Model Risk In Financial Markets: From Financial Engineering To Risk Management: From Financial Engineering to Risk Management

Model Risk In Financial Markets: From Financial Engineering To Risk Management: From Financial Engineering to Risk Management

by Radu Tunaru

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Overview

The financial systems in most developed countries today build up a large amount of model risk on a daily basis. However, this is not particularly visible as the financial risk management agenda is still dominated by the subprime-liquidity crisis, the sovereign crises, and other major political events. Losses caused by model risk are hard to identify and even when they are internally identified, as such, they are most likely to be classified as normal losses due to market evolution.

Model Risk in Financial Markets: From Financial Engineering to Risk Management seeks to change the current perspective on model innovation, implementation and validation. This book presents a wide perspective on model risk related to financial markets, running the gamut from financial engineering to risk management, from financial mathematics to financial statistics. It combines theory and practice, both the classical and modern concepts being introduced for financial modelling. Quantitative finance is a relatively new area of research and much has been written on various directions of research and industry applications. In this book the reader gradually learns to develop a critical view on the fundamental theories and new models being proposed.


Contents:
  • Introduction
  • Fundamental Relationships
  • Model Risk in Interest Rate Modelling
  • Arbitrage Theory
  • Derivatives Pricing Under Uncertainty
  • Portfolio Selection Under Uncertainty
  • Probability Pitfalls of Financial Calculus
  • Model Risk in Risk Measures Calculations
  • Parameter Estimation Risk
  • Computational Problems
  • Portfolio Selection Using Sharpe Ratio
  • Bayesian Calibration for Low Frequency Data
  • MCMC Estimation of Credit Risk Measures
  • Last But Not Least. Can We Avoid the Next Big Systemic Financial Crisis?
  • Notations for the Study of MLE for CIR Process

Readership: Graduate students, researchers, practitioners, senior managers in financial institutions and hedge-funds, regulators and risk managers, who are keen to understand the pitfalls of financial modelling, and also those who are looking for a career in model validation, product control and risk management functions.
Key Features:
  • Some innovative results are presented for the first time
  • Covers a wide range of models, results and applications in financial markets to demonstrate that model risk is generally spread

Product Details

ISBN-13: 9789814663427
Publisher: World Scientific Publishing Company, Incorporated
Publication date: 06/08/2015
Sold by: Barnes & Noble
Format: NOOK Book
Pages: 384
File size: 12 MB
Note: This product may take a few minutes to download.

Table of Contents

Preface vii

List of Notations ix

List of Figures xix

List of Tables xxiii

1 Introduction 1

2 Fundamental Relationships 11

2.1 Introduction 11

2.2 Present Value 11

2.3 Constant Relative Risk Aversion Utility 12

2.4 Risk versus Return: The Sharpe Ratio 14

2.4.1 Issues related to non-normality 14

2.4.2 The Sharpe ratio and negative returns 15

2.5 APT 16

2.6 Notes and Summary 18

3 Model Risk in Interest Rate Modelling 21

3.1 Introduction 21

3.2 Short Rate Models 22

3.3 Theory of Interest Rate Term Structure 31

3.3.1 Expectations Hypothesis 31

3.3.2 A reexamination of Log EH 36

3.3.3 Reconciling the arguments and examples 38

3.4 Yield Curve 39

3.4.1 Parallel shift of a fiat yield curve 39

3.4.2 Another proof that the yield curve cannot be fiat 40

3.4.3 Deterministic maturity independent yields 41

3.4.4 Consol modelling 42

3.5 Interest Rate Forward Curve Modelling 45

3.6 One-factor or Multi-factor models 48

3.7 Notes and Summary 51

4 Arbitrage Theory 55

4.1 Introduction 55

4.2 Transaction Costs 56

4.3 Arbitrage 58

4.3.1 Non-convergence financial gain process 58

4.3.2 Distortion operator with arbitrage 60

4.4 Notes and Summary 63

5 Derivatives Pricing Under Uncertainty 65

5.1 Introduction to Model Risk 65

5.1.1 Parameter estimation risk 68

5.1.2 Model selection risk 70

5.1.3 Model identification risk 71

5.1.4 Computational implementation risk 74

5.1.5 Model protocol risk 75

5.2 Uncertain Volatility 77

5.2.1 An option pricing model with uncertain volatility 78

5.3 Option Pricing under Uncertainty in Complete Markets 80

5.3.1 Parameter uncertainty 81

5.3.2 Model uncertainty 86

5.3.3 Numerical examples 87

5.3.4 Accounting for parameter estimation risk in the Black-Scholes model 88

5.3.5 Accounting for parameter estimation risk in the CEV model 92

5.4 A Simple Measure of Parameter Uncertainty Risk 97

5.5 Bayesian Option Pricing 99

5.5.1 Modelling the future asset value under physical measure 100

5.5.2 Modelling the current asset value under a risk- neutral measure 101

5.6 Measuring Model Uncertainty 102

5.6.1 Worst case risk measure 103

5.7 Cont's framework for Model Uncertainty 104

5.7.1 An axiomatic approach 104

5.7.2 A coherent measure of model risk 106

5.7.3 A convex measure of model risk 109

5.8 Notes and Summary 113

6 Portfolio Selection under Uncertainty 115

6.1 Introduction to Model Risk for Portfolio Analysis 115

6.2 Bayesian Averaging for Portfolio Analysis 118

6.2.1 Empirical Bayes priors 119

6.2.2 Marginal likelihood calculations 120

6.3 Portfolio Optimization 121

6.3.1 Portfolio optimisation with stochastic interest rates 123

6.3.2 Stochastic market price of risk 125

6.3.3 Stochastic volatility 126

6.4 Notes and Summary 127

7 Probability Pitfalls of Financial Calculus 129

7.1 Introduction 129

7.2 Probability Distribution Functions and Density Functions 130

7.3 Gaussian Distribution 131

7.1 Moments 133

7.4.1 Mean-median-mode inequality 133

7.4.2 Distributions are not defined by moments 134

7.4.3 Conditional expectation 135

7.5 Stochastic Processes 136

7.5.1 Infinite returns from finite variance processes 136

7.5.2 Martingales 137

7.6 Spurious Testing 138

7.6.1 Spurious mean reversion 138

7.6.2 Spurious regression 139

7.7 Dependence Measures 140

7.7.1 Problems with the Pearson linear correlation coefficient 140

7.7.2 Pitfalls in detecting breakdown of linear-correlation 141

7.7.3 Copulas 145

7.7.4 More general issues 152

7.7.5 Dependence and Levy processes 153

7.8 Notes and Summary 154

8 Model Risk in Risk Measures Calculations 157

8.1 Introduction 157

8.2 Controlling Risk in Insurance 158

8.2.1 Diversification 158

8.2.2 Variance 159

8.3 Coherent Distortion Risk Measures 160

8.4 Value-at-Risk 163

8.4.1 General observations 163

8.4.2 Expected shortfall and expected tail loss 168

8.4.3 Violations ratio 168

8.4.4 Correct representation 171

8.4.5 VaR may not be subadditive 175

8.4.6 Artificial improvement of VaR 176

8.4.7 Problems at long horizon 177

8.5 Backtesting 179

8.5.1 Uncertainty in risk estimates: A short overview 179

8.5.2 Backtesting VaR 181

8.6 Asymptotic Risk of VaR 187

8.6.1 Normal VaR 187

8.6.2 More general asymptotic standard errors for VaR 191

8.6.3 Exact confidence intervals for VaR 192

8.6.4 Examples 193

8.6.5 VaR at different significance levels 195

8.6.6 Exact confidence intervals 196

8.6.7 Extreme losses estimation and uncertainty 197

8.6.8 Backtesting expected shortfall 199

8.7 Notes and Summary 199

9 Parameter Estimation Risk 205

9.1 Introduction 205

9.2 Problems with Estimating Diffusions 206

9.2.1 A brief review 206

9.2.2 Parameter estimation for the Vasicek model 208

9.2.3 Parameter estimation for the CTR model 212

9.3 Problems with Estimation of Jump-Diffusion Models 215

9.3.1 The Gaussian-Poisson Jump-diffusion model 215

9.3.2 ML Estimation under the Merton Model 216

9.3.3 Inexistence of an unbiased estimator 218

9.4 A Critique of Maximum Likelihood Estimation 218

9.5 Bootstrapping Can Be Unreliable Too 221

9.6 Notes and Summary 224

10 Computational Problems 227

10.1 Introduction 227

10.2 Problems with Monte Carlo Variance Reduction Techniques 228

10.3 Pitfalls in Estimating Greeks with Pathwise Monte Carlo Simulation 232

10.4 Pitfall in Options Portfolio Calculation by Approximation Methods 239

10.5 Transformations and Expansions 242

10.5.1 Edgeworth expansion 242

10.5.2 Computational issues for MLE 244

10.6 Calculating the Implied Volatility 245

10.6.1 Existence and uniqueness of implied volatility under Black-Scholes 245

10.6.2 Approximation formulae for implied volatility 248

10.6.3 An interesting example 249

10.7 Incorrect Implied Volatility for Merton Model 251

10.8 Notes and Summary 253

11 Portfolio Selection Using the Sharpe Ratio 257

12 Bayesian Calibration for Low Frequency Data 263

12.1 Introduction 263

12.2 Problems in Pricing Derivatives for Assets with a Slow Business Time 264

12.3 Choosing the Correct Auxiliary Values 266

12.4 Empirical Exemplifications 268

12.4.1 A mean-reversion model with predictability in the drift 268

12.4.2 Data augmentation 269

12.5 MCMC Inference for the IPD model 270

12.6 Derivatives Pricing 276

12.7 Notes and Summary 281

13 MCMC Estimation of Credit Risk Measures 283

13.1 Introduction 283

13.2 A Short Example 285

13.3 Further Analysis 290

13.3.1 Bayesian inference with Gibbs sampling 291

13.4 Hierarchical Bayesian Models for Credit Risk 294

13.4.1 Model specification of probabilities of default 295

13.4.2 Model estimation 297

13.5 Standard&Poor's Rating Data 301

13.5.1 Data description 301

13.5.2 Hierarchical model for aggregated data 302

13.5.3 Hierarchical time-series model 308

13.5.4 Hierarchical model for disaggregated data 309

13.6 Further Credit Modelling with MCMC Calibration 313

13.7 Estimating the Transition Matrix 316

13.7.1 MCMC estimation 316

13.7.2 MLE estimation 318

13.8 Notes and Summary 319

14 Last But Not Least. Can We Avoid the Next Big Systemic Financial Crisis? 321

14.1 Yes, We Can 321

14.2 No, We Cannot 322

14.3 A Non-technical Template for Model Risk Control 324

14.3.1 Identify the type of model risk that may appear 325

14.3.2 A guide for senior managers 326

14.4 There is Still Work to Do 327

15 Notations for the Study of MLE for CIR process 329

Bibliography 331

Index 351

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