Model Risk in Financial Markets From Financial Engineering to Risk Management Radu Tunaru University of Kent, UK \Yp World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI
Contents Preface List of Notations List of Figures List of Tables vii ix xix 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 ART 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 xi
xii Model Risk in Financial Markets 3.4.1 Parallel shift of a flat yield curve 39 3.4.2 Another proof that the yield curve cannot be flat 40 3.4.3 Deterministic maturity independcnt 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 frnancial 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
Contents xiii 5.5.2 Modelling the current asset value under a riskneutral 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 7.4 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
xiv Model Risk in Financial Markets 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 Diversifikation 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
Contents xv 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 CIR 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
xvi Model Risk in Financial Markets 12.2 Problems in Pricing Derivatives for Assets with a Slow Business Time 264 12.3 Choosing the Correct Auxiliary Vahles 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 probabilitics 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 Callibration... 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
Contents xvii 15. Notations for the Study of MLE for CIR process 329 Bibliography 331 Index 351