A risk measurement and management framework that takes model risk seriously Most financial risk models assume the future will look like the past, but effective risk management depends on identifying fundamental changes in the marketplace as they occur. Bayesian Risk Management details a more flexible approach to risk management, and provides tools to measure financial risk in a dynamic market environment. This book opens discussion about uncertainty in model parameters, model specifications, and model-driven forecasts in a way that standard statistical risk measurement does not. And unlike current machine learning-based methods, the framework presented here allows you to measure risk in a fully-Bayesian setting without losing the structure afforded by parametric risk and asset-pricing models. * Recognize the assumptions embodied in classical statistics * Quantify model risk along multiple dimensions without backtesting * Model time series without assuming stationarity * Estimate state-space time series models online with simulation methods * Uncover uncertainty in workhorse risk and asset-pricing models * Embed Bayesian thinking about risk within a complex organization Ignoring uncertainty in risk modeling creates an illusion of mastery and fosters erroneous decision-making. Firms who ignore the many dimensions of model risk measure too little risk, and end up taking on too much. Bayesian Risk Management provides a roadmap to better risk management through more circumspect measurement, with comprehensive treatment of model uncertainty.
Preface ix Acknowledgments xiii CHAPTER 1 Models for Discontinuous Markets 1 Risk Models and Model Risk 2 Time-Invariant Models and Crisis 3 Ergodic Stationarity in Classical Time Series Analysis 5 Recalibration Does Not Overcome the Limits of a Time-Invariant Model 7 Bayesian Probability as a Means of Handling Discontinuity 8 Accounting for Parameter and Model Uncertainty 9 Responding to Changes in the Market Environment 12 Time-Invariance and Objectivity 14 PART ONE Capturing Uncertainty in Statistical Models CHAPTER 2 Prior Knowledge, Parameter Uncertainty, and Estimation 19 Estimation with Prior Knowledge: The Beta-Bernoulli Model 20 Encoding Prior Knowledge in the Beta-Bernoulli Model 21 Impact of the Prior on the Posterior Distribution 23 Shrinkage and Bias 24 Efficiency 25 Hyperparameters and Sufficient Statistics 30 Conjugate Prior Families 31 Prior Parameter Distributions as Hypotheses: The Normal Linear Regression Model 31 Classical Analysis of the Normal Linear Regression Model 32 Estimation 32 Hypothesis Testing 34 Bayesian Analysis of the Normal Linear Regression Model 35 Hypothesis Testing with Parameter Distributions 39 Comparison 41 Decisions after Observing the Data: The Choice of Estimators 42 Decisions and Loss 43 Loss and Prior Information 44 CHAPTER 3 Model Uncertainty 47 Bayesian Model Comparison 49 Bayes Factors 49 Marginal Likelihoods 50 Parsimony 52 Bayes Factors versus Information Criteria 53 Bayes Factors versus Likelihood Ratios 54 Models as Nuisance Parameters 55 The Space of Models 56 Mixtures of Models 58 Uncertainty in Pricing Models 58 Front-Office Models 59 The Statistical Nature of Front-Office Models 61 A Note on Backtesting 62 PART TWO Sequential Learning with Adaptive Statistical Models CHAPTER 4 Introduction to Sequential Modeling 67 Sequential Bayesian Inference 68 Achieving Adaptivity via Discounting 71 Discounting in the Beta-Bernoulli Model 73 Discounting in the Linear Regression Model 77 Comparison with the Time-Invariant Case 81 Accounting for Uncertainty in Sequential Models 83 CHAPTER 5 Bayesian Inference in State-Space Time Series Models 87 State-Space Models of Time Series 88 The Filtering Problem 90 The Smoothing Problem 91 Dynamic Linear Models 94 General Form 94 Polynomial Trend Components 95 Seasonal Components 96 Regression Components 98 Building DLMs with Components 98 Recursive Relationships in the DLM 99 Filtering Recursion 99 Smoothing Recursion 102 Predictive Distributions and Forecasting 104 Variance Estimation 105 Univariate Case 106 Multivariate Case 107 Sequential Model Comparison 108 CHAPTER 6 Sequential Monte Carlo Inference 111 Nonlinear and Non-Normal Models 113 Gibbs Sampling 113 Forward-Filtering Backward-Sampling 114 State Learning with Particle Filters 116 The Particle Set 117 A First Particle Filter: The Bootstrap Filter 117 The Auxiliary Particle Filter 119 Joint Learning of Parameters and States 120 The Liu-West Filter 122 Improving Efficiency with Sufficient Statistics 124 Particle Learning 125 Sequential Model Comparison 126 PART THREE Sequential Models of Financial Risk CHAPTER 7 Volatility Modeling 131 Single-Asset Volatility 132 Classical Models with Conditional Volatility 132 Rolling-Window-Based Methods 133 GARCH Models 136 Bayesian Models 138 Volatility Modeling with the DLM 139 State-Space Models of Stochastic Volatility 140 Comparison 141 Volatility for Multiple Assets 144 EWMA and Inverted-Wishart Estimates 144 Decompositions of the Covariance Matrix 148 Time-Varying Correlations 149 CHAPTER 8 Asset-Pricing Models and Hedging 155 Derivative Pricing in the Schwartz Model 156 State Dynamics 157 Describing Futures Prices as a Function of Latent Factors 157 Continuous- and Discrete-Time Factor Dynamics 158 Model-Implied Prices and the Observation Equation 161 Online State-Space Model Estimates of Derivative Prices 162 Estimation with the Liu-West Filter 163 Prior Information 165 Estimation Results 166 Estimation Results with Discounting 176 Hedging with the Time-Varying Schwartz Model 188 Connection with Term-Structure Models 190 Models for Portfolios of Assets 191 PART FOUR Bayesian Risk Management CHAPTER 9 From Risk Measurement to Risk Management 195 Results 195 Time Series Analysis without Time-Invariance 196 Preserving Prior Knowledge 196 Information Transmission and Loss 198 Bayesian State-Space Models of Time Series 199 Real-Time Metrics for Model Risk 200 Adaptive Estimates without Recalibration 202 Prior Information as an Instrument of Corporate Governance 204 References 207 Index 213
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