Continuous-Time Finance

"This book is definitely not a light read for the typical reader, but I really had to include it among my five because its author, Bob Merton, has had an enormous impact on my career. When I was a first-year economics graduate student, I was actually quite discouraged about economics. As an undergraduate, I had studied a lot of mathematical economics and while I enjoyed it and was reasonably good at it, I became somewhat disillusioned by all the fancy mathematics applied to situations with relatively unrealistic economic assumptions. This kind of economics didn’t seem especially relevant to the real world, and it was the relevance of The Worldly Philosophers that drew me to economics in the first place. At its worst, economics sometimes seemed like it was about asking how many angels can dance on the head of a pin, not about doing something that could improve people’s lives. I was willing to suspend my disbelief until graduate school because I thought that, as an undergraduate, I might not have the tools to deal with the kinds of things that are more realistic, and perhaps more challenging. Even though I’d taken a first-year graduate course as an undergraduate and found it to be much the same as my other classes, I was still hoping that graduate school would be different. But I quickly realized during the first semester of my first year as a Ph.D. student that this wasn’t the case. What we were learning was exactly the same things I had covered in college, maybe with slightly more mathematical formalism, but with no more relevance than before. So, after the first semester, I reluctantly filled out an application to law school and planned to withdraw from the economics PhD program at year end. By chance, one of my high school classmates who was a student at MIT suggested that I cross-register for a finance course being taught at the MIT Sloan School. At the time, I had no idea what finance was, other than what I read in Malkiel’s book in college, but I remembered how interesting the random walk was when applied to predicting the stock market. I had never heard of the faculty member who was teaching the introductory finance course at MIT—someone named ‘Bob Merton’—but given how easy it was at the time (and still is) for Harvard students to cross-register for MIT classes, I decided to give it a try. I was just blown away. In the first two or three lectures, it became clear to me that this is exactly what I was looking for: rigorous mathematics and beautiful theory applied to some very real and important problems. After two weeks, I decided that I wanted to be a finance professor and forgot about law school. I didn’t realize until much later that Bob Merton was a pretty special individual; at the time, I just figured that all finance professors were like Bob, and that’s what I wanted to be when I grew up. I now realize—as does everyone else in our profession—that Bob’s approach to finance was unique. He applied some extraordinarily sophisticated mathematics to some very practical problems, like pricing options and other derivative securities. And more than any other academic that I’ve ever come across since then, Bob showed me that finance could be a science in every way as rigorous as biology or physics. And in every way as practical. So that’s what drew me into the field. Ultimately, Bob became a role model and mentor to me and his book, Continuous-Time Finance , really embodies his contributions to financial engineering, a field he created almost single-handedly. And that term, financial engineering , really has meaning to me. I know it’s used in some circles pejoratively, but what Bob did was to demonstrate that finance is a science so precise that it does lead to engineering, meaning you can actually apply it to solve important problems in the financial industry. That’s why I had to put this book on the list, despite the fact that it’s probably not for the average reader. It does have lots and lots of equations and it’s a very challenging book that I continue to learn from. Indeed, it would. Anybody who’s pursuing a career in the financial industry—creating new financial securities, new ways to price and hedge them, or new business structures for investors—will need to be familiar with this book. My guess is they’ll already own copy with a lot of dog-eared pages. The title highlights the new approach that Bob introduced to modeling financial markets. He used the kind of mathematics known as ‘stochastic calculus.’ The easiest way to explain it is to start with the observation that, prior to Bob, most economic models of financial markets (or any other kind of market) measured time at discrete intervals—day 1, day 2, and so on. The much more difficult alternative is to model time as flowing continuously so there’s an infinite number of possible time points in between day 1 and day 2 when a trade can occur. In a continuous-time world, events can be arbitrarily spaced apart from each other, three-and-a-half months or 145 hours or a 1000 picoseconds. The distinction between discrete and continuous time may seem minor, but it turns out that the mathematics of continuous-time economies is actually quite different than that of discrete time. It requires an entirely new arsenal of mathematical tools that Bob developed, in many cases from scratch. He relied on other mathematicians for some of those tools, but the way he applied them was original and had never been done before in the history of the social sciences. He opened up whole new areas of research by simply acknowledging that time flows continuously and that, using stochastic calculus, we can model the behavior of financial markets in a variety of economic environments. One of the many by-products of this new approach is the ability to come up with elegant pricing formulas for many complex securities, like options. Using this powerful framework, Bob pioneered a number of other important breakthroughs in financial economics, and the foundations of these breakthroughs are all contained in Continuous-Time Finance ."