Lectures on Financial Mathematics: Discrete Asset Pricing, by G. Anderson and A. Kercheval

Lectures on Financial Mathematics: Discrete Asset Pricing, by G. Anderson and A. Kercheval, Morgan & Claypool (2011), Paperback. ISBN: 1608454959.

Mathematical Finance, also known as Financial Engineering, is a subject that is highly popular with students. Moreover, it has retained its popularity in the aftermath of the 2008 financial crisis and meltdown of Wall Street, and throughout insider trading scandals such as the recent one of the Galleon Group. Typically a subject restricted to graduate students (both masters and PhD students), it is gradually diffusing its way into the undergraduate curriculum. This is in keeping with a long tradition in mathematics; for example, abstract algebra (groups, rings, fields, etc.) was taught exclusively to graduate students as recently as the 1940s, and is now routinely taught to juniors. But Mathematical Finance presents a special challenge, since it is a subject that is inherently interdisciplinary, and despite many proclamations of support and encouragement for interdisciplinary research and teaching in universities, a better metaphor for university subjects would be apartheid, with departments playing the role of homelands. Courage to the person who ventures out of his or her homeland.

The subject of Mathematical Finance involves multiple branches of mathematics, and some profound knowledge of economics, both concrete and intuitive. It also involves ideas not present in other forms of mathematics, so to many it involves ideas that are completely new. The topics in mathematics alone that one needs to know, or at least be familiar with, are probability, statistics, partial differential equations including free boundary problems, Monte Carlo methods, simulations, computer programming on the level of C or C++, Brownian motion, stochastic differential equations, and, on a more advanced level, topics such as numerical analysis and functional calculus, including the Malliavin calculus. Given this, it is hard to imagine a priori teaching such a subject with so many sophisticated prerequisites to undergraduates, especially since most of the demand is not from students whose principal subject is mathematics.

A graduate course will usually begin with Brownian motion and stochastic calculus. The key new idea coming from economics is that of risk-neutral measures, where one constructs a pricing system for financial derivatives as a method of computing expectations of random variables under a risk-neutral measure, which is not the true, or physical, underlying probability measure, but instead an artificial construct that is equivalent to it, in the sense that they both have the same null sets. Experience shows that this artificial construction puzzles students for a long time, and they have trouble relating to it. One way to make sense of such a construction is through equilibrium theory, but that would involve a long sojourn through economics, usually unwelcome in a mathematics or engineering class.

So how does one broach such a subject on a level that undergraduates can both understand and value, in the sense that they will have grasped some of the key ideas? The answer is to restrict oneself to a small subset of the subject, and for that subset, to treat only the simplest possible case. What one gives up in generality, breadth, and applicability, one gains in the potential success of a transfer of knowledge. To follow this recipe the key factors will be the choice of the topic or topics on which to focus, and of course the quality of the presentation and explanations. Even so restricted, this is not a trivial task. Therefore, it is impressive, perhaps amazing, at how well the two authors have succeeded in doing just this, and in only 50 pages.

The key idea of their presentation is to cover only the subject of the pricing of derivatives (and not, for example, the hedging of derivatives), and to do this in discrete time, and in a finite probability space. This approach allows much of the complicated (and interesting) mathematics to disappear, and instead focuses more on the interplay between the mathematical model presented and the underlying economics. In a finite world, all random variables are bounded, all expectations exist, etc. Nevertheless, the authors are still able to treat trading strategies, and justify the presence of a risk-neutral pricing measure through the ability to replicate a claim, in the case of a complete market. Thus they naturally introduce the concept of arbitrage, or, more properly, its absence, and relate this to risk-neutral measures, one of the key ideas of the subject. However, even the authors themselves have not fully conveyed the simplicity of their setting. They still speak of local martingales, and local martingale risk-neutral measures, for example, which do not exist in a space where all random variables are bounded (i.e. all local martingales are martingales). These types of oversights could be confusing to a student who wants to grapple with the subject on a profound level.

An unusual (and welcome) feature of this book is its treatment of forwards and futures, especially the latter, although of course the two are close cousins. Futures form an important area for study, especially in terms of the size of the markets in futures, which is huge. But often they are overlooked in textbooks, probably because the mathematics involved does not really present many interesting challenges. For example, in the lovely elementary book of Williams (2006), her introductory chapter is similar to that of Anderson and Kercheval, but then she goes on to introduce Brownian motion and a more sophisticated mathematical framework. Nevertheless, her book is also aimed at undergraduates. But forwards and futures are not treated. Nor are they treated in the classic text of Baxter and Rennie (1996), also perhaps suitable for an undergraduate engineering class. (They are, of course, treated in the ‘industry standard’ two volumes of Steve Shreve (Shreve 2004, 2005), but only a little space is given to them.) So it is welcome that they are treated here in the book under review, and that the treatment is well done. My only quibble is that, in corollary 4.6 on page 29, the authors state that futures prices and forward prices are equal. This is true, of course, since the authors assume all interest rates are deterministic. But it leaves open the unaddressed question (that a student might well ask): what happens if interest rates are modeled as stochastic processes themselves, and therefore are not deterministic? Then, of course, futures and forwards prices are not equal (see, for example, the classic works of Cox et al. (1981) and Jarrow and Oldfield (1981), or the more recent paper of Jarrow and Protter (2009, section 5)). Such problems, however, are endemic to simplified treatments of complicated subjects, and perhaps are to be expected.

There are many books on the subject, but most of them (such as those of Björk (1998), Mikosch (1999), Duffie (2001), Steele (2001), Cont and Tankov (2004), Shreve (2004), Delbaen and Schachermayer (2006), and Lamberton and Lapeyre (2007)) are dependent on developing or using stochastic calculus, an approach definitively not suitable for undergraduates. The excellent book on Monte Carlo methods by Glasserman (2004) is tangentially related to the subject and is also dependent on stochastic calculus. We have already discussed Baxter and Rennie (1996), Shreve (2005), and Williams (2006), and that leaves the intriguing book by Föllmer and Schied (2004). While this beautiful and profound book does not require the sophistication of prior knowledge of stochastic calculus or even of Brownian motion, it is nevertheless a high-level book, requiring a level of ‘mathematical maturity’ that is beyond American undergraduates. Thus we see that this little book of Anderson and Kercheval has found an ecological niche all its own: it is (to use my favorite IRS adjective) a ‘super simplified’ approach of an introductory nature to the challenging subject of Mathematical Finance, with an unusual but welcome selection of topics. And, in addition, its exposition is clear and well written.

Philip Protter

Department of Statistics, Columbia University