Customer Reviews

Stochastic Calculus and Financial Applications

5 star:		(8)
4 star:		(3)
3 star:		(2)
2 star:		(3)
1 star:		(1)

 

 

Before I write this review, it's only fair to disclose that before even hearing of it I already had a very solid background in (graduate-level) analysis, which as another reader astutely pointed out is often considered "calculus" in the math community (I think the classic Calculus by Shlomo Steinberg, which can be found free online, has been used at Harvard for decades, while Tom Apostol's "Calculus," a misnomer to say the least, is the standard text at Stanford and Cal Tech - both are really books on advanced calculus and elementary real analysis). Part of the reason I am writing this is to clarify the distinction - many people aspiring towards quantitative roles on Wall Street don't know exactly what the mathematical prerequisites are for a particular subject or presentation, and hopefully I can help clarify this for other readers who, like myself, sought books like this one to learn the basics of mathematical finance.

On that note, Steele's book is a MATH book. By contrast, the wonderful book by Baxter & Rennie emphasizes core ideas with emphasis on the relationship between the three primary tools of the discipline (Martingale Representation, Ito-Doeblin Calculus, and the Feynman-Kac formula) while Shreve's classic emphasizes actual development of key models and techniques. Even Oksendal, which is aimed at a slightly more sophisticated mathematical audience, emphasizes applications at the expense of elegance.

In contrast, Steele's book is a math book aimed at Wharton (read: finance and economics doctoral students, likely in their second year) students with varied interests. Students taking this course probably have already taken a rigorous course in asset pricing theory from the academic viewpoint and need to fill in the blanks with the continuous-time techniques to extend these techniques and to understand stochastic calculus at the level necessary for research in economics/finance.

With that in mind, the book is versatile enough to be appreciated by different audiences. Steele certainly takes care give a clear, well-motivated presentation which explains to the reader WHY he is giving a concept, proof, or problem, and breaks the book up into small, digestible chapters. The problems are neither overly difficult nor disconnected from the text, although doing them is not an essential part of understanding the overall view. Furthermore, Steele clearly takes delight in the beauty of stochastic calculus, as demonstrated by Chapter 5 - Richness of Paths, which discusses the "interesting" properties of Brownian motion. For anyone who sat through a difficult analysis class thinking the whole purpose of the course was to annoy and taunt the student with irrelevant counterexamples (remember constructing a continuous yet non-differentiable function using limits?), this chapter will be especially fun.

In the first part of the book, Steele covers the basics of the random walk and martingales, introducing important theorems such as the upcrossing (downcrossing) lemma, submartingales and the Doob Decomposition theorem, the basic martingale inequalities, stopping times, and conditional probability (for those who are familiar with Williams' Probability with Martingales, the treatment is similiar). He then covers Brownian motion from both the standard perspective (a Brownian motion is a process such that...) and more intuitively as a limit of random walks (i.e. the "wavelet" construction/proof), using this subject as an opportunity to extend the martingale concepts to continuous-time.

In what could roughly be called the "second" part of the book, Steele develops the Ito integral as a martingale and as a process. Steele provides a lot of detail to the subject, perhaps in mind with the view that readers using stochastic calculus with more general underlying processes will have to understand the difference between a martingale and "just" a local martingale. He then quickly but sufficiently covers the standard topics of Ito calculus - Ito's lemma, quadratic variation, and the basic SDE, although in the Picard-type existence/uniqueness proof of SDEs he shows why the careful description of the Ito integral is not simply technical.

The next part of the book covers the "standard" topics in financial mathematics that would appeal to quant finance students . The chapter on arbitrage covers the basic Black-Scholes-Merton equation and its generalization to arbitrage pricing, although Steele (appropriately) addresses Black and Scholes CAPM derivation of their options pricing formula, which gives the finance/economics reader a historical perspective. The chapter on diffusions is excellent and gives all of the necessary elements for handling "nice" parabolic second-order equations. He even sneaks in Green's functions, series expansions, and the Maximum Principle without making uninterested readers have to learn them to follow the presentation.

In the last few chapters, he covers Martingale Representation, Girsanov's Theorem and their applications to more advanced topics in pricing, such as forward measures. The problems in this part of the book are nice because they help the reader understand the intuition behind a particular mathematical principle but not necessarily its application to a well-recognized model. The final chapter on the Feynman-Kac formula gives a very intuitive proof of its topic which helps the reader understand what is meant by "killing" a process and hopefully how that translates into finance; other books often just do a coefficient-matching proof, which really doesn't capture what's really going on.

I emphasize again that while the book is designed to serve a different purpose than texts such as Shreve or Baxter & Rennie, it can help readers of different backgrounds understand the basic elements needed for more advanced stochastic analysis and gain an appreciation for both the beauty of the subject and the underlying intuition liking the math to the finance. The prerequisite, though, is at least a (rigorous undergrad) course in real analysis, probably some familiarity with measure theory, probability, and L(p) spaces (or at least L(1,2,inf) spaces), and at least basic familiarity with the elements of stochastic calculus (Ito's lemma and computations with "box calculus", for example). For readers seeking a more comprehensive treatment of quantitative finance, this book is reasonably good mathematical preparation to understand Musiela/Rutkowski, and for doctoral students, understanding most of the topics in this book with a brief introduction to dynamic programming in the continuous-time setting is sufficient background to read Merton's book (consumption-investment problems) as well as understand the basics of derivative pricing.

I knew Mike Steele from my days as a graduate student at Stanford. He is also a Stanford graduate and a first rate probabilist. When I knew him he was doing some post-doctoral teaching at Stanford. He is a great teacher and writer.

Mike Steele has used the material in this text to teach stochastic calculus to business students. The text presupposes knowledge of calculus and advanced probability. However the students are not expected to have had even a first course in stochastic processes. The book introduces the Ito calculus by first teaching about random walks and other discrete time processes. Steele uses a lecturing style and even brings in some humor and philosophy. He also presents results using more than one approach or proof. This can help the student get a deeper appreciation for the probabilitist concepts.

The gambler's ruin problem is one of the first problems that Steele tackles and he uses recursive equations as his way to introduce it.

Brownian Motion, Skorohod embedding and other advanced mathematics is introduced and emphasized. After motivating the stochastic calculus and developing martingales Steele covers arbitrage and stochastic differential equations leading up to the fundamental Black-Scholes theory that is important in financial applications. It is not fair to criticize this book for lack of applicability. It is strickly intended to develop a firm theoretical background for the students that will prepare them for a deep understanding of financial models important in applications.

I am not enough of an expert in this area to know if Professor McCauley's criticism in another amazon review of this book is valid, but I do think he is a little too harsh in criticizing the ideology that Steele presents. The ideology is what makes Steele's lectures stimulating and interesting to the students.

I took the author's course (at Wharton) on the subject when his book was in its early stages. I went very carefully through the notes (chapters of the book), and I learned a great deal (which is why I have purchased the final product). Given that I had previously used Musiela and Rutkowski ("Martingale Methods in Financial Modelling") in a Columbia graduate course, this was a considerable feat.
Steele, a Wharton Statistics professor, uses financial applications to motivate stochastic calculus from a particular perspective. I have no doubt that he sees stochastic calculus as a field that exists outside of finance and that he does not intend to teach the reader finance theory. His goal, I believe, is to offer a text that is more readable than the classic text of Karatzas and Shreve ("Brownian Motion and Stochastic Calculus"). In my opinion, he has accomplished this goal.
Protter ("Stochastic Integration and Differential Equations: a new approach") does an excellent job, as he is clear and develops the theory in greater generality (using semi-martingales). However, his text is highly theoretical and offers no finance applications. Duffie ("Dynamic Asset Pricing Theory") and Musiela and Rutkowski (above) do not offer the reader the necessary stochastic calculus background.
Lastly, this is a non-trivial subject. For people who do not sit down by themselves and put in the required hours, the outcome will be disappointing.

The book says that its only prerequisites are calculus and probability. This is not true. To be able to understand everything that's going on, you'll need to have a very good grasp of subjects like measure-theoretic probability, Hilbert spaces, and functional analysis. I quit reading the book in the early chapters, when Steele starts talking about things like "spans" and "denseness" for function spaces. I don't know where you went to school, but at my school, I didn't learn these subjects in my intro calculus and probability classes. To summarize, don't buy this book if you don't know measure theory.

If you want to learn quant finance at an elementary level, Baxter and Rennie is much, much better. Moreover, if you're comfortable with measure theory,and you want to learn the math that's necessary for option pricing, you'd be better off buying Oksendal's excellent book, which is at least as rigorous as Steele's book but much more clear.

The book is at the interface of three areas, math, statistics, and finance. While connections between the first two have a long history, it was the connection to finance that caught my attention. Coming from math myself, I needed first to take a closer look at the book to orient myself. The mathematical subjects, smooth sailing, include stochastic differential equations (SDE) as they relate to PDEs; and the ideas from probability and statistics include Brownian motion, martingales, stochastic processes, and the Feynman-Kac connection. Browsing the chapters I found them to be a lovely presentation of ideas with which I am familiar. For me, it was chapter 10 that turned out to have stuff that I wasn't familiar with. That is the finance part, and it is based on a model for Option Pricing developed in 1973 by Fischer Black and Myron Scholes. An arbitrage opportunity [simplified] amounts to the simultaneous purchase and sale of related securities which is guaranteed to produce a *riskless* profit. It was after reading more in this chapter I understood why the book is used in a course at the Wharton School at the University of Pennsylvania. I am impressed with the level of math in this course. Part of the motivation in the applications to finance is that arbitrage enforces the price of most derivative securities. And I learned from ch 10 that the SDE of the Black-Scholes model governs the processes which represent the two variables S, the price of a stock, and B the price of a bond, both S and B representing stochastic variables depending of time t, i.e., both stochastic processes. In the model, S is a geometric Brownian motion, and B is a deterministic process with exponential growth. The two are determined as solutions to the SDE of Black-Scholes.

I just started reading Steele's book. It is very well written, the mathematics is very well explained, and uses lots of very down to earth examples and motivations. But what is really refreshing about the book is its style: Informal and full of side remarks and jokes that are actually funny. Look up the three bonus observations on page 291. Even the typos are funny: Is using geometric Brownian motion to model stock prices a 'bold' or a 'bald' assumption? I hope to write a more technical and detailed review in the future.

I have read many books in my quest to understand Stochastic Calculus. Of all the books out there this one along perhaps with Financial Calculus (Baxter&Rennie) is the only book that describes the *essence* or *core* of what you must understand! I only gave this book 5 stars because I could not give it more. Thank you Professor Steele.

Sorry to disagree with my esteemed "colleagues", but this book is not the best choice if you are (aspiring) financial market professional. The author is clearly in love with his subject (cf., p. 61: "One could spend a lifetime exploring the delicate - and fascinating - properties of the paths of Brownian motion. Most of us cannot afford such an investment, so hard choices must be made." Hard indeed!). If you are not a mathematician, a book by Klebaner ("Introduction to Stochastic Calculus with Applications") can teach you everything you need to know about stochastic calculus.

I am completely new to the subject of stochastic calculus and bought this book because it had been recommended as a good introductory text. The exposition is well structured and the author does a good job of introducing things using the first principles. However the author has a tendency to introduce variables and notation without explanation or comment. While there may be a "standard notation" in the corpus of stochastic calculus, no such assumption should be made in an introductory text on the subject. I found this aspect of the book very frustrating. But whenever I could infer the notation, the explanations were very easy to understand.

The goal of this book is to disseminate the knowledge of a very technical subject to a very wide range of audience, including finance professionals. The author did a respectable job in that regard. With some improvement in future revisions, this book seems to be one of the best introductionary texts on stochastic calculus.