Stochastic Calculus and Financial Applications |  | Author: J. Michael Steele
Publisher: Springer
List Price: $99.00
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Rating:  17 reviews
Sales Rank: 470745
Media: Hardcover
Edition: Corrected
Pages: 344
Number Of Items: 1
Shipping Weight (lbs): 1.1
Dimensions (in): 9.2 x 6.1 x 0.8
ISBN: 0387950168
Dewey Decimal Number: 519.2
EAN: 9780387950167
ASIN: 0387950168
Publication Date: June 3, 2003
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Product Description
The Wharton School course on which the book is based is designed for energetic students who have had some experience with probability and statistics, but who have not had advanced courses in stochastic processes. Even though the course assumes only a modest background, it moves quickly and - in the end - students can expect to have the tools that are deep enough and rich enough to be relied upon throughout their professional careers. The course begins with simple random walk and the analysis of gambling games. This material is used to motivate the theory of martingales, and, after reaching a decent level of confidence with discrete processes, the course takes up the more demanding development of continuous time stochastic process, especially Brownian motion. The construction of Brownian motion is given in detail, and enough material on the subtle properties of Brownian paths is developed so that the student should sense of when intuition can be trusted and when it cannot. The course then takes up the It(tm) integral and aims to provide a development that is honest and complete without being pedantic. With the It(tm) integral in hand, the course focuses more on models. Stochastic processes of importance in Finance and Economics are developed in concert with the tools of stochastic calculus that are needed in order to solve problems of practical importance. The financial notion of replication is developed, and the Black-Scholes PDE is derived by three different methods. The course then introduces enough of the theory of the diffusion equation to be able to solve the Black-Scholes PDE and prove the uniqueness of the solution.
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| Customer Reviews:
Showing reviews 1-5 of 17
 deceived by the publisher March 22, 2009
L. CUNHA (Sao Paulo, Brasil)
0 out of 1 found this review helpful
I felt extremely disappointed to look into this book the minute I received it. I usually love all springer books specially for their top quality presentation. This book is an exception. You pay nearly [...] bucks for a book to get a fac-simile of the original with less margin space, larger fonts, and less line-to-line space making it overly populated of characters and ugly to the eye. Not even Dover books are like this, despite their lower prices.
I am definitely contacting springer on this and I am seriously thinking of returning the book to amazon and getting a used one with better presentation quality.
 A Pleasure to Read February 25, 2009
S. L. Starr (New York, USA)
I just got this book. I've been looking at a lot of introductory books on Mathematical Finance, and this seems like one of the best. As other authors have said, the level of this book is consistent with graduate study. I happen to be a mathematician. This is the best book I've found for (graduate student and higher level) mathematicians interested in learning math finance. Also, if you're a mathematician teaching undergraduate math finance (and assuming you're not already an expert) I highly recommend this book for your "bookshelf." Having this type of in-depth knowledge provides the ballast you need to do a proficient job teaching.
Strikes a rare balance May 23, 2008
Alexander Sokol
1 out of 1 found this review helpful
I find myself wishing again and again that I had discovered this little book years ago. Steele goes through the basics of the parts of the stochastic calculus relevant for financial mathematics and applies it to basic financial models. The theory covered is basic martingale theory, the stochastic integral with respect to standard processes - that is, processes of the form X = A + M, where A is continuous and of finite variation and M is a stochastic integral with respect to the Wiener process. He proceeeds to develop the quadratic variation, Itô's formula, the martingale representation theorem, Girsanov's theorem and the Feynman-Kac formula. He also proves results of exactness of SDEs.
Although Steele often manages to pretend that his book is rigorous, it isn't. There are several occasions where Steele jumps over details, particularly about measurability. I don't mind not being given all the details, but it is a bit annoying that he doesn't admit when he's forgetting some details.
In any case, even though the book isn't completely rigorous, the basic structure of the proofs is often the right one. The really great thing about this book, then, is this: It shows how to develop the stochastic calculus relevant to fundamental financial models, and no more. Books like Karatzas & Shreve, Rogers & Williams and Protter are much more advanced than often is necessary. On the other hand, books like Björk or Duffie are much too sloppy. Even if Steele misses af few details here and there, most of the time, he is rigorous, and thus this book offers the rare combination of treating only RELEVANT theory (no general semimartingales here) and doing so rigorously.
A good companion volume is Karatzas & Shreve, which is much more detail-oriented. Whenever you find some holes in Steele's book, Karatzas & Shreve will usually have the answer.
nice treatment of a difficult subject in probability January 22, 2008
Michael R. Chernick (Holland PA)
23 out of 23 found this review helpful
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.
A Beautiful MATH Book June 20, 2006
longhorn24 (New York, NY)
30 out of 30 found this review helpful
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.
Showing reviews 1-5 of 17
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