Finance, Economics, and Mathematics

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Library of Congress Cataloging-in-Publication Data

Vasicek, Oldrich Alfons, 1941- author.

[Essays. Selections]

Finance, economics, and mathematics / Oldrich Alfons Vasicek.

pages cm

Includes index.

ISBN 978-1-119-12220-3 (cloth), ISBN 978-1-119-18620-5 (epub), ISBN 978-1-119-18621-2 (ePDF)

1. Finance. 2. Economics. 3. Finance, Mathematical. I. Title.

HG173.V36 2015

332–dc23

2015026777

Cover Design: Wiley

Cover Image: Tech background © RomanOkopny / iStockphoto

Foreword

About a half century ago began a remarkable intellectual revolution which transformed the field of finance from a collection of anecdotes and accounting identities into a scientific discipline with general principles and rigorous empirical assessments of its hypotheses. In the ensuing decades, finance science was both shaped, and shaped by, the extraordinary transformation of the practice of finance.

Oldrich Vasicek was one of the pioneer scientists to provide foundational contributions in the pricing and risk measurement of fixed income securities–default-free bonds that form the term structure of interest rates, and credit-risky bonds and loans–and to implement them in practice. Here we have the collection of the original papers and articles of his intellectual history of thought–with the content of the models still applicable today.

Whether a master researcher, experienced finance professional or novice student of finance, you are in for a treat.

Bon appétit!

Robert C. Merton

Distinguished Professor of Finance

MIT Sloan School of Management

1997 Nobel Memorial Prize in Economic Sciences

Preface

This book is a selection of my published and unpublished papers written between 1968, when I came to the United States, and 2014. The ideas for them came to me at different times and in different circumstances. I have worked as a theoretical mathematician in the Czech Academy of Sciences, as a vice-president in a large U.S. bank, as an external consultant in a small investment technology software developer, as a full-time professor and a visiting professor at several universities. I have been a founding partner and a managing director in a startup, a special adviser for a large bond rating firm and, since 2010, an independent researcher not associated with any institution.

I cannot say which was the most satisfying. It has all been fun, and still is. I have met, and in many cases worked with, many extremely interesting and capable people. These meetings, discussions, and collaboration have meant a lot to me.

I have usually worked somewhat outside the organizational structure. I have never had a subordinate, which was exactly as I wanted it. And I have been very fortunate that I rarely found much discrepancy between what I wanted to work on, and between what seemed to be needed at the time.

I have enjoyed collaborating with other people. A number of the papers in this collection are joint works. I would like to take this opportunity to thank my coauthors: Gifford Fong, my one-time employer and long-time friend, with whom I wrote six of the papers here; John McQuown, who hired me to Wells Fargo Bank on my arrival to the USA, with whom, along with Stephen Kealhofer, we founded the KMV Corporation years later, and with whom I have been friends the whole time; my colleague at the University of Rochester, the late Professor Julian Keilson; and the tireless and resourceful Professor Helyette Geman, who arranged my pleasant stay at ESSEC. I appreciate their input, insights, and joint work.

I would like to express my thanks and appreciation to many people. There are some, however, that I cannot but mention specifically: my father, JUDr. Oldřich Vašíček, who encouraged and supported my interest in mathematics from early childhood; the late Professor Alois Apfelbeck, much feared for his first-year analysis class, who singlehandedly, and successfully, opposed the Party authorities from expelling me from the university; John McQuown, who introduced me to finance; Professor Richard Roll, whose critique of an earlier draft of my 1977 paper made me to rewrite it for much improvement; and Professor Robert Merton, one of the smartest yet gracious people I know. To these and many other people who helped me by advice, debate, collaboration, or example, I wish to give my gratitude.

Oldrich Alfons Vasicek

August 2014

Part One

Efforts and Opinions

A lot of attention goes to the pricing of various complicated debt instruments because those instruments are becoming more common. That's needed short-term. I think long-term it's important to understand the more basic problem we were talking about before — what exactly goes into the pricing of the straight debt of a firm. That's the economics of credit, not the valuation of assorted derivatives. There is too much mathematics and too little economics in finance nowadays. That may sound funny coming from a mathematician, but nevertheless that's my opinion. We must not forget that the subject of finance is economic decisions". (page 15)

Chapter 1

Introduction to Part I

Risk, 72-73, December 2002

The past fifty years or so have been a time of great bloom in the field of finance. This period has seen the birth of concepts such as variance as a quantitative definition of risk, portfolio diversification as a means of controlling risk, portfolio optimization in the mean/variance framework, expected utility maximization as an investment and consumption decision making criterion. These notions were applied in the development of Capital Asset Pricing Model to describe the market equilibrium, to the concepts of systematic and specific risks and the introduction of asset beta. We have witnessed the revolution brought by the theory of options pricing. We have seen the appearance of the general principle of asset pricing as the present value of the cash flows expected under the risk-neutral probability measure. We have seen the development of the theory of the term structure of interest rates and the pricing of interest rate derivatives.

These theoretical developments have been accompanied by equally exciting changes in investment practices and indeed in the nature of capital markets. Few of us can still envision investment decision making without quantitative risk measurement, without hedging techniques, without deep and efficient markets for futures and options, without swaps and interest rate derivatives, and without computer models to price such instruments. And yet, these are all very recent developments. It has not been much longer than some thirty years ago that the very notion of an index fund was greeted with disbelief, if not outright ridicule!

I had the great fortune to be cast right into the middle of such developments when I joined the Management Science Department of Wells Fargo Bank in 1969. The annual conferences organized by Wells Fargo in the early seventies brought together people such as Franco Modigliani, Merton Miller, Jack Treynor, William Sharpe, Fisher Black, Myron Scholes, Robert Merton, Richard Roll and many others. The second half of the twentieth century was in my eyes as exciting in the field of finance as the first half must have been in physics.

I have worked on a variety of projects at Wells Fargo and later at the University of Rochester and the University of California at Berkeley, but one thing that bothered me for quite a while in the mid-seventies was the absence of solid results on the pricing of bonds. At that time, the CAPM was already in existence, and people had tried to apply it to bonds by measuring their betas to determine the yield, but that did not really lead anywhere. The options pricing theory had also been freshly developed by then, but it did not seem very feasible to apply a theory of pricing derivative assets to assets as primary as government bonds. What would be the underlying?

And yet, it was obvious that there must be some conditions that govern interest rate behavior in efficient markets. You cannot have, for instance, a fixed-income market in which the yield curves are always flat and move up and down in some random fashion through time, because then a barbell portfolio would always outperform a bullet portfolio of the same duration, and therefore it would be possible to set up a profitable riskless arbitrage. But what are these conditions?

The clue came from comparing the return to maturity on a term bond to that of a repeated investment in a shorter bond. The common denominator between bonds of any maturity would be a rollover of the very short bond, and thus it seemed natural to postulate that the pricing of a bond should be a function of the short rate over its term. And once the idea of describing the short rate by a Markov process came to me, it became obvious: the future behavior of the short rate is determined by its current value and therefore the price of the bond must be a function of the short rate! From then on, it's mathematics: in order to exclude riskless arbitrage, this function must be such that the expected excess return on each bond is proportional to its risk, which gives rise to a partial differential equation. The boundary condition of this equation is the maturity value, and the solution is the bond price. This was my 1977 paper. (Curiously, the thing that became known as the Vasicek model was just an example that I put in that paper to illustrate the general theory on a specific case. Well, you never know.)

Since then, it was like opening Pandora's box. Great many papers followed, extending the model in various ways—multiple factors, non-Markov risk sources, development of various specific models for practical use. One paper I have a great respect for is the Cox, Ingersoll and Ross article (for some reason, they did not publish the paper until 1985, although they did the work many years earlier), because it is about more than interest rates: it is about an equilibrium in the bond market.

A big shift came in 1986 with the publication of the Ho and Lee paper. This article presented a simple interest rate model, which was just a special case of my theory. The shift was in the interpretation: Ho and Lee assumed that the current bond prices were given (equal to the actual observed prices) and concerned themselves with pricing interest rate derivatives. This, of course, allows very useful applications for valuation of various instruments from simple callable bonds to the most complex swaptions.

The Ho and Lee paper engendered a great development effort in that direction, including the 1992 paper by Heath, Jarrow, and Morton, which formalized this approach. This direction was in fact taken further: There are models that assume as given not only the current bond prices, but also prices of caps and floors or even more. These models, used then to value other derivatives, have the great virtue of fitting the current market pricing of the more primary assets.

While I appreciate the usefulness of these models, I somewhat regret the direction away from the economics. To ask how derivatives are priced given the pricing of bonds seems to me assuming away the more interesting question: How are bonds priced? I personally hope to see a return to efforts to understand the economics, rather just to aid trading.

A similar situation has arisen in default risk measurement and pricing, another subject dear to my heart. The so-called reduced-form models, which have been advocated for the purpose of credit risk analysis, assume that corporate debt prices are given and use these prices to value debt derivatives. Again, to me it seems that the more interesting question is how to price corporate debt. Fortunately, this is possible given the legacy of Merton, Black, and Scholes, since corporate liabilities are derivatives of the firm's asset value, and a structural model of the firm can price its debt (and debt derivatives) from equity prices.

As appreciative as I am of the past in the field of finance, I am equally enthusiastic about its future. There will be no lack of problems to address, and there will be no lack of talent to solve them. Indeed, it is the professionals in this area of endeavor that are its greatest assets, and I am grateful to have worked with, and learned from, so many of them.

Chapter 2

Lifetime Achievement Award

By Dwight Cass

Risk, 44-45, January 2002

In the late 1960s, Wells Fargo Bank in San Francisco assembled a team of uniquely gifted thinkers who would go on to push the boundaries of financial theory. Working alongside William Sharpe, Myron Scholes, Fisher Black, and Robert Merton at the time was Oldrich Vasicek, who is Risk's lifetime achievement award recipient. Like his Wells Fargo colleagues, Vasicek has had a profound effect on both financial theory and practice. His equilibrium model of the term structure of interest rates is widely acknowledged as the landmark work in the field, and many credit it for setting off the series of modeling innovations that paved the way for the rapid growth of the interest rate derivatives market. Ten years later, he developed a groundbreaking credit portfolio risk model that paved the way for the approaches incorporated in the Basel II capital Accord.

Among market practitioners, he is perhaps best known for co-founding KMV, the San Francisco credit analysis firm, and for using Scholes, Black, and Merton's insights on option pricing to develop the expected default frequency (EDF) credit pricing system—a so-called Merton model approach—at the heart of KMV's product line. The company has been extremely successful, with KMV claiming more than 70 percent of the world's largest financial institutions as clients. It is hard to find a major credit derivatives dealer or loan house that does not use it. The success of the approach has prompted other companies, including Moody's Investors Service and JPMorgan Chase, to add a Merton model-based default probability estimator to their offerings.

This combination of theoretical and business accomplishments alone might be enough to warrant Risk's lifetime achievement award. But 60-year-old Vasicek has shown no interest in resting on his laurels to free more time for his enthusiasms, which range from playing classical flute music to windsurfing in the cold, windy waters surrounding San Francisco. He continues to tackle new challenges, such as the tricky problem of modeling spot and derivatives price behavior of nonstorable commodities such as electricity and telecoms bandwidth (on which he co-authored a technical article with Hélyette Geman published in the August 2001 issue of Risk). And, according to his colleagues at KMV, he remains the driving force behind the evolution of that firm's product line.

Vasicek did not originally intend to pursue a career as a financial theorist. He trained in his native Czechoslovakia as a mathematician, earning a PhD with honors in probability theory from Charles University in 1968. The first event that placed him on the road to his career in finance was the Soviet invasion in August of that year. Vasicek had been at the Czechoslovak Academy of Science in Prague, working in pure mathematics, when the Soviet tanks rolled in. He and his wife left for Vienna a few days later.

He made his way to San Francisco and began applying for jobs as a mathematician. He was interviewed for several positions—including a job at Stanford University's marine biology department doing spectral analyses of dolphins' songs. But fate lent a hand again, and he was interviewed by John (Mac) McQuown, head of Wells Fargo's Management Science Department, who was looking to hire several mathematicians. I'm a mathematician by profession, and only went into finance because my first job here was in a bank, Vasicek jokes.

McQuown's group already included Scholes, Merton, and Black. Also, Sharpe was consulting for the bank. This was before the Black-Scholes option pricing model had been published. Mac hired these guys before they were famous, Vasicek says. Wells Fargo was one of the first banks to embrace Sharpe's capital asset pricing model (CAPM), and McQuown's group was looking at ways to apply it. Vasicek worked on this project and on index fund construction.

McQuown, who would later launch KMV with Vasicek and Stephen Kealhofer, said Vasicek's talents quickly became obvious. I vividly remember Fischer Black saying to me, on a couple of occasions, that when he had a really intractable mathematical problem, he would go to Oldrich, he says.

Myron Scholes, the Nobel laureate in economics who is now a finance professor at Stanford University in Palo Alto, California, says: He's got tremendous mathematical and engineering abilities, and is also a very good listener. He articulates his views and holds to his path if he thinks he is right, but he's willing to appreciate other people's views, which makes him a good scientist. Indeed, in an arena where oversized egos are often the norm, many of Vasicek's past and current colleagues praise his tolerance and humility. He has ideas that seem quite simple in retrospect, so much so that they're quickly borrowed by everyone else. But they reflect a deep and powerful mind. says Kealhofer. He has a wonderful old world/new world charm—a mixture of Prague and San Francisco, he adds.

In 1974, Vasicek left Wells Fargo to teach at the University of Rochester (New York) Graduate School of Management, where he would stay for two years. During this period, his attention turned to the problem of interest rate behavior, which he would continue to pursue when he returned to California as a visiting professor at UC Berkeley's business school. The pricing of bonds and behavior of interest rates was an open question at that point, Vasicek says. The work on the CAPM was exciting, but the research that had been done wasn't applicable to bonds.

Vasicek realized that arbitrage would link bond prices up and down the term structure, so, for example, there had to be a relationship between investing in a one-year bond twice in succession and investing in a two-year bond straight off. He found the common denominator to be the short-term interest rate. If you postulate that the pricing of a long bond is a function of the short rate over the term of the bond, or more accurately, of your probabilistic description of the short rate's behavior over the term of the term bond, you have a common variable for the pricing of bonds, he notes. In other words, you have a state variable for the pricing of bonds with all terms from the current value of the short rate—that was the point at which the idea really broke for me at the time, he says.

Inspiration

Vasicek's paper, titled An Equilibrium Characterization of the Term Structure, was published in the Journal of Financial Economics in 1977. It was either the basis for, or inspired many of the theoretical advances that came in the years that followed. Among these was the influential 1985 Cox-Ingersoll-Ross model. One of that model's authors, Stephen Ross, professor of finance and economics at MIT's Sloan School of Management in Boston, says of Vasicek's work: It is a wonderfully simple, empirically amenable model. It guides us in a lot of our intuitions about the subject. McQuown argues that Vasicek's model was the critical catalyst that spurred development of the interest rate derivatives market. The interest rate swap market relies on that model, he says. If it wasn't for Oldrich's contribution, the interest rate swap market may have taken longer to develop. I dare say the problem would have been solved sometime, but Oldrich solved it first.

It was like opening a Pandora's box as far as academic research goes, Vasicek says. But, he notes with some modesty that he had not actually aimed to redefine how the world viewed interest rate products. But what's kind of funny is that my paper focused on the theory, he says. What has become known as the Vasicek model was just an example. I developed the theory and I wanted to illustrate it on a particular type of process and go through the calculation and determine the final equation. But it's the example I used that's been remembered.

The shift in his emphasis to credit risk came when McQuown recruited Vasicek for an ill-fated scheme in the early 1980s. I persuaded him to join me in a venture that ultimately went down the tubes called Diversified Corporate Loans in 1983. We wanted to create a pool of credit from major US banks where the banks could swap qualified loans into a pool in return for a pool interest. The idea was to give the banks liquidity and portfolio diversification, but to do so, the firm needed a way to value credit risk, to aid the participating banks in valuing the loans they put into the pool, and the pool itself.

So, Vasicek says, I started to work on credit. Up to then, credit was strictly a judgment call. So I developed an application of option pricing theory—the Black-Scholes and Merton work. Kealhofer says: He laid down the theoretical footprint in a short time that we've been using for 17 years now. He laid the groundwork for both the basic credit technology and the portfolio technology. We've been laboring in those two veins ever since.

When DCL went out of business in 1989, Kealhofer (who had joined several years before), McQuown and Vasicek launched KMV to further develop and market the credit evaluation tool. Myron Scholes says: It set the stage for using more modern technology than the rating agencies have used, and it led to more people thinking about using option technology to do credit pricing. Others had used the option framework to price debt. But his work at KMV took the lead in developing something that was usable by a vast number of people. Indeed, having a widely available set of reliable credit pricing tools was a necessary precondition for the development of the credit derivatives market, Scholes notes.

Pioneering

MIT's Ross says: Vasicek's work on credit risk is a different kind of pioneering. It's a demonstration of how one can take high-quality academic work and turn it into a solid business without compromising the academic stuff or the understanding of the needs of the marketplace. Marrying those two is never easy.

Vasicek says the launch of competing Merton-model products—JPMorgan Chase's being the most recent, the specifications of which were published in the November 2001 issue of Risk—is good for the markets. I'm glad that people are becoming interested, because any effort in this direction will make the market for these securities more liquid and more efficient, he says. There has been a bit of confusion about what a Merton model is. I guess people keep forgetting it is not a formula, it is a framework. It is a structure that allows you to get a specific mathematical solution to the value of a firm. But it would depend on the assumptions you make about the firm's financial structure and the capital statements and the payments the firm is making—coupons, dividends, and the nature of the debt, convertibility and optionality. It's a very complicated thing. We give a lot of attention to how we characterize the firm, on top of the mathematical problem, then it's a fair amount of work.

As co-head of KMV's research group, Vasicek continues to guide development of the firm's products. We just rolled out a new method of calculating a loss on credit instruments that uses the empirical distribution of default risk, Kealhofer says. That was suggested by Oldrich. As factors such as the Basel Accord and the rapid growth of the credit derivatives and synthetic collateralized debt obligation markets have kept the credit risk modeling arms race in full swing, and new pricing challenges such as those in the electricity and bandwidth markets have arisen, Vasicek's work has remained at the vital crux between theory and practice.

Chapter 3

One-on-One Interview with Oldrich Alfons Vasicek

By Nina Mehta

Financial Engineering News, May 2005

Oldrich Alfons Vasicek, a mathematician, is one of the leading lights in modern finance. His 1977 paper, An Equilibrium Characterization of the Term Structure, described the behavior of interest rates across maturities and paved the way for the development of the interest rate derivatives market. He is one of the co-founders of KMV Corp., the granddaddy of credit risk modeling firms, which Moody's Investors Service bought in 2002 (and renamed Moody's KMV). Throughout his career Vasicek has been at the forefront of credit risk modeling. Earlier this year he increased his stockpile of awards when he was named the 2004 IAFE/SunGard Financial Engineer of the Year. The author of more than 30 papers in mathematical and finance journals, Vasicek received a PhD in probability theory from Charles University in Prague, in what is now the Czech Republic, in 1968. This interview was conducted in February.

FEN:

You've worked on models to value credit risk for a large part of your career, but you said recently that valuing the plain debt of a single issuer is