Table of Contents Title Page Copyright Preface Acknowledgments Chapter 1: Models for Discontinuous Markets Risk Models and Model Risk Time-Invariant Models and Crisis Bayesian Probability as a Means of Handling Discontinuity Time-Invariance and Objectivity Part One: Capturing Uncertainty in Statistical Models Chapter 2: Prior Knowledge, Parameter Uncertainty, and Estimation Estimation with Prior Knowledge: The Beta-Bernoulli Model Prior Parameter Distributions as Hypotheses: The Normal Linear Regression Model Decisions after Observing the Data: The Choice of Estimators Chapter 3: Model Uncertainty Bayesian Model Comparison Models as Nuisance Parameters Uncertainty in Pricing Models A Note on Backtesting Part Two: Sequential Learning with Adaptive Statistical Models Chapter 4: Introduction to Sequential Modeling Sequential Bayesian Inference Achieving Adaptivity via Discounting Accounting for Uncertainty in Sequential Models Chapter 5: Bayesian Inference in State-Space Time Series Models State-Space Models of Time Series Dynamic Linear Models Recursive Relationships in the DLM Variance Estimation Sequential Model Comparison Chapter 6: Sequential Monte Carlo Inference Nonlinear and Non-Normal Models State Learning with Particle Filters Joint Learning of Parameters and States Sequential Model Comparison Part Three: Sequential Models of Financial Risk Chapter 7: Volatility Modeling Single-Asset Volatility Volatility for Multiple Assets Chapter 8: Asset-Pricing Models and Hedging Derivative Pricing in the Schwartz Model Online State-Space Model Estimates of Derivative Prices Models for Portfolios of Assets Part Four: Bayesian Risk Management Chapter 9: From Risk Measurement to Risk Management Results Prior Information as an Instrument of Corporate Governance References Index End User License Agreement List of Illustrations Chapter 2: Prior Knowledge, Parameter Uncertainty, and Estimation Figure 2.1 Posterior Distribution of Success Probability: Random Data with s = 0.3 Figure 2.2 Posterior Distribution of Success Probability: Random Data with s = 0.5 Figure 2.3 Posterior Distribution of Success Probability: Random Data with s = 0.7 Chapter 4: Introduction to Sequential Modeling Figure 4.1 Sequential Inference on Bernoulli Data with Oscillatory Success Probability Figure 4.2 Sequential Inference on Bernoulli Data with Discount Factor = 0.99 Figure 4.3 Sequential Inference on Bernoulli Data with Discount Factor = 0.98 Figure 4.4 Sequential Inference on Time-Invariant Bernoulli Process with Discount Factor = 0.99 Figure 4.5 Sequential Inference on Time-Invariant Bernoulli Process with Discount Factor = 0.98 Figure 4.6 Time-Varying Coefficients Used to Generate Data for Regression Model Figure 4.7 Sequential Inference on Regression Intercept under Assumption of Time-Invariance Figure 4.8 Sequential Inference on Regression Beta under Assumption of TimeInvariance Figure 4.9 Sequential Inference on Regression Intercept with Discount Factor = 0.99 Figure 4.10 Sequential Inference on Regression Beta with Discount Factor = 0.99 Figure 4.11 Sequential Inference on Standard Error of Regression with Discount Factor = 0.99 Chapter 7: Volatility Modeling Figure 7.1 Rolling Standard-Deviation Estimates of S&P 500 Volatility for Three Choices of Window Length Figure 7.2 Exponentially Weighted Moving-Average Estimates of S&P 500 Volatility for Three Choices of Lambda Figure 7.3 GARCH(1,1) Estimates of S&P 500 Volatility for Three Choices of Window Length Figure 7.4 GARCH(1,1) Model Parameters: Daily Recalibration of S&P 500 Volatility Model Figure 7.5 S&P 500 Volatility Estimates from a Local-Level DLM with Discount Factor = 0.95 Figure 7.6 S&P 500 Volatility Estimates from a State-Space Volatility Model: LiuWest Filter with Discount Factor = 0.95 Figure 7.7 Posterior Model Probabilities: State-Space Volatility Model versus Rolling Standard-Deviation Models Figure 7.8 Posterior Model Probabilities: State-Space Volatility Model versus Rolling EWMA Models Figure 7.9 Posterior Model Probabilities: State-Space Volatility Model versus GARCH Models Figure 7.10 Posterior Model Probabilities: State-Space Volatility Model versus DLM Figure 7.11 Loadings of Major Stock Market Indices on Market, Size, and Value Factors Figure 7.12 Evolution of Market, Size, and Value Factor Volatilities Figure 7.13 Implied Correlations from Factor Stochastic Volatility Model, Discount Factor = 0.95 Figure 7.14 Implied Correlations from EWMA Stochastic Volatility Model, Lambda = 0.95 Figure 7.15 Comparison of Implied Correlations from Both Models Chapter 8: Asset-Pricing Models and Hedging Figure 8.1 Spot Price Estimates and One-Month Futures Price, Flexible Parameters (2%) 2000–2013 Figure 8.2 Market Price of Convenience Yield Risk, Fixed Parameters, 2000–2002 Figure 8.3 Long-Run Convenience Yield and Mean-Reversion Rates, Flexible Parameters (1%) 2000–2002 Figure 8.4 Long-Run Convenience Yield and Mean-Reversion Rates, Flexible Parameters (2%) 2000–2013 Figure 8.5 Spot Price Estimates and One-Month Futures Price, Fixed Parameters, 2012–2013 Figure 8.6 Long-Run Convenience Yield and Mean-Reversion Rates, Fixed Parameters, 2000–2002 Figure 8.7 Long- and Short-Term Interest Rate Estimates, Flexible Parameters (2%) 2000–2013 Figure 8.8 Long- and Short-Term Interest Rate Estimates, Flexible Parameters (1%) 2000–2002 Figure 8.9 State Variable Volatility Estimates, Flexible Parameters (2%) 2000– 2013 Figure 8.10 State Variable Correlation Estimates, Flexible Parameters (1%) 2012– 2013 Figure 8.11 Spot Price Estimates and One-Month Futures Price, Flexible Parameters (1%), 2000–2002 Figure 8.12 Convenience Yield State Variable Estimates, Fixed Parameters, 2012– 2013 Figure 8.13 Market Price of Convenience Yield Risk, Flexible Parameters (2%) 2000–2013 Figure 8.14 State Variable Volatility Estimates, Flexible Parameters (1%) 2012– 2013 Figure 8.15 Convenience Yield State Variable Estimates, Flexible Parameters (2%) 2000–2013 Figure 8.16 Long-Run Convenience Yield and Mean-Reversion Rates, Flexible Parameters (1%) 2012–2013 Figure 8.17 Long- and Short-Term Interest Rate Estimates, Flexible Parameters (1%) 2012–2013 Figure 8.18 Market Price of Convenience Yield Risk, Flexible Parameters (1%) 2000–2002 Figure 8.19 State Variable Volatility Estimates, Flexible Parameters (1%) 2000– 2002 Figure 8.20 Spot Price Estimates and One-Month Futures Price, Fixed Parameters, 2000–2002 Figure 8.21 State Variable Volatility Estimates, Fixed Parameters, 2000–2002 Figure 8.22 State Variable Correlation Estimates, Fixed Parameters, 2012–2013 Figure 8.23 State Variable Correlation Estimates, Fixed Parameters, 2000–2002 Figure 8.24 Market Price of Convenience Yield Risk, Flexible Parameters (1%) 2012–2013 Figure 8.25 Market Price of Convenience Yield Risk, Fixed Parameters, 2012–2013 Figure 8.26 Convenience Yield State Variable Estimates, Flexible Parameters (1%) 2012–2013 Figure 8.27 State Variable Correlation Estimates, Flexible Parameters (2%) 2000– 2013 Figure 8.28 Convenience Yield State Variable Estimates, Flexible Parameters (1%) 2000–2002 Figure 8.29 State Variable Volatility Estimates, Fixed Parameters, 2012–2013 Figure 8.30 Long- and Short-Term Interest Rate Estimates, Fixed Parameters, 2000–2002 Figure 8.31 Long- and Short-Term Interest Rate Estimates, Fixed Parameters, 2012–2013 Figure 8.32 Spot Price Estimates and One-Month Futures Price, Flexible Parameters (1%) 2012–2013 Figure 8.33 Long-Run Convenience Yield and Mean-Reversion Rates, Fixed Parameters, 2012–2013 Figure 8.34 Convenience Yield State Variable Estimates, Fixed Parameters, 2000– 2002 Figure 8.35 State Variable Correlation Estimates, Flexible Parameters (1%) 2000– 2002 List of Tables Chapter 7: Volatility Modeling Table 7.1 Exception Counts for 95% 1-Day VaR Calculated with Each Volatility Model Chapter 8: Asset-Pricing Models and Hedging Table 8.1 RMSEs for Schwartz Model Estimates The Wiley Finance series contains books written 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or DVD that is not included in the version you purchased, you may download this material at http://booksupport.wiley.com For more information about Wiley products, visit www.wiley.com Library of Congress Cataloging-in-Publication Data: Sekerke, Matt Bayesian risk management : a guide to model risk and sequential learning in financial markets / Matt Sekerke pages cm — (The Wiley finance series) Includes bibliographical references and index ISBN 978-1-118-70860-6 (cloth) – ISBN 978-1-118-74745-2 (epdf) – ISBN 978-1-118-74750-6 (epub) Finance—Mathematical models Financial risk management—Mathematical models Bayesian statistical decision theory I Title HG106.S45 2015 332′.041501519542–dc23 2015013791 Cover Design: Wiley Cover Image: Abstract background © iStock.com/matdesign24 Bond prices, closed-form expressions Bootstrap filter Capital asset pricing model (CAPM) Cholesky decomposition Classical statistics estimation maximum-likelihood ordinary least squares hypothesis testing “objectivity” of time series batch analysis ergodic stationarity Clifford-Hammersley theorem Commodity markets derivatives markets, Black model forward price volatility Conditional independence Conditional volatility, usage Conjugate prior families Convenience yield dynamics long-run level mean reversion rate spot prices, correlation with stochastic process for Corporate governance Correlation constancy, assumption estimation structure Covariance matrix, decompositions Credit crisis time-invariant models and Credit risk modeling DCC See Dynamic conditional correlations Decisions, loss (relationship) Derivative prices, online state-space model estimates estimation results discounting, usage Liu-West filter, usage prior information, impact Dimensionality, curse mitigation Dirichlet distribution Discounting in Beta-Bernoulli model in DLMs in Liu-West filter in normal linear regression model DLMs See Dynamic linear models Dynamic conditional correlations (DCC) method Dynamic linear models (DLMs) filtering recursion forecasting general form modular construction polynomial trend components predictive distributions regression components seasonal components sequential model comparisons smoothing recursion variance, multivariate case variance, univariate case Dynamic regression model Ergodic stationarity Estimators, selection Evolution equation Expectations, revisions Exponentially weighted moving average (EWMA) covariance matrix estimate correlations parameters estimates See also S&P 500 volatility; Volatility filter estimate usage formulation method stochastic volatility model, implied correlations volatility estimate classification comparison Factor stochastic volatility model, implied correlations online estimates Fama-French factors Feedback, usage Filtering adaptivity inference problem operation problem recursion computation Kalman gain state-space problems Forward-filtering backward sampling (FFBS) algorithm weaknesses Fourier-form seasonality Frequentist risk Front-office models statistical nature Gaussian distributions, mixtures Generalized autoregressive conditional heterosckedasticity (GARCH) models parameters See S&P 500 volatility recalibration state-space volatility model, contrast Generalized method of moments (GMM) Gibbs sampling Hannon-Quinn Information Criterion (HQIC) Hedging strategy, stability time-varying Schwartz model, usage Hyperbolic distributions Hyperparameters definition sequential updating Hypothesis tests, error rates Implied volatility Improper priors, usage Interest rate markets, Black model (usage) modeling one-factor Vasicek model, usage processes incorporation parameters, estimation Inverted-gamma distribution Inverted-Wishart distribution estimates hyperparameters prior Kalman filter batch estimation expression usage Kalman gain Knightian risk Latent factors, futures prices description (function) LIBOR Market Model Likelihood principle Likelihood ratio Bayes factors, contrast formation usage Linear regression model See Normal linear regression model Liu-West filter convergence discount factor, usage estimates of Schwartz model parameters and states parameter evolution in Logit models Loss function prior information, interaction with selection suppression use in decisions Marginal likelihood Market microstructure noise Markov Chain Monte Carlo (MCMC) batch analysis methods Markov process Markov switching process Model risk definition hedging metrics Models as nuisance parameters comparison completeness of model space complexity front-office models mixtures model-implied prices parametric selection, classical statistics specification use in forming expectations Model uncertainty Monte Carlo error Multiple asset volatility covariance matrix, decompositions exponentially weighted moving average (EWMA) inverted-Wishart estimates time-varying correlations Nonstationarity Normal linear regression model Bayesian analysis classical analysis estimation hypothesis testing parameter distributions sequential inference in Normals, location-scale mixture Nuisance parameters models Null hypothesis overrejection rejection Objective Bayesians Observation equation Observation matrix Ockham factor comparisons Odds ratios One-factor Cox-Ingersoll-Ross model One-factor Vasicek model Overfitting, identification Parameter uncertainty Parsimony Particle filters methods, representation parameter learning state learning Particle learning Particle set impoverishment Periodic differencing, impact Polynomial trends Posterior model probabilities defining state-space volatility model, dynamic linear models (contrast) state-space volatility model, GARCH models (contrast) state-space volatility model, rolling EWMA models (contrast) state-space volatility model, rolling standard deviation models (contrast) Posterior odds ratios Posterior parameter distribution Pricing error Pricing models Bayesian analysis observational error/overparametrization See Risk-neutral pricing models parameter uncertainty statistical models, relationship uncertainty Prior knowledge treatment in classical methods use in estimation Prior parameter distributions as hypothesis conjugate families hyperparameters in filtering problems integration role in marginal likelihoods Prior model probabilities defining distribution odds Probit models Proportional hazard models Recalibration Reduced-form models Reduction, problem Risk factors, identification governance measurement modeling, usage models tail risk uncertainty, Knightian distinction RiskMetrics system Risk-neutral parameters Risk-neutral pricing models Observational error/overparametrization Risk transfer markets Rolling-window-based methods Root mean square errors (RMSEs) SABR model Sampling See Forward filtering backward sampling; Gibbs sampling; Markov Chain Monte Carlo error, reduction Schwartz model derivative pricing estimation latency of factors prior elicitation state dynamics time-varying Schwartz model, usage Schwarz Information Criterion (SIC) Sequential Monte Carlo (SMC) inference approach estimates methods nonlinear and non-normal models sequential model comparison Shrinkage Single-asset volatility Bayesian models classical models, conditional volatility (usage) comparison GARCH models rolling-window-based methods SMC See Sequential Monte Carlo Smoothing distribution problem recursion See Dynamic linear models retrospective analysis S&P 500 volatility DLM estimates exponentially weighted moving average estimates GARCH (1,1) model estimates rolling standard deviation estimates state-space model estimates State equation State-space models basic notions filtering problem smoothing problem State-space volatility model dynamic linear models, contrast GARCH models, contrast rolling EWMA models, contrast rolling standard deviation models, contrast Stationary time series Statistical arbitrage hedge funds Statistical inference, points (development) Statistical models, pricing models (relationship) Storvik filter Structural breaks Sufficient statistics System matrix Tail events, occurrence Term-structure models Three-factor Schwartz model Time series analysis assumptions batch analysis perspective description ergodic stationarity filtering problem fundamental concepts inference, adjustment observations, joint distribution (assumption) sample moments, convergence smoothing problem state space models time-invariance Unconditional asset returns Uninformative priors Updates, visualization Value-at-risk (VaR) exceptions 1-day VaR, computation Volatility See Multiple asset volatility; Single-asset volatility; Volatility modeling Volatility modeling dynamic linear models, usage single-asset volatility state-space models, usage WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley's ebook EULA ... investors and their financial advisors Book topics range from portfolio management to e-commerce, risk management, financial engineering, valuation and financial instrument analysis, as well as... present and justify alternative tools to measure financial risk without assuming that time-invariant stochastic processes drive financial phenomena Discarding time-invariance as a modeling assumption... personal knowledge and understanding Bayesian Risk Management A Guide to Model Risk and Sequential Learning in Financial Markets MATT SEKERKE Copyright © 2015 by Matt Sekerke All rights reserved