364.1.2.2 Measuring the counterparty exposure . . . . 2404.1.2.3 Practical implementation for calculating coun-

terparty exposure . . . . . . . . . . . . . . . 2424.1.3 Regulatory capital . . . . . . . . . . . . . . . . . . . . 245

4.1.3.1 Internal model method . . . . . . . . . . . . 2464.1.3.2 Non-internal models methods (Basel II) . . 2474.1.3.3 SA-CCR method (Basel IV) . . . . . . . . . 249

4.1.4 Impact of wrong-way risk . . . . . . . . . . . . . . . . 2544.2 Credit valuation adjustment . . . . . . . . . . . . . . . . . . 254

4.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 2544.2.1.1 Difference between CCR and CVA . . . . . 2544.2.1.2 CVA, DVA and bilateral CVA . . . . . . . . 255

4.2.2 Regulatory capital . . . . . . . . . . . . . . . . . . . . 2584.2.2.1 Advanced method . . . . . . . . . . . . . . . 2584.2.2.2 Standardized method . . . . . . . . . . . . . 2594.2.2.3 Basel IV standardized approach (SA-CVA) . 260

4.2.3 Impact of CVA on the banking industry . . . . . . . . 2604.3 Collateral risk . . . . . . . . . . . . . . . . . . . . . . . . . . 2604.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

4.4.1 Impact of netting agreements in counterparty creditrisk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

4.4.2 Calculation of the effective expected positive exposure 2624.4.3 Calculation of the capital charge for counterparty credit

risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2624.4.4 Illustration of the wrong-way risk . . . . . . . . . . . 2634.4.5 Counterparty exposure of interest rate swap . . . . . 2634.4.6 Derivation of SA-CCR formulas . . . . . . . . . . . . 2634.4.7 Examples of SA-CCR calculation . . . . . . . . . . . 2634.4.8 Calculation of CVA and DVA measures . . . . . . . . 263

5 Operational Risk 2655.1 Definition of operational risk . . . . . . . . . . . . . . . . . . 2655.2 Basel approaches for calculating the regulatory capital . . . 268

5.2.1 The basic indicator approach . . . . . . . . . . . . . . 2685.2.2 The standardized approach . . . . . . . . . . . . . . . 2695.2.3 Advanced measurement approaches . . . . . . . . . . 2715.2.4 Basel IV approach . . . . . . . . . . . . . . . . . . . . 273

5.3 Loss distribution approach . . . . . . . . . . . . . . . . . . . 273

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5.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 2735.3.2 Parametric estimation of F and P . . . . . . . . . . . 276

5.3.2.1 Estimation of the loss severity distribution . 2765.3.2.2 Estimation of the loss frequency distribution 285

5.3.3 Calculating the capital charge . . . . . . . . . . . . . 2915.3.3.1 Monte Carlo approach . . . . . . . . . . . . 2925.3.3.2 Analytical approaches . . . . . . . . . . . . 2955.3.3.3 Aggregation issues . . . . . . . . . . . . . . 302

5.3.4 Incorporating scenario analysis . . . . . . . . . . . . . 3055.3.4.1 Probability distribution of a given scenario . 3055.3.4.2 Calibration of a set of scenarios . . . . . . . 306

5.3.5 Stability issue of the LDA model . . . . . . . . . . . . 3085.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

5.4.1 Estimation of the loss severity distribution . . . . . . 3095.4.2 Estimation of the loss frequency distribution . . . . . 3105.4.3 Using the method of moments in operational risk mod-

els . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3115.4.4 Calculation of the Basel II required capital . . . . . . 3125.4.5 Parametric estimation of the loss severity distribution 3135.4.6 Mixed Poisson process . . . . . . . . . . . . . . . . . 3135.4.7 Loss frequency distribution with data truncation . . . 3145.4.8 Moments of compound distribution . . . . . . . . . . 3145.4.9 Characteristic functions and fast Fourier transform . 3145.4.10 Derivation and implementation of the Panjer recursion 3145.4.11 The Bcker-Klppelberg-Sprittulla approximation for-

mula . . . . . . . . . . . . . . . . . . . . . . . . . . . 3145.4.12 Frequency correlation, severity correlation and loss ag-

gregation . . . . . . . . . . . . . . . . . . . . . . . . . 3145.4.13 Loss aggregation using copula functions . . . . . . . . 3145.4.14 Scenario analysis and stress testing . . . . . . . . . . 314

6 Liquidity Risk 3156.1 Market liquidity . . . . . . . . . . . . . . . . . . . . . . . . . 315

6.1.1 Transaction cost versus volume-based measures . . . 3166.1.1.1 Bid-ask spread . . . . . . . . . . . . . . . . . 3166.1.1.2 Trading volume . . . . . . . . . . . . . . . . 3186.1.1.3 Liquidation ratio . . . . . . . . . . . . . . . 3196.1.1.4 Liquidity ordering . . . . . . . . . . . . . . . 322

6.1.2 Other liquidity measures . . . . . . . . . . . . . . . . 3226.2 Funding liquidity . . . . . . . . . . . . . . . . . . . . . . . . 324

6.2.1 Asset/liability mismatch and maturity transformation 3246.2.2 Open market operations . . . . . . . . . . . . . . . . 324

6.3 Regulation of the liquidity risk . . . . . . . . . . . . . . . . . 3256.3.1 Liquidity coverage ratio . . . . . . . . . . . . . . . . . 325

6.3.1.1 Definition . . . . . . . . . . . . . . . . . . . 325

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6.3.1.2 Monitoring tools . . . . . . . . . . . . . . . 3306.3.2 Net stable funding ratio . . . . . . . . . . . . . . . . . 330

6.3.2.1 Definition . . . . . . . . . . . . . . . . . . . 3316.3.2.2 ASF and RSF factors . . . . . . . . . . . . . 331

6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

7 Asset/Liability Management 3357.1 Risk management of the banking book . . . . . . . . . . . . 3357.2 Interest rate risk . . . . . . . . . . . . . . . . . . . . . . . . . 335

7.2.1 Duration gap analysis . . . . . . . . . . . . . . . . . . 3357.2.2 Net interest margin . . . . . . . . . . . . . . . . . . . 3357.2.3 Funds transfer pricing . . . . . . . . . . . . . . . . . . 3357.2.4 Simulation approach . . . . . . . . . . . . . . . . . . . 335

7.3 Pricing implicit embedded options . . . . . . . . . . . . . . . 3357.3.1 Mortgage prepayment risk . . . . . . . . . . . . . . . 3357.3.2 Non-maturity deposits . . . . . . . . . . . . . . . . . 335

7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

II Risk Management in Other Financial Sectors 337

12 Systemic Risk and Shadow Banking System 33912.1 Defining systemic risk . . . . . . . . . . . . . . . . . . . . . . 340

12.1.1 Systemic risk, systematic risk and idiosyncratic risk . 34012.1.2 Sources of systemic risk . . . . . . . . . . . . . . . . . 342

12.1.2.1 Systematic shocks . . . . . . . . . . . . . . . 34312.1.2.2 Propagation mechanisms . . . . . . . . . . . 345

12.1.3 Supervisory policy responses . . . . . . . . . . . . . . 34712.1.3.1 A new financial regulatory structure . . . . 34812.1.3.2 A myriad of new standards . . . . . . . . . . 350

12.2 Systemic risk measurement . . . . . . . . . . . . . . . . . . . 35112.2.1 The supervisory approach . . . . . . . . . . . . . . . . 352

12.2.1.1 The G-SIB assessment methodology . . . . . 35212.2.1.2 Identification of G-SIIs . . . . . . . . . . . . 35512.2.1.3 Extension to NBNI SIFIs . . . . . . . . . . . 356

12.2.2 The academic approach . . . . . . . . . . . . . . . . . 35612.2.2.1 Marginal expected shortfall . . . . . . . . . 35712.2.2.2 Delta conditional value-at-risk . . . . . . . . 35912.2.2.3 Systemic risk measure . . . . . . . . . . . . 36212.2.2.4 Network measures . . . . . . . . . . . . . . . 367

12.3 Shadow banking system . . . . . . . . . . . . . . . . . . . . . 36912.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 36912.3.2 Measuring the shadow banking . . . . . . . . . . . . . 370

12.3.2.1 The broad measure . . . . . . . . . . . . . . 37012.3.2.2 The narrow measure . . . . . . . . . . . . . 372

12.3.3 Regulatory developments of shadow banking . . . . . 377

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12.3.3.1 Data gaps . . . . . . . . . . . . . . . . . . . 37712.3.3.2 Mitigation of interconnectedness risk . . . . 37712.3.3.3 Money market funds . . . . . . . . . . . . . 37812.3.3.4 Complex shadow banking activities . . . . . 379

12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

III Mathematical and Statistical Tools 381

13 Model Risk of Exotic Derivatives 38313.1 Basics of option pricing . . . . . . . . . . . . . . . . . . . . . 38313.2 Volatility risk . . . . . . . . . . . . . . . . . . . . . . . . . . 38313.3 Other model risk topics . . . . . . . . . . . . . . . . . . . . . 38313.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

13.4.1 Option pricing and martingale measure . . . . . . . . 38313.4.2 The Vasicek model . . . . . . . . . . . . . . . . . . . 38413.4.3 The Black model . . . . . . . . . . . . . . . . . . . . . 38413.4.4 Change of numraire and Girsanovs theorem . . . . . 38513.4.5 The HJM model and the forward probability measure 38713.4.6 Equivalent martingale measure in the Libor market

model . . . . . . . . . . . . . . . . . . . . . . . . . . . 38813.4.7 Equivalent martingale measure in the swap market

model . . . . . . . . . . . . . . . . . . . . . . . . . . . 38913.4.8 Displaced diffusion option pricing . . . . . . . . . . . 38913.4.9 CMS convexity adjustment in the shifted log-normal

model . . . . . . . . . . . . . . . . . . . . . . . . . . . 39013.4.10 Dupire local volatility model . . . . . . . . . . . . . . 39013.4.11 The stochastic normal model . . . . . . . . . . . . . . 39213.4.12 The quadratic Gaussian model . . . . . . . . . . . . . 39313.4.13 Pricing two-asset basket options . . . . . . . . . . . . 394

14 Statistical Inference and Model Estimation 39714.1 Estimation methods . . . . . . . . . . . . . . . . . . . . . . . 39714.2 Time series modeling . . . . . . . . . . . . . . . . . . . . . . 39714.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

14.3.1 Linear regression without a constant . . . . . . . . . . 39714.3.2 Linear regression with linear constraints . . . . . . . . 397

15 Copulas and Dependence 40115.1 Canonical representation of multivariate distributions . . . . 401

15.1.1 Sklars theorem . . . . . . . . . . . . . . . . . . . . . 40115.1.2 Expression of the copula density . . . . . . . . . . . . 40315.1.3 Frchet classes . . . . . . . . . . . . . . . . . . . . . . 404

15.1.3.1 The bivariate case . . . . . . . . . . . . . . . 40415.1.3.2 The multivariate case . . . . . . . . . . . . . 40515.1.3.3 Concordance ordering . . . . . . . . . . . . . 406

15.2 Copula functions and random vectors . . . . . . . . . . . . . 409

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15.2.1 Countermonotonicity, comonotonicity and scale invari-ance property . . . . . . . . . . . . . . . . . . . . . . 409

15.2.2 Dependence measures . . . . . . . . . . . . . . . . . . 41215.2.2.1 Concordance measures . . . . . . . . . . . . 41215.2.2.2 Linear correlation . . . . . . . . . . . . . . . 41615.2.2.3 Tail dependence . . . . . . . . . . . . . . . . 418

15.3 Parametric copula functions . . . . . . . . . . . . . . . . . . 42115.3.1 Archimedean copulas . . . . . . . . . . . . . . . . . . 421

15.3.1.1 Definition . . . . . . . . . . . . . . . . . . . 42115.3.1.2 Properties . . . . . . . . . . . . . . . . . . . 42215.3.1.3 Two-parameter Archimedean copulas . . . . 42315.3.1.4 Extension to the multivariate case . . . . . . 423

15.3.2 Normal copula . . . . . . . . . . . . . . . . . . . . . . 42515.3.3 Students t copula . . . . . . . . . . . . . . . . . . . . 427

15.4 Statistical inference and estimation of copula functions . . . 43115.4.1 The empirical copula . . . . . . . . . . . . . . . . . . 43215.4.2 The method of moments . . . . . . . . . . . . . . . . 43515.4.3 The method of maximum likelihood . . . . . . . . . . 436

15.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44015.5.1 Gumbel logistic copula . . . . . . . . . . . . . . . . . 44015.5.2 Farlie-Gumbel-Morgenstern copula . . . . . . . . . . . 44015.5.3 Survival copula . . . . . . . . . . . . . . . . . . . . . 44115.5.4 Method of moments . . . . . . . . . . . . . . . . . . . 44115.5.5 Correlated loss given default rates . . . . . . . . . . . 44115.5.6 Calculation of correlation bounds . . . . . . . . . . . 44215.5.7 The bivariate Pareto copula . . . . . . . . . . . . . . 443

16 Extreme Value Theory 44516.1 Order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 445

16.1.1 Main Properties . . . . . . . . . . . . . . . . . . . . . 44516.1.2 Extreme order statistics . . . . . . . . . . . . . . . . . 44816.1.3 Inference statistics . . . . . . . . . . . . . . . . . . . . 45116.1.4 Extension to dependent random variables . . . . . . . 454

16.2 Univariate extreme value theory . . . . . . . . . . . . . . . . 45616.2.1 Fisher-Tippet theorem . . . . . . . . . . . . . . . . . 45716.2.2 Maximum domain of attraction . . . . . . . . . . . . 459

16.2.2.1 MDA of the Gumbel distribution . . . . . . 46116.2.2.2 MDA of the Frchet distribution . . . . . . . 46116.2.2.3 MDA of the Weibull distribution . . . . . . 46216.2.2.4 Main results . . . . . . . . . . . . . . . . . . 462

16.2.3 Generalized extreme value distribution . . . . . . . . 46516.2.3.1 Definition . . . . . . . . . . . . . . . . . . . 46516.2.3.2 Estimating the value-at-risk . . . . . . . . . 466

16.2.4 Peak over threshold . . . . . . . . . . . . . . . . . . . 46916.2.4.1 Definition . . . . . . . . . . . . . . . . . . . 469

xi

16.2.4.2 Estimating the expected shortfall . . . . . . 47116.3 Multivariate extreme value theory . . . . . . . . . . . . . . . 474

16.3.1 Multivariate extreme value distributions . . . . . . . 47416.3.1.1 Extreme value copulas . . . . . . . . . . . . 47416.3.1.2 Deheuvels-Pickands representation . . . . . 476

16.3.2 Maximum domain of attraction . . . . . . . . . . . . 47916.3.3 Tail dependence of extreme values . . . . . . . . . . . 481

16.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48116.4.1 Uniform order statistics . . . . . . . . . . . . . . . . . 48116.4.2 Order statistics and return period . . . . . . . . . . . 48216.4.3 Extreme order statistics of exponential random vari-

ables . . . . . . . . . . . . . . . . . . . . . . . . . . . 48216.4.4 Construction of a stress scenario with the GEV distri-

bution . . . . . . . . . . . . . . . . . . . . . . . . . . 48316.4.5 Extreme value theory in the bivariate case . . . . . . 48416.4.6 Max-domain of attraction in the bivariate case . . . . 484

17 Monte Carlo Simulation Methods 48717.1 Random variate generation . . . . . . . . . . . . . . . . . . . 487

17.1.1 Generating uniform random numbers . . . . . . . . . 48717.1.2 Generating non-uniform random numbers . . . . . . . 490

17.1.2.1 Method of inversion . . . . . . . . . . . . . . 49017.1.2.2 Method of transformation . . . . . . . . . . 49317.1.2.3 Rejection sampling . . . . . . . . . . . . . . 49517.1.2.4 Method of mixtures . . . . . . . . . . . . . . 498

17.1.3 Generating random vectors . . . . . . . . . . . . . . . 50117.1.3.1 Method of conditional distributions . . . . . 50117.1.3.2 Method of transformation . . . . . . . . . . 505

17.1.4 Generating random matrices . . . . . . . . . . . . . . 51117.1.4.1 Orthogonal and covariance matrices . . . . . 51217.1.4.2 Correlation matrices . . . . . . . . . . . . . 51417.1.4.3 Wishart matrices . . . . . . . . . . . . . . . 517

17.2 Simulation of stochastic processes . . . . . . . . . . . . . . . 51817.2.1 Discrete-time stochastic processes . . . . . . . . . . . 518

17.2.1.1 Correlated Markov chains . . . . . . . . . . 51817.2.1.2 Time-series . . . . . . . . . . . . . . . . . . 522

17.2.2 Univariate continuous-time processes . . . . . . . . . 52317.2.2.1 Brownian motion . . . . . . . . . . . . . . . 52317.2.2.2 Geometric Brownian motion . . . . . . . . . 52417.2.2.3 Ornstein-Uhlenbeck process . . . . . . . . . 52517.2.2.4 Stochastic differential equations without an

explicit solution . . . . . . . . . . . . . . . . 52617.2.2.5 Poisson processes . . . . . . . . . . . . . . . 53017.2.2.6 Jump-diffusion processes . . . . . . . . . . . 53417.2.2.7 Processes related to Brownian motion . . . . 536

xii

17.2.3 Multivariate continuous-time processes . . . . . . . . 54117.2.3.1 Multi-dimensional Brownian motion . . . . 54117.2.3.2 Multi-dimensional geometric Brownian mo-

tion . . . . . . . . . . . . . . . . . . . . . . . 54317.2.3.3 Euler-Maruyama and Milstein schemes . . . 543

17.3 Monte Carlo methods . . . . . . . . . . . . . . . . . . . . . . 54717.3.1 Computing integrals . . . . . . . . . . . . . . . . . . . 547

17.3.1.1 A basic example . . . . . . . . . . . . . . . . 54717.3.1.2 Theoretical framework . . . . . . . . . . . . 54717.3.1.3 Extension to the calculation of mathematical

expectations . . . . . . . . . . . . . . . . . . 55017.3.2 Variance reduction . . . . . . . . . . . . . . . . . . . . 553

17.3.2.1 Antithetic variates . . . . . . . . . . . . . . 55417.3.2.2 Control variates . . . . . . . . . . . . . . . . 56017.3.2.3 Importance sampling . . . . . . . . . . . . . 56817.3.2.4 Other methods . . . . . . . . . . . . . . . . 573

17.3.3 MCMC methods . . . . . . . . . . . . . . . . . . . . . 58517.3.3.1 Gibbs sampling . . . . . . . . . . . . . . . . 58517.3.3.2 Metropolis-Hastings algorithm . . . . . . . . 58917.3.3.3 Sequential Monte Carlo methods and particle

filters . . . . . . . . . . . . . . . . . . . . . . 59317.3.4 Quasi-Monte Carlo simulation methods . . . . . . . . 597

17.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60117.4.1 Simulating random numbers using the inversion

method . . . . . . . . . . . . . . . . . . . . . . . . . . 60117.4.2 Simulating random numbers using the transformation

method . . . . . . . . . . . . . . . . . . . . . . . . . . 60317.4.3 Simulating random numbers using rejection sampling 60417.4.4 Simulation of Archimedean copulas . . . . . . . . . . 60617.4.5 Simulation of conditional random variables . . . . . . 60717.4.6 Simulation of the bivariate Normal copula . . . . . . 60717.4.7 Computing the capital charge for operational risk . . 60817.4.8 Simulation of CIR and CEV processes . . . . . . . . . 60917.4.9 The Lamperti transform . . . . . . . . . . . . . . . . 60917.4.10 The Ozaki method . . . . . . . . . . . . . . . . . . . . 60917.4.11 Simulation of Poisson processes . . . . . . . . . . . . 60917.4.12 Simulating a Brownian bridge . . . . . . . . . . . . . 60917.4.13 Estimating the value-at-risk of credit portfolios using

variance reduction techniques . . . . . . . . . . . . . 60917.4.14 Optimal importance sampling . . . . . . . . . . . . . 60917.4.15 Rare event simulation . . . . . . . . . . . . . . . . . . 61017.4.16 Efficient simulation of operational risk losses . . . . . 61017.4.17 Importance sampling and option pricing . . . . . . . 61017.4.18 Control variates and option pricing . . . . . . . . . . 610

xiii

17.4.19 Conditional Monte Carlo and stochastic volatility mod-els . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610

17.4.20 Gibbs sampling and conditional distributions . . . . . 61017.4.21 Simulating copula functions with the Metropolis-

Hastings algorithm . . . . . . . . . . . . . . . . . . . 61017.4.22 The independence sampler . . . . . . . . . . . . . . . 610

A Technical Appendix 611A.1 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . 611

A.1.1 Linear algebra . . . . . . . . . . . . . . . . . . . . . . 611A.1.1.1 Eigendecomposition . . . . . . . . . . . . . . 611A.1.1.2 Schur decomposition . . . . . . . . . . . . . 612

A.1.2 Approximation methods . . . . . . . . . . . . . . . . 613A.1.2.1 Spline functions . . . . . . . . . . . . . . . . 613A.1.2.2 Semidefinite approximation . . . . . . . . . 613A.1.2.3 Numerical integration . . . . . . . . . . . . . 613A.1.2.4 Finite difference methods . . . . . . . . . . . 613

A.1.3 Numerical optimization . . . . . . . . . . . . . . . . . 619A.1.3.1 Quadratic programming problem . . . . . . 619A.1.3.2 Non-linear unconstrained optimization . . . 620A.1.3.3 Sequential quadratic programming algorithm 622

A.2 Statistical and probability analysis . . . . . . . . . . . . . . . 624A.2.1 Probability distributions . . . . . . . . . . . . . . . . 624

A.2.1.1 The Bernoulli distribution . . . . . . . . . . 624A.2.1.2 The binomial distribution . . . . . . . . . . 624A.2.1.3 The geometric distribution . . . . . . . . . . 624A.2.1.4 The Poisson distribution . . . . . . . . . . . 624A.2.1.5 The negative binomial distribution . . . . . 625A.2.1.6 The gamma distribution . . . . . . . . . . . 625A.2.1.7 The beta distribution . . . . . . . . . . . . . 626

A.2.2 The noncentral chi-squared distribution . . . . . . . . 626A.2.2.1 The exponential distribution . . . . . . . . . 627A.2.2.2 The normal distribution . . . . . . . . . . . 627A.2.2.3 The Students t distribution . . . . . . . . . 628A.2.2.4 The log-normal distribution . . . . . . . . . 629A.2.2.5 The Pareto distribution . . . . . . . . . . . 629A.2.2.6 The generalized extreme value distribution . 630A.2.2.7 The generalized Pareto distribution . . . . . 630A.2.2.8 The skew normal distribution . . . . . . . . 630A.2.2.9 The skew t distribution . . . . . . . . . . . . 632A.2.2.10 The Wishart distribution . . . . . . . . . . . 634

A.2.3 Special results . . . . . . . . . . . . . . . . . . . . . . 635A.2.3.1 Affine transformation of random vectors . . 635A.2.3.2 Change of variables . . . . . . . . . . . . . . 636

xiv

A.2.3.3 Relationship between density and quantilefunctions . . . . . . . . . . . . . . . . . . . . 636

A.2.3.4 Conditional expectation in the case of theNormal distribution . . . . . . . . . . . . . . 637

A.2.3.5 Calculation of a useful integral function incredit risk models . . . . . . . . . . . . . . . 637

A.3 Stochastic analysis . . . . . . . . . . . . . . . . . . . . . . . . 638A.3.1 Brownian motion and Wiener process . . . . . . . . . 639A.3.2 Stochastic integral . . . . . . . . . . . . . . . . . . . . 641A.3.3 Stochastic differential equation and Its lemma . . . 642

A.3.3.1 Existence and uniqueness of a stochastic dif-ferential equation . . . . . . . . . . . . . . . 642

A.3.3.2 Relationship with diffusion processes . . . . 643A.3.3.3 It calculus . . . . . . . . . . . . . . . . . . 644A.3.3.4 Extension to the multi-dimensional case . . 645

A.3.4 Feynman-Kac formula . . . . . . . . . . . . . . . . . . 646A.3.5 Girsanov theorem . . . . . . . . . . . . . . . . . . . . 647A.3.6 Fokker-Planck equation . . . . . . . . . . . . . . . . . 648A.3.7 Reflexion principle and stopping times . . . . . . . . 649A.3.8 Some diffusion processes . . . . . . . . . . . . . . . . 651

A.3.8.1 Geometric Brownian motion . . . . . . . . . 651A.3.8.2 Ornstein-Uhlenbeck process . . . . . . . . . 652A.3.8.3 Cox-Ingersoll-Ross process . . . . . . . . . . 653A.3.8.4 Multi-dimensional processes . . . . . . . . . 654

A.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655A.4.1 Discrete-time random process . . . . . . . . . . . . . 655A.4.2 Properties of Brownian motion . . . . . . . . . . . . . 655A.4.3 Stochastic integral for random step functions . . . . . 656A.4.4 Power of Brownian motion . . . . . . . . . . . . . . . 656A.4.5 Exponential of Brownian motion . . . . . . . . . . . . 657A.4.6 Exponential martingales . . . . . . . . . . . . . . . . 657A.4.7 Existence of solutions to stochastic differential equa-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . 657A.4.8 It calculus and stochastic integration . . . . . . . . . 657A.4.9 Solving a PDE with the Feynman-Kac formula . . . . 658A.4.10 Fokker-Planck equation . . . . . . . . . . . . . . . . . 659A.4.11 Dynamic strategy based on the current asset price . . 659A.4.12 Strong Markov property and maximum of Brownian

motion . . . . . . . . . . . . . . . . . . . . . . . . . . 660A.4.13 Moments of the Cox-Ingersoll-Ross process . . . . . . 661A.4.14 Probability density function of Heston and SABR mod-

els . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661

Bibliography 663

xv

Subject Index 689

Author Index 701

List of Figures

1.1 Notional amount of exchange-traded derivatives ((in $ tn) . 81.2 Notional amount of OTC derivatives (in $ tn) . . . . . . . . 91.3 Number of bank defaults in the US . . . . . . . . . . . . . . 121.4 Probability distribution of the portfolio loss . . . . . . . . . 181.5 Minimum capital requirements in the Basel II framework . . 221.6 Solvency I capital requirement . . . . . . . . . . . . . . . . . 271.7 Solvency II capital requirement . . . . . . . . . . . . . . . . 28

2.1 Calculation of the required capital with the VaR . . . . . . 582.2 Two different periods to compute the VaR and the SVaR . . 602.3 Color zones of the backtesting procedure ( = 99%) . . . . . 642.4 Density of the VaR estimator . . . . . . . . . . . . . . . . . 692.5 Kernel estimation of the historical VaR . . . . . . . . . . . . 732.6 Cash flows of two bonds and two short exposures . . . . . . 792.7 Convergence of the VaR with PCA risk factors . . . . . . . 822.8 Weights of the EWMA estimator . . . . . . . . . . . . . . . 832.9 Comparison of GARCH and EWMA volatilities (S&P 500) . 852.10 Examples of skewed and fat tailed distributions . . . . . . . 872.11 Estimated distribution of S&P 500 daily returns (2007-2014) 872.12 Derivatives and definition domain of the Cornish-Fisher ex-

pansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.13 Skew normal distribution of asset returns . . . . . . . . . . . 912.14 Convergence of the Monte Carlo VaR when asset returns are

skew normal . . . . . . . . . . . . . . . . . . . . . . . . . . . 952.15 Probability density function of the daily P&L with credit risk 972.16 Relationship between the asset return RS and the option re-

turn RC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012.17 Approximation of the option price with the Greek coefficients 1042.18 Probability density function of the different risk contribution

estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1252.19 Probability density function of the RC1 estimator for the VaR

99% and ES 99.5% . . . . . . . . . . . . . . . . . . . . . . . 128

3.1 Notional outstanding of credit in the United States (in $ tn) 1413.2 Credit to the private non-financial sector (in $ tn) . . . . . . 1423.3 Amounts outstanding US bond market debt (in $ tn) . . . . 144

xvii

xviii

3.4 Issuance in the US bond markets (in $ tn) . . . . . . . . . . 1453.5 Average daily trading volume in US bond markets (in $ bn) 1463.6 Cash flows of a bond with a fixed coupon rate . . . . . . . . 1463.7 Movements of the yield curve . . . . . . . . . . . . . . . . . 1483.8 Cash flows of a bond with default risk . . . . . . . . . . . . 1503.9 Difference between market and credit risk for a bond . . . . 1533.10 Securitization in Europe and US (in etn) . . . . . . . . . . 1543.11 Structure of pass-through securities . . . . . . . . . . . . . . 1553.12 Structure of pay-through securities . . . . . . . . . . . . . . 1563.13 Amounts outstanding of credit default swaps (in $ tn) . . . 1583.14 Cash flows of a single-name credit default swap . . . . . . . 1593.15 Evolution of some sovereign CDS spreads . . . . . . . . . . . 1633.16 Evolution of some financial and corporate CDS spreads . . . 1633.17 Examples of CDS spread curve as of 2015-09-17 . . . . . . . 1643.18 Hedging a defaultable bond with a credit default swap . . . 1653.19 An example of CDS offsetting . . . . . . . . . . . . . . . . . 1673.20 Structure of a collateralized debt obligation . . . . . . . . . 1763.21 Probability functions of the credit portfolio loss . . . . . . . 1983.22 Relationship between the risk contribution RCi and model

parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1993.23 Examples of piecewise exponential model . . . . . . . . . . . 2093.24 Estimated hazard function . . . . . . . . . . . . . . . . . . . 2103.25 Calibrated survival function from CDS prices . . . . . . . . 2123.26 Estimated hazard function i (t) from the credit migration

matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2173.27 Density function fi (t) of S&P ratings . . . . . . . . . . . . . 2233.28 Hazard function (t) (in bps) estimated respectively with the

piecewise exponential model and the Markov generator . . . 2303.29 Hazard functions (n) and (t) (in bps) . . . . . . . . . . . 232

4.1 Density function of the counterparty exposure after sixmonths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

4.2 Density function of the counterparty exposure after ninemonths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

4.3 Evolution of the counterparty exposure . . . . . . . . . . . . 2404.4 Counterparty exposure profile of option . . . . . . . . . . . . 2444.5 Counterparty exposure profile of interest rate swap . . . . . 2444.6 Impact of negative mark-to-market on the PFE multiplier . 250

5.1 Compound distribution when N P (50) and X LN (8, 5) 2765.2 Impact of the threshold H on the severity distribution . . . 2835.3 Comparison of the estimated severity distributions . . . . . 2835.4 An example of QQ plot where extreme events are underesti-

mated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2845.5 PMF of the Poisson distribution P (60) . . . . . . . . . . . . 287

xix

5.6 PMF of the negative binomial distribution . . . . . . . . . . 2895.7 PDF of the parameter . . . . . . . . . . . . . . . . . . . . 2895.8 Histogram of the MC estimator CaR . . . . . . . . . . . . . 2935.9 Convergence of the accuracy ratio R (ns) when = 1 . . . . 2945.10 Convergence of the accuracy ratio R (ns) when = 2.5 . . . 2955.11 Impact of an insurance contract in an operational risk loss . 2965.12 Comparison between the Panjer and MC compound distribu-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2995.13 Relationship between CaR and Severity . . . . . . . . . . . 3015.14 Numerical illustration of the single loss approximation . . . 3015.15 Upper bound + of the aggregate loss correlation . . . . . . 3045.16 Simulation of the Poisson process N (t) and peak-over-

threshold events . . . . . . . . . . . . . . . . . . . . . . . . . 307

6.1 An example of a limit order book . . . . . . . . . . . . . . . 3176.2 Comparing the liquidation ratio (in %) between index fund

portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

12.1 Illustration of tail risk . . . . . . . . . . . . . . . . . . . . . 34512.2 Impact of the TLAC on capital requirements . . . . . . . . . 35612.3 Impact of the uniform correlation on CoVaRi . . . . . . . 36212.4 A completely connected network . . . . . . . . . . . . . . . . 36712.5 A sparse network . . . . . . . . . . . . . . . . . . . . . . . . 36812.6 A partially dense network . . . . . . . . . . . . . . . . . . . 36812.7 Credit assets (in $ tn) . . . . . . . . . . . . . . . . . . . . . 37212.8 Calculation of the shadow banking narrow measure . . . . . 37412.9 Breakdown by country of shadow banking assets (2014) . . . 37512.10 Interconnectedness between banks and OFIs . . . . . . . . . 376

15.1 Example of a bivariate probability distribution with givenmarginals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

15.2 The triangle region of the contour lines C (u1, u2) = . . . 40715.3 The three copula functions C, C and C+ . . . . . . . . . 40815.4 Concordance ordering of the Frank copula . . . . . . . . . . 40815.5 Contour lines of bivariate densities (normal copula) . . . . . 41415.6 Contour lines of bivariate densities (Frank copula) . . . . . . 41515.7 Contour lines of bivariate densities (Gumbel copula) . . . . 41515.8 Bounds of (, %) statistics . . . . . . . . . . . . . . . . . . . 41615.9 Bounds of the linear correlation between two log-normal ran-

dom variables . . . . . . . . . . . . . . . . . . . . . . . . . . 41815.10 Quantile-quantile dependence measures + () and () . 42115.11 Tail dependence + () for the normal copula . . . . . . . . 42715.12 Relationship between and % of the Students t copula . . . 42915.13 Tail dependence + () for the Students t copula ( = 1) . 43015.14 Tail dependence + () for the Students t copula ( = 4) . 430

xx

15.15 Comparison of the empirical copula (blue line) and the normalcopula (red line) . . . . . . . . . . . . . . . . . . . . . . . . . 433

15.16 Dependogram of EU and US equity returns . . . . . . . . . 43415.17 Dependogram of simulated Gaussian returns . . . . . . . . . 434

16.1 Distribution function Fi:n when the random variablesX1, . . . , Xn are Gaussian . . . . . . . . . . . . . . . . . . . . 447

16.2 Density function fn:n of the Gaussian random variableN (0, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

16.3 Density function of the maximum order statistic (daily returnof the MSCI USA index, 1995-2015) . . . . . . . . . . . . . 450

16.4 Annualized volatility (in %) calculated from the order statis-tics Ri:10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

16.5 Density function of the first-to-default 1:10 . . . . . . . . . 45616.6 Max-convergence of the exponential distribution E (1) to the

Gumbel distribution . . . . . . . . . . . . . . . . . . . . . . 45816.7 Density function of , 1 and 1 . . . . . . . . . . . . . . . 46016.8 Density function of the Frchet probability distribution . . . 46016.9 The graphical verification of the regular variation property for

the N (0, 1) distribution function . . . . . . . . . . . . . . . 46316.10 Probability density function of the GEV distribution . . . . 46716.11 Probability density function of the maximum return R22:22 . 46716.12 Mean residual life plot . . . . . . . . . . . . . . . . . . . . . 47216.13 Multivariate extreme value distributions . . . . . . . . . . . 476

17.1 Lattice structure of the linear congruential generator . . . . 48917.2 Inversion method when X is a discrete random variable . . 49217.3 Rejection sampling applied to the Normal distribution . . . 49717.4 Comparison of the exact and simulation densities . . . . . . 49917.5 Simulation of the Normal copula . . . . . . . . . . . . . . . 50617.6 Simulation of the t1 copula . . . . . . . . . . . . . . . . . . . 50717.7 Simulation of the Clayton copula . . . . . . . . . . . . . . . 50817.8 Simulation of the correlated random vector ( ,LGD) . . . . 50917.9 Convergence of the method of the empirical quantile function 51017.10 Simulation of the random variables Z1 and Z2 . . . . . . . . 51117.11 Simulation of the random vector (Z1, Z2) . . . . . . . . . . . 51217.12 Distribution of the eigenvalues of simulated random orthogo-

nal matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 51317.13 Comparison of the Bendel-Mickey and covariance algorithms 51517.14 Price of the basket option . . . . . . . . . . . . . . . . . . . 51717.15 Simulation of rating dynamics (correlation matrix 1) . . . . 52017.16 Simulation of rating dynamics (correlation matrix 2) . . . . 52017.17 Simulation of rating dynamics (correlation matrix 1) . . . . 52117.18 Simulation of rating dynamics (correlation matrix 2) . . . . 52117.19 Simulation of the geometric Brownian motion . . . . . . . . 525

xxi

17.20 Simulation of the Ornstein-Uhlenbeck process . . . . . . . . 52617.21 Comparison of exact, Euler-Maruyama and Milstein schemes

(monthly discretization) . . . . . . . . . . . . . . . . . . . . 52817.22 Simulation of a non-homogenous Poisson process with cyclical

intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53217.23 Simulation of a jump-diffusion process . . . . . . . . . . . . 53517.24 Simulation of the Brownian bridge B1 (t) using the time re-

versibility property . . . . . . . . . . . . . . . . . . . . . . . 53717.25 Simulation of the Brownian bridge B (t) . . . . . . . . . . . 53817.26 Simulation of the diffusion bridge X (t) . . . . . . . . . . . . 53917.27 Density of the maximum estimators M and M . . . . . . . . 54017.28 Brownian motion in the plane (independent case) . . . . . . 54217.29 Brownian motion in the plane (1,2 = 85%) . . . . . . . . . 54217.30 Computing with 1 000 simulations . . . . . . . . . . . . . 54817.31 Density function of nS . . . . . . . . . . . . . . . . . . . . . 54917.32 Convergence of the estimator InS . . . . . . . . . . . . . . . 55117.33 Computing the lock-back option price . . . . . . . . . . . . . 55217.34 Computing with normal random numbers . . . . . . . . . 55417.35 Functions 1 (x), 2 (x) and 3 (x) . . . . . . . . . . . . . . 55717.36 Antithetic simulation of the GBM . . . . . . . . . . . . . . . 55917.37 Probability density function of CMC and CAV (nS = 1 000) . 56117.38 Understanding the variance reduction in control variates . . 56317.39 CV estimators of the arithmetic Asian call option . . . . . . 56617.40 Histogram of the MC and IS estimators (nS = 1 000) . . . . 57017.41 Standard deviation (in %) of the estimator pIS (nS = 1 000) 57117.42 Density function of the estimators PMC and PIS (nS = 1 000) 57317.43 Function (x) . . . . . . . . . . . . . . . . . . . . . . . . . . 57917.44 Intra-strata variance 2 (j) (in bps) . . . . . . . . . . . . . . 58117.45 Optimal allocation q? (j) (in %) . . . . . . . . . . . . . . . . 58117.46 Strata for different random variables . . . . . . . . . . . . . 58217.47 Variance of the two estimators I(1)STR (m) and I

(2)STR (m) for dif-

ferent values of m . . . . . . . . . . . . . . . . . . . . . . . . 58417.48 Illustration of the Gibbs sampler . . . . . . . . . . . . . . . 58817.49 Illustration of the random walk sampler . . . . . . . . . . . 59217.50 Simulating bivariate probability distributions with the MH

algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59317.51 An example of a SMC run with 1 000 particles . . . . . . . . 59717.52 Density of the RMSE statistic for 1 000 particles . . . . . . . 59817.53 Density of the RMSE statistic for the SIS algorithm . . . . . 59817.54 Comparison of different low discrepancy sequences . . . . . 60017.55 The Sobol generator . . . . . . . . . . . . . . . . . . . . . . 60017.56 Quasi-random points on the unit sphere . . . . . . . . . . . 601

List of Tables

1.1 Amounts outstanding of exchange-traded derivatives . . . . 71.2 Amounts outstanding of OTC derivatives . . . . . . . . . . . 71.3 The supervision institutions in finance . . . . . . . . . . . . 141.4 The three pillars of the Basel II framework . . . . . . . . . . 191.5 Basel III capital requirements . . . . . . . . . . . . . . . . . 24

2.1 Value of the penalty coefficient for a sample of 250 observa-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.2 Probability distribution (in %) of the number of exceptions(n = 250 trading days) . . . . . . . . . . . . . . . . . . . . . 64

2.3 Computation of the market risk factors R1,s and R2,s . . . . 702.4 Computation of the simulated P&L s (w) . . . . . . . . . . 712.5 Number of exceptions per year for long and short exposures

on the S&P 500 index . . . . . . . . . . . . . . . . . . . . . 852.6 Value of the Cornish-Fisher quantile z (99%; 1, 2) . . . . . 892.7 Value of the multiplication factormc deduced from the Cheby-

shevs inequality . . . . . . . . . . . . . . . . . . . . . . . . . 932.8 Daily P&L of the long position on the call option when the

risk factor is the underlying price . . . . . . . . . . . . . . . 1002.9 Daily P&L of the long position on the call option when the

risk factors are the underlying price and the implied volatility 1022.10 Calculation of the P&L based on the Greek sensitivities . . 1052.11 Calculation of the P&L using the vega coefficient . . . . . . 1062.12 The 10 worst scenarios identified by the hybrid method . . . 1072.13 Scaling factors cVaR and cES . . . . . . . . . . . . . . . . . . 1142.14 Risk decomposition of the 99% Gaussian value-at-risk . . . . 1182.15 Risk decomposition of the 99% Gaussian expected shortfall 1182.16 Risk decomposition of the 99% historical value-at-risk . . . 1212.17 Risk contributions calculated with regularization techniques 1232.18 Risk decomposition of the 99% historical expected shortfall 1272.19 Risk decomposition of the 97.5% historical expected shortfall 127

3.1 Debt securities by residence of issuer (in $ bn) . . . . . . . . 1433.2 Price, yield to maturity and sensitivity of bonds . . . . . . . 1493.3 Impact of a parallel shift of the yield curve on the bond with

five-year maturity . . . . . . . . . . . . . . . . . . . . . . . . 149

xxiii

xxiv

3.4 Computation of the credit spread s . . . . . . . . . . . . . . 1523.5 US mortgage-backed securities . . . . . . . . . . . . . . . . . 1573.6 Global collateralized debt obligations . . . . . . . . . . . . . 1573.7 US asset-backed securities . . . . . . . . . . . . . . . . . . . 1573.8 Price, spread and risky PV01 of CDS contracts . . . . . . . 1693.9 Price, spread and risky PV01 of CDS contracts (without the

accrued premium) . . . . . . . . . . . . . . . . . . . . . . . . 1693.10 Calibration of the CDS spread curve using the exponential

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1703.11 List of Markit CDX main indices . . . . . . . . . . . . . . . 1743.12 List of Markit iTraxx main indices . . . . . . . . . . . . . . 1743.13 Historical spread of CDX/iTraxx indices (in bps) . . . . . . 1753.14 List of Markit credit default tranches . . . . . . . . . . . . . 1783.15 Worlds largest banks in 1981 and 1988 . . . . . . . . . . . . 1813.16 Risk weights by category of on-balance-sheetOn-balance-sheet

asset asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1823.17 Risk weights of the SA approach (Basel II) . . . . . . . . . . 1833.18 Comparison of risk weights between Basel I and Basel II . . 1843.19 Credit rating system of S&P, Moodys and Fitch . . . . . . 1853.20 Examples of countrys S&P rating . . . . . . . . . . . . . . . 1853.21 Examples of banks S&P rating . . . . . . . . . . . . . . . . 1853.22 Examples of corporates S&P rating . . . . . . . . . . . . . . 1863.23 Standardized supervisory haircuts for collateralized transac-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1883.24 Example of Internal Rating Systems . . . . . . . . . . . . . 1923.25 Numerical values of f (`), F (`) and F1 () when is equal

to 10% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1973.26 Percentage change in capital requirements under CP2 propos-

als . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2003.27 IRB Risk weights (in %) for corporate exposures . . . . . . 2043.28 IRB Risk weights (in %) for retail exposures . . . . . . . . . 2053.29 Common survival functions . . . . . . . . . . . . . . . . . . 2073.30 Calibrated piecewise exponential model from CDS prices . . 2113.31 Example of credit migration matrix (in %) . . . . . . . . . . 2143.32 Two-year transition matrix P 2 (in %) . . . . . . . . . . . . . 2153.33 Five-year transition matrix P 5 (in %) . . . . . . . . . . . . . 2153.34 Markov generator (in bps) . . . . . . . . . . . . . . . . . . 2203.35 Markov generator (in bps) . . . . . . . . . . . . . . . . . . 2203.36 Markov generator (in bps) . . . . . . . . . . . . . . . . . . 2213.37 Markov generator (in bps) . . . . . . . . . . . . . . . . . . 2213.38 207-day transition matrix (in %) . . . . . . . . . . . . . . . 222

4.1 Counterparty exposure of Bank A . . . . . . . . . . . . . . . 2354.2 Counterparty exposure of Bank B . . . . . . . . . . . . . . . 2364.3 Mark-to-market and counterparty exposure of the call option 237

xxv

4.4 Capital charge of counterparty credit risk under the FIRBapproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

4.5 Regulatory add-on factors for the current exposure method 2474.6 Supervisory credit conversion factors for the SM-CCR ap-

proach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2484.7 Supervisory parameters for the SA-CCR approach . . . . . . 252

5.1 Internal losses larger than e20 000 by year . . . . . . . . . . 2675.2 Mapping of business lines for operational risk . . . . . . . . 2705.3 Distribution of annualized operational losses (in %) . . . . . 2725.4 Density function, mean and variance of parametric probability

distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 2785.5 Comparison of the capital-at-risk calculated with Panjer re-

cursion and Monte Carlo simulations . . . . . . . . . . . . . 298

6.1 An example of a limit order book . . . . . . . . . . . . . . . 3166.2 Statistics of the liquidation ratio (size = $10 bn, liquidation

policy = 10% of ADV) . . . . . . . . . . . . . . . . . . . . . 3206.3 Statistics of the liquidation ratio (size = $10 bn, liquidation

policy = 30% of ADV) . . . . . . . . . . . . . . . . . . . . . 3216.4 Stock of HQLA . . . . . . . . . . . . . . . . . . . . . . . . . 3266.5 Cash outflows of the LCR . . . . . . . . . . . . . . . . . . . 3286.6 Cash inflows of the LCR . . . . . . . . . . . . . . . . . . . . 329

12.1 List of global systemically important banks (November 2015) 34912.2 List of global systemically important insurers (November

2015) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34912.3 Scoring system of G-SIBs . . . . . . . . . . . . . . . . . . . . 35212.4 An example of calculating the G-SIB score . . . . . . . . . . 35312.5 Risk decomposition of the 95% systemic expected shortfall . 35912.6 Calculation of the 95% CoVaR measure . . . . . . . . . . . 36212.7 Calculation of the SRISK measure (S = 40%) . . . . . . . 36512.8 Impact of the stress S on SRISK . . . . . . . . . . . . . . . 36512.9 Systemic risk contributions in America (2015-11-27) . . . . . 36612.10 Systemic risk contributions in Europe (2015-11-27) . . . . . 36612.11 Systemic risk contributions in Asia (2015-11-27) . . . . . . . 36612.12 Assets of financial institutions (in $ tn) . . . . . . . . . . . . 37012.13 Assets of OFIs (in $ tn) . . . . . . . . . . . . . . . . . . . . 37112.14 Classification of the shadow banking system by economic func-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37312.15 Size of the narrow shadow banking (in $ tn) . . . . . . . . . 374

14.1 Numerical example . . . . . . . . . . . . . . . . . . . . . . . 398

15.1 Examples of Archimedean copula functions . . . . . . . . . . 42215.2 Values in % of the upper tail dependence + for the t copula 431

xxvi

15.3 Matrix of linear correlations i,j . . . . . . . . . . . . . . . . 43615.4 Matrix of parameters i,j estimated using Kendalls tau . . 43615.5 Matrix of parameters i,j estimated using Spearmans rho . 43615.6 Omnibus estimate (Gaussian copula) . . . . . . . . . . . . 43915.7 Omnibus estimate (Students t copula with = 1) . . . . 44015.8 Omnibus estimate (Students t copula with = 4) . . . . 440

16.1 ML estimate of (in bps) for the distribution t1 . . . . . . 45216.2 ML estimate of (in bps) for the distribution t6 . . . . . . 45216.3 ML estimate of (in bps) for the distribution t . . . . . . 45216.4 Maximum domain of attraction and normalizing constants of

some distribution functions . . . . . . . . . . . . . . . . . . 46416.5 Comparing Gaussian, historical and GEV value-at-risk . . . 46916.6 Estimation of the generalized Pareto distribution . . . . . . 47216.7 Estimating value-at-risk and expected shortfall using the gen-

eralized Pareto distribution . . . . . . . . . . . . . . . . . . 47416.8 List of extreme value copulas . . . . . . . . . . . . . . . . . 478

17.1 Simulation of 10 uniform pseudorandom numbers . . . . . . 48917.2 Simulation of the piecewise exponential model . . . . . . . . 49317.3 Simulation of the standard Gaussian distribution using the

acceptance-rejection algorithm . . . . . . . . . . . . . . . . . 49917.4 Simulation of the Clayton copula . . . . . . . . . . . . . . . 50517.5 Linear regression between the Asian call option and the con-

trol variates . . . . . . . . . . . . . . . . . . . . . . . . . . . 56717.6 Variance ratio (in %) when a = 1 . . . . . . . . . . . . . . . 57517.7 Comparison between MC and STR estimators . . . . . . . . 58017.8 Pricing of the spread option using quasi-Monte Carlo methods 602

List of Symbols and Notations

Symbol Description

Scalar multiplication Hadamard product:

(x y)i = xiyi Kronecker product AB|E| Cardinality of the set E Concordance ordering1 Vector of ones1 {A} The indicator function is

equal to 1 if A is true, 0otherwise

1A {x} The characteristic functionis equal to 1 if x A, 0otherwise

0 Vector of zeros(Ai,j) Matrix A with entry Ai,j in

row i and column jA1 Inverse of the matrix AA1/2 Square root of the matrix AA> Transpose of the matrix AA+ Moore-Penrose pseudo-

inverse of the matrix Ab Vector of weights

(b1, . . . , bn) for the bench-mark b

Bt (T ) Price of the zero-couponbond at time t for the ma-turity T

B (t, T ) Alternative form of Bt (T )B (p) Bernoulli distribution with

parameter pB (n, p) Binomial distribution with

parameter n and p

i Beta of asset i with respectto portfolio w

i (w) Another notation for thesymbol i

(w | b) Beta of portfolio w whenthe benchmark is b

B (, ) Beta distribution with pa-rameter and

B (, ) Beta function defined as 10 t

1 (1 t)1 dtB (x;, ) Incomplete beta function x

0 t1 (1 t)1 dt

ccc Coupon rate of the CDSpremium leg

C (or ) Correlation matrixC (u1, u2) Copula functionC Frchet lower bound copulaC Product copulaC+ Frchet upper bound cop-

ulaCt Price of the call option at

time tC (tm) Coupon paid at time tmCn () Constant correlation ma-

trix (n n) with i,j = CE (t0) Current exposure at time t0cov (X) Covariance of the random

vector X2 () Chi-square distribution

with degrees of freedomD Covariance matrix of id-

iosyncratic risks

xxvii

xxviii

det (A) Determinant of the matrixA

diag v Build a diagonal matrixwith elements (v1, . . . , vn)

t Delta of the option at timet

h Difference operator withlag h, e.g. hVt = VtVth

CoVaRi Delta CoVaR of Institu-tion i

tm Time interval tm = tm1ei The value of the vector is

1 for the row i and 0 else-where

E [X] Mathematical expectationof the random variable X

E () Exponential probabilitydistribution with param-eter

e (t) Potential future exposureat time t

EE (t) Expected exposure at timet

EEE (t) Effective expected expo-sure at time t

EEPE (0; t) Effective expected posi-tive exposure for the timeperiod [0, t]

EPE (0; t) Expected positive exposurefor the time period [0, t]

ES (w) Expected shortfall of port-folio w at the confidencelevel

expA Exponential of the matrixA

f (x) Probability density func-tion (pdf)

F (x) Cumulative distributionfunction (cdf)

F1 () Quantile functionFn? n-fold convolution of the

probability distribution Fwith itself

F Vector of risk factors(F1, . . . ,Fm)

Fj Risk factor jFt Filtrationft (T ) Instantaneous forward rate

at time t for the maturityT

f (t, T ) Alternative form of ft (T )Ft (T1, T2)Forward interest rate at

time t for the period[T1, T2]

F (t, T1, T2) Alternative form ofFt (T1, T2)

G (p) Geometric distributionwith parameter p

G () Standard gamma distribu-tion with parameter

G (, ) Gamma distribution withparameters and

1 Skewness2 Excess kurtosist Gamma of the option at

time t () Gamma function defined as

0 t1et dt

(, x) Lower incomplete gammafunction defined as x

0 t1et dt

(, x) Upper incomplete gammafunction defined asxt1et dt

GEV (, , ) GEV distribution withparameters , and

GPD (, ) Generalized Pareto distri-bution with parameters and

h Holding periodhhh Kernel or smoothing pa-

rameteri Asset (or component) iIn Identity matrix of dimen-

sion nK Regulatory capital` () Log-likelihood function

with the vector of pa-rameters to estimate

xxix

`t Log-likelihood function forthe observation t

L or L (w) Loss of portfolio wlnA Logarithm of the matrix ALG (, ) Log-gamma distribution

with parameters and LL (, ) Log-logistic distribution

with parameters and LN

(, 2

)Log-normal distribution

with parameters and Parameter of exponential

survival times (t) Hazard function (t) Markov generatorMDA (G) Maximum domain of at-

traction of the extremevalue distribution G

MESi Marginal expected shortfallof Institution i

MPE (0; t) Maximum peak exposurefor the time period [0, t]with a confidence level

MRi Marginal risk of asset iMtM Mark-to-market of the

portfolio Vector of expected returns

(1, . . . , n)i Expected return of asset imkt Expected return of the

market portfolio Empirical mean (w) Expected return of portfo-

lio w (X) Mean of the random vector

Xm (X) m-th centered moment of

the random vector Xm (X) m-th moment of the ran-

dom vector XN(, 2

)Normal distribution withmean and standard devi-ation

N (,) Multivariate normal distri-bution with mean and co-variance matrix

nS Number of scenarios orsimulations

N (t) Poisson counting processfor the time interval [0, t]

N (t1; t2) Poisson counting processfor the time interval [t1, t2]

NB (r, p) Negative binomial distribu-tion with parameters r andp

Covariance matrix of riskfactors

P Markov transition matrixP Historical probability mea-

sureP () Poisson distribution with

parameter p (k) Probability mass function

of an integer-valued ran-dom variable

Pt Price of the put option attime t

P (, x) Pareto distribution withparameters and x

P (, ) Pareto distribution withparameters and

PE (t) Peak exposure at time twith a confidence level

PVt (L) Present value of the leg L or (w)P&L of the portfolio w (x) Probability density func-

tion of the standardizednormal distribution

2 (x1, x2; ) Probability densityfunction of the bivariatenormal distribution withcorrelation

n (x; ) Probability density func-tion of the multivariatenormal distribution withcovariance matrix

(x) Cumulative density func-tion of the standardizednormal distribution

1 () Inverse of the cdf of the

xxx

standardized normal distri-bution

2 (x1, x2; ) Cumulative densityfunction of the bivariatenormal distribution withcorrelation

n (x; ) Cumulative density func-tion of the multivariatenormal distribution withcovariance matrix

X (t) Characteristic function ofthe random variable X

q (nS) Integer part of nSq (nS) Integer part of (1 )nSQ Risk-neutral probability

measureQT Forward probability mea-

sureR (t) Rating of the entity at time

t

r Return of the risk-free assetR Vector of asset returns

(R1, . . . , Rn)Ri Return of asset iRi,t Return of asset i at time tR (w) Return of portfolio wR (w) Risk measure of portfolio wR (L) Risk measure of loss LR () Risk measure of P&L Rt (T ) Zero-coupon rate at time t

for the maturity TRCi Risk contribution of asset iRC?i Relative risk contribution

of asset iR Recovery rateRPV01 Risky PV01 (or C) Correlation matrix of asset

returnsi,j Correlation between asset

returns i and j (x, y) Correlation between port-

folios x and ys Credit spreadS Stress scenario

St Price of the underlying as-set at time t

St (T ) Survival function of T attime t

SESi Systemic expected shortfallof Institution i

SN (,, ) Skew normal distribu-tion

SRISKi Systemic risk contributionof Institution i

ST (,, , ) Skew t distributionSVt (L) Stochastic discounted value

of the leg L Covariance matrix Empirical covariance ma-

trixi Volatility of asset imkt Volatility of the market

portfolioi Idiosyncratic volatility of

asset i Empirical volatility (w) Volatility of portfolio w (X) Standard deviation of the

random variable Xt Students t distribution

with degrees of freedomt (x) Probability density func-

tion of the univariate t dis-tribution with number ofdegrees of freedom

tn (x; , ) Probability density func-tion of the multivariate tdistribution with parame-ters and

t2 (x1, x2; , ) Probability densityfunction of the bivariatet distribution with param-eters and

T Maturity dateT (x) Cumulative density func-

tion of the univariate t dis-tribution with number ofdegrees of freedom

T1 () Inverse of the cdf of the

xxxi

Students t distributionwith the number of de-grees of freedom

Tn (x; , ) Cumulative density func-tion of the multivariate tdistribution with parame-ters and

T2 (x1, x2; , ) Cumulative densityfunction of the bivariate tdistribution with parame-ters and

T return periodtr (A) Trace of the matrix A Vector of parameters Estimator of t Theta of the option at time

t Default time

Time to maturity T tvar (X) Variance of the random

variable XVaR (w) Value-at-risk of portfolio w

at the confidence level t Vega of the option tw Vector of weights

(w1, . . . , wn) for portfoliow

wi Weight of asset i in portfo-lio w

W (t) Wiener processX Random variablex+ Maximum value between x

and 0Xi:n i

th order statistic of a sam-ple of size n

y Yield to maturity

Abbreviations

ABCP Asset-backed commercialpaper

ABS Asset-backed securityADV Average daily volumeAFME Association for Financial

Markets in EuropeAIFM Alternative investment fund

managers (directive)AIFMD Alternative investment fund

managers directiveAIRB Advanced internal rating-

based approach (credit risk)AMA Advanced Measurement Ap-

proaches (operational risk)AT1 Additional tier 1BAC Binary asset-or-nothing call

optionBAP Binary asset-or-nothing put

optionBCBS Basel Committee on Bank-

ing SupervisionBCC Binary cash-or-nothing call

option

BCP Binary cash-or-nothing putoption

BCVA Bilateral CVABIA Basic indicator approach

(operational risk)BIS Bank for International Set-

tlementBS Black-Scholes modelCAD Capital adequacy directiveCaR Capital-at-riskCB Conservation buffer (CET1)CBO Collateralized bond obliga-

tionCCB Countercyclical capital

buffer (CET1)CCF Credit conversion factorCCP Central counterparty clear-

ingCCR Counterparty credit riskCDF Cumulative distribution

functionCDS Credit default swapCDT Credit default trancheCDX Credit default index

xxxii

CDO Collateralized debt obliga-tion

CE Current exposureCEM Current exposure method

(CEM)CET1 Common equity tier 1CIR Cox-Ingersoll-Ross processCLO Collateralized loan obliga-

tionCMBS Commercial mortgage-

backed securityCoVaR Conditional value-at-riskCP Consultation paperCRA Credit rating agencyCRD Capital requirements direc-

tiveCRR Capital requirements regula-

tionCRM Comprehensive risk measureCVA Credit valuation adjustmentDIC Down-and-in call optionDIP Down-and-in put optionDOC Down-and-out call optionDOP Down-and-out put optionDVA Debit valuation adjustmentEAD Exposure at defaultEBA European Banking Author-

ityECB European Central BankEE Expected exposureEEE Effective expected exposureEEPE Effective expected positive

exposureEL Expected lossEMIR European market infrastruc-

ture regulationENE Expected negative exposureEPE Expected positive exposureESMA European Securities and

Markets AuthorityETF Exchange traded fundEVT Extreme value theoryFASB Financial Accounting Stan-

dards Board

FIRB Foundation internal rating-based approach (credit risk)

FRTB Fundamental review of thetrading book

FSB Financial Stability BoardFtD First-to-default swapFVA Founding valuation adjust-

mentGAAP Generally accepted account-

ing principles (US)GBM Geometric Brownian motionGEV Generalized extreme value

distributionGFC Global financial crisisGMM Generalized method of mo-

mentsGPD Generalized Pareto distribu-

tionHF Hedge fundHJM Heath-Jarrow-Morton

modelHLA Higher loss absorbencyHPP Homogeneous Poisson pro-

cessHQLA High-quality liquid assetsHY High yield entitiesIAIS International Association of

Insurance SupervisorsIAS International accounting

standardICAAP Internal capital adequacy

assessment processICP Insurance Core PrinciplesICPF Insurance companies and

pension fundsIFRS International financial re-

porting standardIG Investment grade entitiesIMA Internal model-based ap-

proach (market risk)IMF International Monetary

FundIMM Internal model method

(counterparty credit risk)

xxxiii

IOSCO International Organizationof Securities Commissions

IPP Integration by partsIRB Internal rating-based ap-

proach (credit risk)IRC Incremental risk chargeIRS Interest rate swapIRRBB Interest rate risk of the

banking bookISDA International Swaps and

Derivatives AssociationKIC Knock-in call optionKIP Knock-in put optionKOC Knock-out call optionKOP Knock-out put optionKRI Key risk indicatorLCG Linear congruential genera-

torLCR Liquidity coverage ratioLDA Loss distribution approachLDCE Loss data collection exerciseLEE Loan equivalent exposureLGD Loss given defaultLMM Libor market modelM Effective maturityMBS Mortgage-backed securityMC Monte CarloMCMC Monte Carlo Markov chainMCR Minimum capital require-

mentMDA Maximum domain of attrac-

tionMES Marginal expected shortfallMF Mutual fundMiFID Markets in financial instru-

ments directiveMiFIR Markets in financial instru-

ments regulationML Maximum likelihoodMM Method of momentsMMF Money market fundMPE Maximum peak exposureMPP Mixed Poisson processMtM Mark-to-market

NHPP Non-homogeneous Poissonprocess

NQD Negative quadrant depen-dence

NSFR Net stable funding ratioODE Ordinary differential equa-

tionORSA Own risk and solvency as-

sessmentOTC Over-the-counterOU Ornstein-Uhlenbeck processPD Probability of DefaultPDE Partial differential equationPDF Probability density functionPE Peak exposurePFE Potential future exposurePMF Probability mass functionPOT Peak over thresholdPQD Positive quadrant depen-

denceQIS Quantitative impact studyQMC Quasi-Monte CarloRBC Risk-based capital (US in-

surance)RMBS Residential mortgage-

backed securityRW Risk weightRWA Risk-weighted assetSA Standardized approach

(credit risk)SA-CCRStandardized approach

(counterparty credit risk)SBE Shadow banking entitySCR Solvency capital require-

mentSDE Stochastic differential equa-

tionSES Systemic expected shortfallSIFMA Securities Industry and Fi-

nancial Markets AssociationSIFI Systemically important fi-

nancial institutionSIV Structured investment vehi-

cleSLA Single loss approximation

xxxiv

SLN Shifted log-normal modelSM-CCR Standardized method

(counterparty credit risk)SMC Sequential Monte CarloSME Small and medium-sized en-

terprisesSMM Standardized measurement

method (market risk)SMM Swap market modelSRC Specific risk chargeSREP Supervisory review and eval-

uation processSRISK Systemic risk contributionSRP Supervisory review processSSM Single supervisory mecha-

nismSSM State space modelSVaR Stressed value-at-riskSPV Special purpose vehicleT1 Tier 1

T2 Tier 2TLAC Total loss absorbing capac-

ityTSA The standardized approach

(operational risk)UCITS Undertakings for collective

investment in transferablesecurities (directive)

UCVA Unilateral CVAUDVA Unilateral DVAUIC Up-and-in call optionUIP Up-and-in put optionUOC Up-and-out call optionUOP Up-and-out put optionUL Unexpected lossUVM Uncertain volatility modelVaR Value-at-riskXO Crossover (or sub-

investment grade) entities

Other scientific conventionsYYYY-MM-DD We use the international standard date notation where

YYYY is the year in the usual Gregorian calendar, MMis the month of the year between 01 (January) and 12(December), and DD is the day of the month between01 and 31.

$1 mn One million dollars.$1 bn One billion dollars.$1 tn One trillion dollars.

Chapter 1Introduction

The idea that risk management creates value is largely accepted today. How-ever, this has not always been the case in the past, especially in the financialsector (Stulz, 1996). Rather, it has been a long march marked by a number ofdecisive steps. In this introduction, we present an outline of the most impor-tant achievements from a historical point of view. We also give an overviewof the current financial regulation, which is a cornerstone in financial riskmanagement.

1.1 The need for risk managementThe need for risk management is the title of the first section of the leader-

ship book by Jorion (2007), who shows that risk management can be justifiedat two levels. At the firm level, risk management is essential for identifyingand managing business risk. At the industry level, risk management is a cen-tral factor for understanding and preventing systemic risk. In particular, thissecond need is the raison dtre of the financial regulation itself.

1.1.1 Risk management and the financial system

The concept of risk management has evolved considerably since its cre-ation, which is believed to be in the early fifties1. In November 1955, WayneSnider gave a lecture entitled The Risk Manager where he proposed creat-ing an integrated department responsible for risk prevention in the insuranceindustry (Snider, 1956). Some months later, Gallagher (1956) published anarticle to outline the most important principles of risk management and topropose the hiring of a full-time risk manager in large companies. For a longtime, risk management was systematically associated with insurance manage-ment, both from a practical point of view and a theoretical point of view. Forinstance, the book of Mehr and Hedges (1963) is largely dedicated to the fieldof insurance with very few applications to other industries. This is explained

1See Crockford (1982) or Snider (1991) for a retrospective view on the risk managementdevelopment.

1

2 Lecture Notes on Risk Management & Financial Regulation

by the fact that the collective risk model has helped to apply the mathemati-cal and statistical tools for measuring risk in insurance companies since 1930.A new discipline known as actuarial science has been developed at the sametime outside the other sciences and has supported the generalization of riskmanagement in the insurance industry.

Simultaneously, risk became an important field of research in economicsand finance. Indeed, Arrow (1964) made an important step by extending theArrow-Debreu model of general equilibrium in an uncertain environment2. Inparticular, he showed the importance of hedging and introduced the conceptof payoff. By developing the theory of optimal allocation for a universe offinancial securities, Markowitz (1952) pointed out that the risk of a financialportfolio can be diversified. These two concepts, hedging and diversification,together with insurance, are the main pillars of modern risk management.These concepts will be intensively used by academics in the 1960s and 1970s.In particular, Black and Scholes (1973) will show the interconnection betweenhedging and pricing problems. Their work will have a strong impact on thedevelopment of equity, interest rates, currency and commodity derivatives,which are today essential for managing the risk of financial institutions. Withthe Markowitz model, a new era had begun in portfolio management andasset pricing. First, Sharpe (1964) showed how risk premia are related tonon-diversifiable risks and developed the first asset pricing model. Then, Ross(1976) extended the CAPM model of Sharpe and highlighted the role of riskfactors in arbitrage pricing theory. These academic achievements will supportthe further development of asset management, financial markets and invest-ment banking.

In commercial and retail banking, risk management was not integrateduntil recently. Even though credit scoring models have existed since the fifties,they were rather designed for consumer lending, especially credit cards. Whenbanks used them for loans and credit issuances, they were greatly simplifiedand considered as a decision-making tool, playing a minor role in the finaldecision. The underlying idea was that the banker knew his client better thana statistical model could. However, Banker Trust introduced the concept ofrisk-adjusted return on capital or RAROC under the initiative of CharlesSanford in the late 1970s for measuring risk-adjusted profitability. Gene Guillmentions a memorandum dated February 1979 by Charles Sanford to the headof bank supervision at the Federal Reserve Board of New York that helps tounderstand the RAROC approach:

We agree that one banks book equity to assets ratio has lit-tle relevance for another bank with a different mix of businesses.Certain activities are inherently riskier than others and more riskcapital is required to sustain them. The truly scarce resource isequity, not assets, which is why we prefer to compare and measure

2This paper was originally presented in 1952 and was also published in Cahiers du CNRS(1953).

Introduction 3

businesses on the basis of return on equity rather than return onassets (Guill, 2009, page 10).

RAROC compares the expected return to the economic capital and has be-come a standard model for combining performance management and risk man-agement. Even if RAROC is a global approach for allocating capital betweenbusiness lines, it has been mainly used as a credit scoring model. Anothermilestone was the development of credit portfolio management when Vasicek(1987) adapted the structural default risk model of Merton (1974) to modelthe loss distribution of a loan portfolio. He then jointly founded KMV Cor-poration with Stephen Kealhofer and John McQuown, which specializes inquantitative credit analysis tools and is now part of Moodys Analytics.

In addition to credit risk, commercial and retail banks have to manageinterest rate and liquidity risks, because their primary activity is to do as-set, liquidity and maturity transformations. Typically, a commercial bank haslong-term and illiquid liabilities (loans) and short-term and liquid assets (de-posits). In such a situation, a bank faces a loss risk that can be partiallyhedged. This is the role of asset-liability management(ALM). But depositorsalso face a loss risk that is virtually impossible to monitor and manage. Con-sequently, there is an information asymmetry between banks and depositors.

In the banking sector, the main issue centered therefore around the depositinsurance. How can we protect depositors against the failure of the bank? The100% reserve proposal by Fisher in 1935 required banks to keep 100% of de-mand deposit accounts in cash or government-issued money like bills. Diamondand Dybvig (1983) argued that the mixing policy of liquid and illiquid assetscan rationally produce systemic risks, such as bank runs. A better way toprotect the depositors is to create a deposit insurance guaranteed by the gov-ernment. According to the Modigliani-Miller theorem on capital structure3,this type of government guarantee implied a higher cost of equity capital. Sincethe eighties, this topic has been highly written about (Admati and Hellwig,2014). Moreover, banks also differ from other companies, because they cre-ate money. Therefore, they are at the heart of the monetary policy. These twocharacteristics (implicit guarantee and money creation) imply that banks haveto be regulated and need regulatory capital. This is all the more valid withthe huge development of financial innovations, which has profoundly changedthe nature of the banking system and the risk.

1.1.2 The development of financial marketsThe development of financial markets has a long history. For instance, the

Chicago Board of Trade (CBOT) listed the first commodity futures contract

3Under some (unrealistic) assumptions, Modigliani and Miller (1958) showed that themarket value of a firm is not affected by how that firm is financed (by issuing stock ordebt). They also established that the cost of equity is a linear function of the firms leveragemeasured by its debt/equity ratio.

4 Lecture Notes on Risk Management & Financial Regulation

in 1864 (Carlton, 1984). Some authors even consider that the first organizedfutures exchange was the Dojima Rice Market in Osaka in the 18th century(Schaede, 1989). But the most important breakthrough came in the seventieswith two major financial innovations. In 1972, the Chicago Mercantile Ex-change (CME) launched currency futures contracts after the US had decidedto abandon the fixed exchange rate system of Bretton Woods (1946). The oilcrisis of 1973 and the need to hedge currency risk have considerably helpedin the development of this market. After commodity and currency contracts,interest rate and equity index futures have consistently grown. For instance,US Treasury bond, S&P 500, German Bund, and EURO STOXX 50 futureswere first traded in 1977, 1982, 1988 and 1998 respectively. Today, the Bundfutures contract is the most traded product in the world.

The second main innovation in the seventies concerned option contracts.The CBOT created the Chicago Board of Options (CBOE) in 1973, which wasthe first exchange specialized in listed stock call options. That same year, Blackand Scholes (1973) published their famous formula for pricing a Europeanoption. It has been the starting point of the intensive development of academicresearch concerning the pricing of financial derivatives and contingent claims.The works of Fisher Black, Myron Scholes and Robert Merton4 are all the moresignificant in that they consider the pricing problem in terms of risk hedging.Many authors had previously found a similar pricing formula, but Black andScholes introduced the revolutionary concept of the hedging portfolio. In theirmodel, they derived the corresponding dynamic trading strategy to hedgethe option contract, and the option price is therefore equivalent to the costof the hedging strategy. Their pricing method had a great influence on thedevelopment of the derivatives market and more exotic options, in particularpath-dependent options5.

Whereas the primary goal of options is to hedge a directional risk, they willbe largely used as underlying assets of investment products. In 1976, HayneLeland and Mark Rubinstein developed the portfolio insurance concept, whichallows for investing in risky assets while protecting the capital of the invest-ment. In 1980, they founded LOR Associates, Inc. with John OBrien and pro-posed structured investment products to institutional investors (Tufano andKyrillos, 1995). They achieved very rapid growth until the 1987 stock marketcrash6, and were followed by Wells Fargo, J.P. Morgan and Chase Manhattanas well as other investment banks. This period marks the start of financialengineering applied to structured products and the development of popular

4As shown by Bernstein (1972), the works of Black and Scholes cannot be dissociatedfrom the research of Merton (1973). This explains that they both received the 1997 NobelPrize in Economics for their option pricing model.

5See Box 1 for more information about th