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Bayesian Risk Management: A Guide to Model Risk and Sequential Learning in Financial Markets



Bayesian Risk Management: A Guide to Model Risk and Sequential Learning in Financial Markets

Matt Sekerke

ISBN: 978-1-118-74750-6 August 2015 240 Pages

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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