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This book is an introduction to financial engineering from the standpoint of martingales, and assumes the reader knows only elementary calculus and probability theory. After giving a motivating example entitled "the parable of the bookmaker" the authors clarify in the introduction the difference between pricing derivatives by expected value versus using the concept of arbitrage. Vowing then never to use the strong law of large numbers to price derivatives, discrete processes are take up in the next chapter. The authors do an excellent job of discussing the binomial tree model using only elementary mathematics. Interestingly, they introduce the concept of a filtration in constructing the binomial tree model for pricing. Filtrations are usually introduced formally in other books as a measure theory concept. They then define a martingale using a filtration and a choice of measure. The use of martingales pretty much dominates the rest of the book. They emphasize that a martingale can be a martingale with respect to one measure but not to another. Continuous models are the subject of the next chapter, where the ubiquitous Brownian motion is introduced. The discussion is very lucid and easy to understand, and they explain why the conditions in the definition of Brownian motion make its use nontrivial; namely, one must pay attention to all the marginals conditioned on all the filtrations (or histories). The Ito calculus is then appropriately introduced along with stochastic differential equations. The authors do a good job of discussing the difference between stochastic calculus and Newtonian calculus. Recognizing that the Brownian motion they have defined is with respect to a given measure, they then ask the reader to consider the effect of changing the measure, thus motivating the idea of a Radon-Nikodym derivative. Their discussion is very intuitive and promotes a clear understanding rather than giving a mere formal measure-theoretic definition. Many interesting examples of changes are given. Portfolio construction and the Black-Scholes model follows. Basing their treatment of the Black-Scholes model of martingales gives an interesting and enlightening alternative to the usual ones based on partial differential equations (they do however later show how to obtain the usual equations). The next chapter discusses how to use the Black-Scholes equations to price market securities and how to assess the market price of risk. The discussion is very understandable but not enough exercises are given. Modeling interest rates is the subject of the next chapter. The mathematical treatiment is somewhat more involved than the rest of the the book. Several models of interest rate dynamics are discussed here very clearly, including the Ho/Lee, Vasicek, Cox-Ingersoll-Ross, Black-Karasinski, and Brace-Gatarek-Musiela models. A few of these models were unfamliar to me so I appreciated the author's detailed discussion. The book ends with a discussion of extensions to the Black-Scholes model. The emphasis is on multiple stock and foreign currency interest-rate models. A brief discussion of the Harrison/Pliska theorem is given with references indicated for the proof. An excellent book and recommended for beginning students or mathematicians interested in entering the field. My sole objection is the paucity of exercises in the last few chapters.
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on June 1, 1999
This is an elegant book for students of financial mathematics. You won't see the tedious Theorem/Proof format so common in other similar textbooks. But what it lacks in rigor it more than makes up for in other more important areas: superb writing, clear explanations and brilliant insight into the most popular valuation models. For instance, the concise but very clear derivation of the Black-Scholes formula should impress anyone who has studied the PDE-based derivation covered by Hull and others.
There is little discussion of empirical issues. This, in my opinion, was a wise choice by the authors. Any such discussion would severely dilute the strength of the book -- namely, the fundamentals of stochastic calculus applied to arbitrage pricing. For those interested in empirical features of the markets, I'd suggest "Econometrics of Financial Markets" (Andy Lo, et al).
I find it ironic that the punchline for the whole book -- a chapter on exotic option valuation where probabilistic techniques such as the reflection principle naturally come into play -- did not make it to production. But this excellent chapter is available on the errata Web page under [...]
This book is a great place to begin study for quantitative MBA students or math students with an interest in option valuation. Supplement this book with Oksendal or Karatzas / Shreve, perhaps, for more in-depth material on stochastic calculus.
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on March 1, 2000
Having been a student of this subject for a short 18 months now, and having looked at many books on the same subject, this is by far THE BEST. What this book does is simple. It lays the groundwork for pricing derivative securities using stochastic calculus. It helps build the intuition behind the stochastics. Then, from this intuition and foundation, you are equipped to read more advanced treatments of the subject.
This is not a book on solving partial differential equations, nor is it supposed to be. If you are looking for a book on solving and creating financial PDE's, then buy Wilmott's books. Rather, this book uses discounted expectations under the risk neutral measure to price securities. What does that mean? Well, all I can say is "READ THE BOOK".
The first three chapters of this book are so fundamental and necessary to building a firm and solid intuition that I have read them over three or four times now. The reason I have read it so many times is because it is so well written and new things pop out at you every time. It really is a delight to read.
Moreover, the section on fixed income models is extremely well written as well.
I can't stress enough how great this text is.
You should buy it even if you already know the material.
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on June 23, 2005
Baxter/Rennie deserves its reputation as the best introductory book to modern financial engineering because of its terse, elegant description of the martingale approach to asset pricing. The basic machinery of financial engineering comes from fields in applied analysis such as stochastic calculus and martingale theory, whose presentation is often weighed down with overbearing technical considerations. While necessary from a mathematician's perspective (and often relevant even in many real-world cases), a reader with mathematical maturity but without formal training in measure theory or probability can nonetheless appreciate and understand the basic tools used over and over again in Black-Scholes, and more generally, in martingale methods for pricing.

The quantitative analyst often needs to thorougly understand both the theory and the techical aspects of implementation of these models, and often studies these subjects in graduate programs. Many participants in quantitative fields such as fixed income and derivative trading/pricing often want an intuitive notion of what these "quants" are doing. Baxter/Rennie serves as a good book that is far more advanced than the excellent book by Hull; the latter is a wonderful introduction to quantitative fields, but Baxter/Rennie is the first book that truly introduces the math in a unified presentation that encompasses the three most important tools of mathematical finance (assuming, of course, the Ito-Doeblin calculus driven by Brownian motion); the Cameron-Martin-Girsanov (CMG) Theorem that permits changes of measure, the Feynman-Kac formula and the Martingale Representation Theorem.

Like most books on the subject, Baxter/Rennie attempts (but actually succeeds unlike many competing texts) to give a relatively harmless, intuitive introduction to the Ito-Doeblin calculus and martingales; the two are linked through the martingale properties of Brownian motion, which later permits martingale pricing under a risk-neutral measure.

Chapter 3, the core of the book, introudces Black-Scholes-Merton theory as a simple, special case of martingale pricing. For those who are sick of reading PDE proofs of the Black-Scholes formula that are technically correct but don't actually teach anything about finance, this chapter is for you.

Chapter 5, which introduces the market models of modern fixed income pricing, provides a presentation different from most other books by introducing the general methods (Heath-Jarrow-Morton, or HJM) models first, and giving the short-rate models as a special case of the HJM model. While this is chronologically out-of-order and arguably gives the "harder" case first, it is consistent with the no-arbitrage (or martingale) approach to asset pricing and emphasizes the approach that now dominates the feld.

The final chapter provides an extension of the models in the previous chapter, i.e. by generalizing to the multidimensional cases, etc. These are intuitively similar (although technically more complex) to the cases presented in the earlier chapters, and in keeping with the simple, elegant presentation of the book, are treated appropriately as relatively basic extensions.

Baxter/Rennie emphasizes the models, not the pricing of individual securities; the elegance and economic meaning are stressed at the expense of practical considerations in many cases. For this reason, the book would be a good text for a motivated undergradate studying math/"hard" sciences or an MBA student, quantitative traders or other practitioners responsible for using, but not building models.

For those learning quantitative analysis more seriously, this book would be a good complement to Shreve's Stochastic Calculus in Finance Vol II (not to be confused with the MUCH more technical monograph by the same author). The latter covers many more mathematical topics and their applications in pricing actual instruments, and has an exhaustive list of excellent problems; nonetheless, the book is meant for a full-year sequence in mathematical finance at the Masters' level and by the very nature of being so complete it lacks the elegance of Baxter/Rennie.

My recommendation for those learning mathematical finance is to buy both Baxter/Rennie and Shreve Vol. II. Work through the problems in the latter text and you'll know a little bit of everything from stochastic volatility to jump proceses. Review the former before and interview and you'll be able to quickly answer basic questions in simple English (or whatever language you speak other than math). While it doesn't have many good problems (and they are relatively basic compared to Shreve, for example), that's not the point of the book - it's meant to develop intuition, not technical skill or theoretical abstraction. Buy both - if you're entering this field or are already studying/working in it, you can afford both of them.
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..I realized just how badly stochastic differential equations and stochastic integration are treated in Hull's popular book Option, Futures, and other Derivatives. The problem with Hull is that Ito's lemma is only stated, not proven, and it's the proof that shows one how to formulate correctly the stochastic integral equations that Hull calls 'stochastic difference equations'. When volatility depends on returns and/or time, then the errors made from following Hull's oversimplified treatment become serious.

My first impression of Baxter & Rennie's 'Financial Calculus' was that it was unnecessary and a waste of money. My opinion reversed completely after realizing (under prodding by a physics colleague who's an expert on sde's) how badly Hull's approach to sde's really is. Also, the systematic derivation of Black-Scholes from the assumption of a replicating, self-financing strategy is very nice. As Feynman said, we don't really understand a result until
we can derive it from many different viewpoints. The method is not really different in principle from the standard short derivation given in Hull, but it does provide a nice, clear example of what is meant by replication and self-financing in the terminology of Brownian motion/sde's. A problem with the book is that one must first learn the rudiments of options elsewhere (Hull, Bodie & Merton): this is not a text for beginners.

A word of warning: empirically seen, the results presented by the book (Black-Scholes and near-B-S) are empirically wrong. The authors present the theory as if it would be biblical, handed down by god, giving the reader no hint that the economic-financial problems discussed there merely abstractly-mathematically are not at all solved by the models presented in the text. For example, the empirical returns distribution is very far from Gaussian and is volatile (the empirical returns diffusion coefficient depends on both returns x and time t) whereas the returns pde in the B-S model has a constant diffusion coeficient. In other words, typical of mathematicians and 'financial engineers' who are not concerned with fundamentals, B&R seem not to be bothered by the fact that the B-S theory cannot be patched up and saved by a perturbative approach. Instead, a completely different starting point than lognormal pricing is required (see my paper with Gunaratne on the empirical distribution of returns and correspondingly correct option pricing).

Note added: the treatment of Girsanov's theorem is restricted to Gaussian processes. See Durrett (1984) for the correct general treatment.
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on August 21, 2002
More books are needed on this topic. Perhaps it was Duffie that started this needlessly complicated approach to texts on quantitative finance? Baxter and Rennie attempt to simplify the learning process but their effort remains too terse to be self-contained. Be prepared to have a couple of good references on probability and calculus handy if you don't happen to remember how to perform Erf(x) substitutions and apply DeMoivre-Laplace.
Neftci's book is a better place to start this material. He's criticized for too much hand waving but it is an *INTRODUCTORY* text! Though not sufficient, it is far more accessible than other offerings.
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on May 12, 2001
Baxter & Rennie are the clear descendents of Silvanus P. Thompson(Calculus Made Easy). A pair of Brits with an engaging style and clear expository prose. They take the reader on a tour of some heavy duty mathematics, without excessive formalism. The basics of arbitrage pricing, binomial branch models, the Ito calculus, and the martingale representation theorem are contained in 100 pages. The second hundred pages are devoted to the construction of models for contemporary financial derivatives in foreign exchange, stock, and fixed income markets. This book is a must read for practitioners of, and students in financial engineering.
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on December 14, 1999
You may find the recent book by Paul Wilmott on option pricing a more comprehensive or a standard reference book by Hull a more direct accessible. However, by this small book, Baxter and Rennie has present the main theme of derivative pricing theory from a very elegant point of view. The reader can get a shortcut and clear insight of the state of modern derivative pricing theory which serious readers may have to consult in the old day the difficult seminal papers or a bunch of mathematical texts combined together. The authors has started with a 2 important concepts of modern pricing theory : no arbitrage and risk neutral approach. The concept of the abstract continuous time stochastic processes, filtration, change of measure, or Ito calculus are presented using a very touchable counter examples in a subsequent chapters. The authors also try devote a few lines to discuss a motive of underlying mathematical principles which is invaluable in a model ramification while most of the text largely ignore this. In the last chapters, the vast world of interest rate models especially in the HJM framework was impressively presented. All-in-all, the book successfully bridge a gap between elementary text and a more abstruse research works.
However, there is nothing perfect in this world. This beautiful book only skimmingly touch the concept of no-arbitrage condition. It also better serves a serious readers or gradute courses on derivatives rather than practitioners or model developers.
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on December 6, 2006
Baxter starts off with a formula familiar to anyone who has read even one text on mathematical finance. Set up the basic terminology and core mathematical expectations, introduce discret time times, evolve the discret time models to now-classic continuous time models, and then adapt for the continuous time models for specific products and situations (American option expiration, interest rate models, jump diffusion, etc...).

What I liked best about the book was the skillful narrative and witty presentation of the foundational concepts. Chapter 3 -- the introduction to continuous time models and Black Scholes -- is by far the best introduction that I have read -- to why stochastic models do not follow the classic rules of Newtonian calculus and heuristic proof that explain the "why" behind the basics of stochatsic integration, Ito's lemma, and Brownian motion. Although I was familiar with the material before reading it, the clarity and insight of his presentation fundementally changed by view of the craft of quantitative finance and simplified the manner in which these topics were organized in my mind.

This is not a Paul Willmott book -- full of pseudo-code and real-code implementations, sweeping in its coverage of the many topical specifics that practitioners reply on day-to-day. It does not replace the need for these other approaches. Like Willmott, though, it is geared for the new student of computational finance.

The lack of implementations is disappointing because his pedagolical style is so basic that the final expressions evolve right up to the edge of practical use but then stops. Yet the discipline is this evolution keeps the book moving at a fast pace from start to finish and keeps the entirity of the book very short.

Additionally, there is a lack of applications and material relevant to the credit markets and fixed income worlds that would have been nice to have covered with the same style. Perhaps he will follow up with a second book that explores a larger universe of financial markets and products.
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on May 22, 2002
I read this book when I was an undergrad (in business) and it sparked my interest in quantitative finance so much that I went on to study it in depth at grad school. The book is easy to read for everyone who knows basic calculus and probability (i.e. even business undergrads). There is ABSOLUTELY NO BLAH-BLAH ABOUT SIGMA ALGEBRAS and whatever [stuff]mathematicians tend to include in those books to confuse us all. Still, this book covers everything from equity derivatives (binomial and continuous time models) to the theory of fixed income and interest rate products. I still think that the explanation in this book about Girsanov's Theorem is better (i.e. less technical and more intuitive) than in any other book I have read (and I had to read lots of them). The transition from discrete to continuous time is smooth, intuitive, and very understandable. As I said, all the measure theory, real analysis and whatnot that makes other books useless and confusing is left out here. After having read and studied this book in college, there were no surprises anymore in grad school.
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