Rating: 
Summary: Would be better without the ideology
Review: Good intro to stochastic calculus and sde's, can compare roughly with Baxter and Rennie in readability. However, the book unnecessarily propagates ideology. First, it makes excuses for the fact that the empirically wrong notion of utility ('maximizing behavior') is totally disconnected from the Black-Scholes model. Second, the text propagates Black and Schole's original mistaken claim that CAPM produces the same option pricing pde as does the delta hedge. A careful and correct calculation shows that this claim is wrong, that with the wrong assumption made by B-S the fractions invested in both the stock and the option are zero! For the correct result, including the difference in option pricing via delta hedge and CAPM, see my recent paper "An Empirical Model for Volatiliy of Returns and Option Pricing' with Gunaratne. A third criticism is that only Gaussian returns are discussed in this text, but the empiciral distribution is far from Gaussian and is approximately exponential, with nontrivial volatility.
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Summary: Good explanation but bad notation
Review: 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.
Rating: 
Summary: the best play to start to learn stochastic calculus
Review: I bought this book rather accidentally. I was looking for a crash course in stochastic calculus, I tried a few different textbooks, but I was not satisfied with any of them: most were overloaded with technical details and I could not afford to spend time on it at that point. I was interested in applications of stochastic calculus in finance (I was taking Financial Economics course and we were supposed to be familiar with stochastic calculus), but most applied textbooks that I saw only supplied Ito's formula without developing any good intuition behind it. Then, following excellent reviews at Amazon and "Finance" in the title, I decided to try Steele's book. And I am glad I did. It turned out to be not exactly a "crash course", but it moves quickly without skipping crucial details. The author presents material very carefully, he knows that his audience are not mathematicians, so he invests a lot of efforts in trying to develop a good intuition about the subject. At the same time, almost all results are given with proofs, and he tries to be always precise and rigorous. He starts with martingales, then presents Brownian Motion in very detailed manner, then he moves to Ito's integral, stochastic differential equations, diffusion processes. He demonstrates how these methods can be applied in finance to options pricing and derives Black-Scholes formula in a number of different ways. He pays particular attention to martingale representation theorems and equivalent martingale measures results. He presents material following the lecture style which makes this book a very enjoyable reading. The bottom line: this is a great book, and I am glad I have started to learn stochastic calculus using Steele's book. One final comment. Editorial review says that the book assumes only a modest background. I do not know what they mean by modest, but in order to read and fully appreciate it, you have to be familiar with probability measure theory, convergence results and L-spaces, but, actually, may be it is a modest background.
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Summary: Finally!
Review: 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.
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Summary: First book on mathematics that is actually funny
Review: 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.
Rating: 
Summary: Very good intro to stochastic calculus and applications
Review: 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.
Rating:
Summary: It is a great book!
Review: It is a wonderful book. It was written by my teacher Dr. Steele. This book contains not only theoretical proof but also links to financil model. It is a good bridge from Probability to Finance. You will find it useful and interesting. And this book will teach you how to use Ito integration, which is a most beautiful tool!
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Summary: graduate course taught at Wharton
Review: 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.
Rating: 
Summary: Review from a grad student not at Wharton
Review: Reading Steele's book without attending has classes at Wharton leaves the reader looking for explanations to equations. Ideas are not clearly explained and problems are not worked out in detail with a descriptive process of how to solve the problem. The brief explanations in this book intended for a reader with knowledge of calculus and probability but not having a background in Stochastic calculus do not provide a sufficient basis for the reader to learn the material.
Rating:
Summary: Too mathematically involved, low in applicability
Review: 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.
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