Bayesian Risk Management E-Book

Matt Sekerke

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Editions
• Sprache: Englisch

Seitenzahl: 322

Veröffentlichungsjahr: 2015

Beschreibung

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.

Leseprobe

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

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Guide

Cover

Table of Contents

Preface

Begin Reading

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 Time-Invariance

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: Liu-West 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

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Bayesian Risk Management

A Guide to Model Risk and Sequential Learning in Financial Markets

MATT SEKERKE

Copyright © 2015 by Matt Sekerke. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

Published simultaneously in Canada.

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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)

1. Finance—Mathematical models. 2. Financial risk management—Mathematical models. 3. Bayesian statistical decision

theory. I. Title.

HG106.S45 2015

332′.041501519542–dc23

2015013791

Cover Design: Wiley

Cover Image: Abstract background © iStock.com/matdesign24

Preface

Most financial risk models assume that the future will look like the past. They don't have to. This book sketches a more flexible risk-modeling approach that more fully recognizes our uncertainty about the future.

Uncertainty about the future stems from our limited ability to specify risk models, estimate their parameters from data, and be assured of the continuity between today's markets and tomorrow's markets. Ignoring any of these dimensions of model risk creates an illusion of mastery and fosters erroneous decision making. It is typical for financial firms to ignore all of these sources of uncertainty. Because they measure too little risk, they take on too much risk.

The core concern of this book is to 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 makes uncertainty about parameters, models, and forecasts accessible and irreducible in a way that standard statistical risk measurements do not. The constructive alternative offered here under the slogan Bayesian Risk Management is an online sequential Bayesian modeling framework that acknowledges all of these sources of uncertainty, without giving up the structure afforded by parametric risk models and asset-pricing models.

Following an introductory chapter on the far-reaching consequences of the time-invariance assumption, Part One of the book shows where Bayesian analysis opens up uncertainty about parameters and models in a static setting. Bayesian results are compared to standard statistical results to make plain the strong assumptions embodied in classical, “objective” statistics. Chapter 2 begins by discussing prior information and parameter uncertainty in the context of the binomial and normal linear regression models. I compare Bayesian results to classical results to show how the Bayesian approach nests classical statistical results as a special case, and relate prior distributions under the Bayesian framework to hypothesis tests in classical statistics as competing methods of introducing nondata information. Chapter 3 addresses uncertainty about models and shows how candidate models may be compared to one another. Particular focus is given to the relationship between prior information and model complexity, and the manner in which model uncertainty applies to asset-pricing models.

Part Two extends the Bayesian framework to sequential time series analysis. Chapter 4 introduces the practice of discounting as a means of creating adaptive models. Discounting reflects uncertainty about the degree of continuity between the past and the future, and prevents the accumulation of data from destroying model flexibility. Expanding the set of available models to entertain multiple candidate discount rates incorporates varying degrees of memory into the modeling enterprise, avoiding the need for an a priori view about the rate at which market information decays. Chapters 5 and 6 then develop the fundamental tools of sequential Bayesian time series analysis: dynamic linear models and sequential Monte Carlo (SMC) models. Each of these tools incorporates parameter uncertainty, model uncertainty, and information decay into an online filtering framework, enabling real-time learning about financial market conditions.

Part Three then applies the methods developed in the first two parts to the estimation of volatility in Chapter 7 and the estimation of a commodity forward curve under the risk-neutral measure subject to arbitrage restrictions in Chapter 8. My goal here is to show the applicability of the methods developed to two problems which represent two extremes in our level of modeling knowledge. Additional applications are also possible. In Chapter 8 especially, I discuss how other common models may be reformulated and estimated using the same sequential Bayesian toolkit.

Chapter 9, the sole chapter of Part Four, synthesizes the results of the first three parts and begins the transition from a risk measurement framework based on Bayesian principles to a properly Bayesian risk management. I argue that the sequential Bayesian framework offers a coherent mechanism for organizational learning in environments characterized by incomplete information. Bayesian models allow senior management to make clear statements of risk policy and test elements of strategy against market outcomes in a direct and rigorous way. One may wish to begin reading at the final chapter: A glimpse of the endgame could provide useful orientation while reading the rest of the text.

The genesis of this book is multifold. As an undergraduate student in economics, I was impressed by the divide between the information-processing capacity assumed for individuals and firms in economic theory and the manner in which empirical individuals and firms actually learn. While economics provided many powerful results for the ultimate market outcomes, the field had less to say about the process by which equilibria were reached, or the dynamic stability of equilibrium given large perturbations from fixed points. Given a disruption to the economy, it seemed as though economic agents would have to find their way back to equilibrium over time, and on the basis of incomplete and uncertain information. With the notable exception of Fisher (1983) and some works by the Austrian economists, I quickly discovered that the field furnished few ready answers.

As I began my career consulting in economic litigations, I had two further experiences that find their theme in this book. The first involved litigation over a long-term purchase contract, which included a clause for renegotiation in the event that a “structural change” in the subject market had occurred. In working to find econometric evidence for such a structural change, I was struck, on the one hand, by the dearth of methods for identifying structural change in a market as it happened; identification seemed to be possible mainly as a forensic exercise, though there were obvious reasons why a firm would want to identify structural change in real time. On the other hand, after applying the available methods to the data, it seemed that it was more likely than not to find structural change wherever one looked, particularly in financial time series data at daily frequency. If structural change could occur at any time, without the knowledge of those who have vested interests in knowing, the usual methods of constructing forecasts with classical time series models seemed disastrously prone to missing the most important events in a market. Worse, their inadequacy would not become evident until it was probably too late.

The second experience was my involvement in the early stages of litigation related to the credit crisis. In these lawsuits, a few questions were on everyone's mind. Could the actors in question have seen significant changes in the market coming? If so, at what point could they have known that a collapse was imminent? If not, what would have led them to believe that the future was either benign or unknowable? The opportunity to review confidential information obtained in the discovery phase of these litigations provided innumerable insights into the inner workings of the key actors with respect to risk measurement, risk management, and financial instrument valuation. I saw two main things. First, there was an overwhelming dependence on front-office information—bid sheets, a few consummated secondary-market trades, and an overwhelming amount of “market color,” the industry term for the best rumor and innuendo on offer—and almost no dependence on middle-office modeling. Whereas certain middle-office modeling efforts could have reacted to changes in market conditions, the traders on the front lines would not act until they saw changes in traded prices. Second, there were interminable discussions about how to weigh new data on early-stage delinquencies, default rates, and home prices against historical data. Instead of asking whether the new data falsified earlier premises on which expectations were built, discussions took place within the bounds of the worst-known outcomes from history, with the unstated assurance that housing market phenomena were stable and mean-reverting overall. Whatever these observations might imply about the capacity of the actors involved, it seemed that a better balance could be struck between middle-office risk managers and front-office traders, and that gains could be had by making the expectations of all involved explicit in the context of models grounded in the relevant fundamentals.

However, it was not until I began my studies at the University of Chicago that these themes converged around the technical means necessary to make them concrete. Nick Polson's course in probability theory was a revelation, introducing the Bayesian approach to probability within the context of financial markets. Two quarters of independent study with him followed immediately in which he introduced me to the vanguard of Bayesian thinking about time series. A capstone elective on Bayesian econometrics with Hedibert Lopes provided further perspective and rigor. His teaching was a worthy continuation of a tradition at the University of Chicago going back to Arnold Zellner.

The essay offered here brings these themes together by offering sequential Bayesian inference as the technical integument, which allows an organization to learn in real time about “structural change.” It is my provisional and constructive answer to how a firm can behave rationally in a dynamic environment of incomplete information.

My intended audience for this book includes senior management, traders and risk managers in banking, insurance, brokerage, and asset management firms, among other players in the wider sphere of finance. It is also addressed to regulators of financial firms who are increasingly concerned with risk measurement and risk governance. Advanced undergraduate and graduate students in economics, statistics, finance, and financial engineering will also find much here to complement and challenge their other studies within the discipline. Those readers who have spent substantial time modeling real data will benefit the most from this book.

Because it is an essay and not a treatise or a textbook, the book is pitched at a relatively mature mathematical level. Readers should already be comfortable with probability theory, classical statistics, matrix algebra, and numerical methods in order to follow the exposition and, more important to appreciate the recalcitrance of the problems addressed. At the same time, I have sought to avoid writing a mathematical book in the usual sense. Math is used mainly to exemplify, calculate, and make a point rather than to reach a painstaking level of rigor. There is also more repetition than usual so the reader can keep moving ahead, rather than constantly referring to previous formulas, pages, and chapters. In almost every case, I provide all steps and calculations in an argument, hoping to provide clarity without becoming tedious, and to avoid referring the reader to a list of hard-to-locate materials for the details necessary to form an understanding. That said, I hardly expect to have carried out my self-imposed mandates perfectly and invite readers to email me at [email protected] with typos and other comments.

Acknowledgments

It is hard to express my gratitude to Nick Polson adequately. Certainly, this book would not exist without him. His intellectual fingerprints are all over it, and I hope I have proven myself a worthy student. More than any lecture or guidance in the thicket of the statistical literature, the many hours spent with Nick (more often than not, over burgers at Medici on 57th in Hyde Park) thinking through the ways in which people attempt to learn about financial markets from data helped me not only to grasp the Bayesian manner of thinking about probability but also to gain the confidence necessary to test it against the prevailing orthodoxy. His intuitive way of proceeding and his fantastic sense of humor also made it great fun to set off in an exciting new field. For all I have absorbed from him, I am still overwhelmed by the many brilliant new directions of his thinking, and will have much to learn from him for many years to come.

This book also bears traces of many years working with Steve Hanke, first as his research assistant and as an ongoing collaborator in writing and consulting. Professor Hanke first introduced me to the importance of time and uncertainty in economic analysis by encouraging me to read the Austrians, and especially Hayek. These pages are part of an ongoing process of coming to grips with the wealth of ideas to which Professor Hanke exposed me. Professor Hanke has also supported my writing efforts from the very beginning and continues to be a source of encouragement and wise counsel to me in virtually all matters of importance.

Chris Culp has been incredibly supportive to me for nearly 15 years as a mentor and a colleague. His boundless productive energy and generosity of spirit have been an inspiration to me from the beginning. (“Ask Culp” was one of the more common prescriptions heard in Professor Hanke's office.) The insightful ways in which Chris connects problems in risk management with fundamental problems in economics and corporate finance were decisive in sparking my interest in the subject. More directly, without his introduction to Bill Falloon at Wiley, this project would have remained in the realm of wishful thinking.

Bill Falloon has shown me a staggering degree of support with this book and more generally in developing as an author. I look forward to more projects with him and his fantastic team, especially Meg Freeborn, who kept my developing manuscript on the rails despite multiple interruptions and radical, wholesale revisions.

Most important, I am grateful for the unflagging support of my incredible wife, Nancy. She kept me going on this project whenever the going got tough, and patiently auditioned my many attempts to distill my thesis to a simple and forthright message. Whatever clarity may be found in a book dense with mathematics and quantitative finance is probably due to her. All of the shortcomings of the book are, however, mine alone.

Chapter 1Models for Discontinuous Markets

The broadening and deepening of markets for risk transfer has marked the development of financial services perhaps more than any other trend. The past 30 years have witnessed the development of secondary markets for a wide variety of financial assets and the explosion of derivative instruments made possible by financial engineering. The expansion of risk transfer markets has liquefied and transformed the business of traditional financial firms such as banks, asset managers, and insurance companies. At the same time, markets for risk transfer have enabled nontraditional players to enter financial services businesses, invigorating competition, driving down prices, and confounding the efforts of regulators. Such specialist risk transfer firms occupy a number of niches in which they can outperform their more diversified counterparts in the regulated financial system by virtue of their specialized knowledge, transactional advantages, and superior risk management.

For all firms operating in risk transfer markets, traditional and nontraditional alike, the ability to create, calibrate, deploy, and refine risk models is a core competency. No firm, however specialized, can afford to do without models that extract information from market prices, measure the sensitivity of asset values to any number of risk factors, or forecast the range of adverse outcomes that might impact the firm's financial position.

The risk that a firm's models may fail to capture shifts in market pricing, risk sensitivities, or the mix of the firm's risk exposures is thus a central operational risk for any financial services business. Yet many, if not most, financial services firms lack insight into the probabilistic structure of risk models and the corresponding risk of model failures. My thesis is that most firms lack insight into model risk because of the way they practice statistical modeling. Because generally accepted statistical practice provides thin means for assessing model risk, alternative methods are needed to take model risk seriously. Bayesian methods allow firms to take model risk seriously—hence a book on Bayesian risk management.

Risk Models and Model Risk

Throughout this book, when I discuss risk models, I will be talking about parametric risk models. Parametric risk models are attempts to reduce the complexity inherent in large datasets to specific functional forms defined completely by a relatively low-dimensional set of numbers known as parameters. Nonparametric risk models, by contrast, rely exclusively on the resampling of empirical data, so no reduction of the data is attempted or accomplished. Such models ask: Given the risk exposures I have today, what is the distribution of outcomes I can expect if the future looks like a random draw from some history of market data? Nonparametric risk models lack model specification in the way we would normally understand it, so that there is no risk of misspecification or estimation error by construction. Are such models therefore superior? Not at all. A nonparametric risk model cannot represent any outcome different from what has happened, including any outcomes more extreme than what has already happened. Nor can it furnish any insight into the ultimate drivers of adverse risk outcomes. As a result, nonparametric risk models have limited use in forecasting, though they can be useful as a robustness check for a parametric risk model.