From Financial Engineering to Risk Management. Radu Tunaru University of Kent, UK

Transcription

1 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

2 Contents Preface List of Notations List of Figures List of Tables vii ix xix xxiii 1. Introduction 1 2. Fundamental Relationships Introduction Present Value Constant Relative Risk Aversion Utility Risk versus Return: The Sharpe Ratio Issues related to non-normality The Sharpe ratio and negative returns ART Notes and Summary Model Risk in Interest Rate Modelling Introduction Short Rate Models Theory of Interest Rate Term Structure Expectations Hypothesis A reexamination of Log EH Reconciling the arguments and examples Yield Curve 39 xi

3 xii Model Risk in Financial Markets Parallel shift of a flat yield curve Another proof that the yield curve cannot be flat Deterministic maturity independcnt yields Consol modelling Interest Rate Forward Curve Modelling One-factor or Multi-factor models Notes and Summary Arbitrage Theory Introduction Transaction Costs Arbitrage Non-convergence frnancial gain process Distortion operator with arbitrage Notes and Summary Derivatives Pricing Under Uncertainty Introduction to Model Risk Parameter estimation risk Model selection risk Model identification risk Computational Implementation risk Model protocol risk Uncertain Volatility An option pricing model with uncertain volatility Option Pricing under Uncertainty in Complete Markets Parameter uncertainty Model uncertainty Numerical examples Accounting for parameter estimation risk in the Black-Scholes model Accounting for parameter estimation risk in the CEV model A Simple Measure of Parameter Uncertainty Risk Bayesian Option Pricing Modelling the future asset value under physical measure 100

4 Contents xiii Modelling the current asset value under a riskneutral measure Measuring Model Uncertainty Worst case risk measure Cont's Framework for Model Uncertainty An axiomatic approach A coherent measure of model risk A convex measure of model risk Notes and Summary Portfolio Selection under Uncertainty Introduction to Model Risk for Portfolio Analysis Bayesian Averaging for Portfolio Analysis Empirical Bayes priors Marginal likelihood calculations Portfolio Optimization Portfolio optimisation with stochastic interest rates Stochastic market price of risk Stochastic volatility Notes and Summary Probability Pitfalls of Financial Calculus Introduction Probability Distribution Functions and Density Functions Gaussian Distribution Moments Mean-median-mode inequality Distributions are not defined by moments Conditional expectation Stochastic Processes Infinite returns from finite variance processes Martingales Spurious Testing Spurious mean reversion Spurious regression Dependence Measures 140

5 xiv Model Risk in Financial Markets Problems with the Pearson linear correlation coefficient Pitfalls in detecting breakdown of linear correlation Copulas More general issues Dependence and Levy processes Notes and Summary Model Risk in Risk Measures Calculations Introduction Controlling Risk in Insurance Diversifikation Variance Coherent Distortion Risk Measures Value-at-Risk General observations Expected shortfall and expected tail loss Violations ratio Correct representation VaR may not be subadditive Artificial improvement of VaR Problems at long horizon Backtesting Uncertainty in risk estimates: A short overview Backtesting VaR Asymptotic Risk of VaR Normal VaR More general asymptotic Standard errors for VaR Exact confidence intervals for VaR Examples VaR at different significance levels Exact confidence intervals Extreme losses estimation and uncertainty Backtesting expected shortfall Notes and Summary 199

6 Contents xv 9. Parameter Estimation Risk Introduction Problems with Estimating Diffusions A brief review Parameter estimation for the Vasicek model Parameter estimation for the CIR model Problems with Estimation of Jump-Diffusion Models The Gaussian-Poisson jump-diffusion model ML Estimation under the Merton Model Inexistence of an unbiased estimator A Critique of Maximum Likelihood Estimation Bootstrapping Can Be Unreliable Too Notes and Summary Computational Problems Introduction Problems with Monte Carlo Variance Reduction Techniques Pitfalls in Estimating Greeks with Pathwise Monte Carlo Simulation Pitfall in Options Portfolio Calculation by Approximation Methods Transformations and Expansions Edgeworth expansion Computational issues for MLE Calculating the Implied Volatility Existence and uniqueness of implied volatility under Black-Scholes Approximation formulae for implied volatility An interesting example Incorrect Implied Volatility for Merton Model Notes and Summary Portfolio Selection Using the Sharpe Ratio Bayesian Calibration for Low Frequency Data Introduction 263

7 xvi Model Risk in Financial Markets 12.2 Problems in Pricing Derivatives for Assets with a Slow Business Time Choosing the Correct Auxiliary Vahles Empirical Exemplifications A mean-reversion model with predictability in the drift Data Augmentation MCMC Inference for the IPD model Derivatives Pricing Notes and Summary MCMC Estimation of Credit Risk Measures Introduction A Short Example Further Analysis Bayesian inference with Gibbs sampling Hierarchical Bayesian Models for Credit Risk Model specification of probabilitics of default Model estimation Standard&Poor's Rating Data Data description Hierarchical model for aggregated data Hierarchical time-series model Hierarchical model for disaggregated data Further Credit Modelling with MCMC Callibration Estimating the Transition Matrix MCMC estimation MLE estimation Notes and Summary Last But Not Least. Can We Avoid the Next Big Systemic Financial Crisis? Yes, We Can No, We Cannot A Non-technical Template for Model Risk Control Identify the type of model risk that may appear A guide for senior managers There is Still Work to Do 327

8 Contents xvii 15. Notations for the Study of MLE for CIR process 329 Bibliography 331 Index 351