Bayesian Risk Management. Sekerke Matt
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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