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Stochastic Calculus and Financial Applications (Stochastic Modelling and Applied Probability) [Englisch] [Gebundene Ausgabe]

J. Michael Steele
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Kurzbeschreibung

27. Juni 2003 0387950168 978-0387950167 1st ed. 2001. Corr. 3rd printing 2003

Stochastic calculus has important applications to mathematical finance. This book will appeal to practitioners and students who want an elementary introduction to these areas.

From the reviews: "As the preface says, ‘This is a text with an attitude, and it is designed to reflect, wherever possible and appropriate, a prejudice for the concrete over the abstract’. This is also reflected in the style of writing which is unusually lively for a mathematics book." --ZENTRALBLATT MATH


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Produktinformation

  • Gebundene Ausgabe: 302 Seiten
  • Verlag: Springer; Auflage: 1st ed. 2001. Corr. 3rd printing 2003 (27. Juni 2003)
  • Sprache: Englisch
  • ISBN-10: 0387950168
  • ISBN-13: 978-0387950167
  • Größe und/oder Gewicht: 2,5 x 15,8 x 23,3 cm
  • Durchschnittliche Kundenbewertung: 4.0 von 5 Sternen  Alle Rezensionen anzeigen (2 Kundenrezensionen)
  • Amazon Bestseller-Rang: Nr. 425.368 in Fremdsprachige Bücher (Siehe Top 100 in Fremdsprachige Bücher)
  • Komplettes Inhaltsverzeichnis ansehen

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From the reviews:

MATHEMATICAL REVIEWS

"…on the whole, the results are presented carefully and thoroughly, and I expect that readers will find that this combination of a careful development of stochastic calculus with many details and examples is very useful and will enable them to apply the whole theory confidently."

SHORT BOOK REVIEWS

"This is a world of 'lovely exercises' that are 'very good good for the soul', 'honest martingales', 'bedrock approximations', portfolios that are 'born to lose', 'intuitive but bogus arguments', and 'embarrassingly crude insights'. In short, this is a book on stochastic calculus of a different flavour. Intuition is not sacrificed for rigour nor rigour for intuition.The main results are reinforced with simple special cases, and only when the intuitive foundations are laid does the auhtor resort to the formalism of probability. The coverage is limited to the essentials but nevertheless includes topics that will catch the eye of experts (such as the wavelet construction of Brownian motion). This is one of the most interesting and easiest reads in the discipline; a gem of a book."

JOURNAL OF THE AMERICAN STOCHASTIC ASSOCIATION

"The book is indeed well written, with many insightful comments. I certainly would recommend it to students wishing to learn stochastic calculus and its applications to the Black-Sholes option-pricing theory…I thoroughly enjoyed reading this book. The author is to be complimented for his efforts in providing many useful insights behind the various theories. It is a superb introduction to stochastic calculus and Brownian motion…An interesting feature in this book is its coverage of partial differential equations."

"It is clear that this is a fairly comprehensive introduction to the tools of (classical) mathematical finance. … the text has much to offer. … In addition, the writing style is refreshingly informal and makes a book about a rather technical subject surprisingly enjoyable to read. In short, despite the recent deluge of textbooks in this area, I know of no better book for self-study." (Christian Kleiber, Statistical Papers, Vol. 46 (2), 2005)

"Steele’s book is a sophisticated introduction to stochastic calculus with applications from basic Black-Scholes theory. … I highly recommend the book. His style is wonderful, and concepts really build on one another. … it offers one of the most elegant treatments of the subject that I know of." (www.riskbook.com, May, 2006)

"As is clear from the title of this book, it is concerned with applications of stochastic calculus to finance. … one naturally judges the book by three criteria: topic selection, organization, and exposition. In all three domains the book succeeds. The topics selected are rich enough … he or she will benefit from the book. … there are innovations as well … from the pedagogic standpoint." (Philip Protter, SIAM Review, Vol. 43 (4), 2001)

"This book offers rich information and a mathematically honest treatment of stochastic calculus and of its use in the theory of finance … . The author gradually builds the reader’s ability to grasp stochastic concepts and techniques … . the author’s presentation of stochastic models in finance and economy is precise and extensive … . Each chapter is accompanied by a collection of rather challenging exercises … ." (EMS Newsletter, December, 2002)

"The present book ‘is designed for students who want to develop professional skill in stochastic calculus and its application to problems in finance’. … the textbook … retains a lovely lecture style focusing basic ideas and not formalities and technical details of stochastic processes needed for finance. I can strongly recommend this book to students of mathematics and physics as well as non-experts in probability theory who are interested in stochastic finance." (H. –J. Girlich, Zeitschrift für Analysis und ihre Anwendungen, Vol. 21 (4), 2002)

"The last few years have been a fertile period for books on stochastic calculus and its financial implications, but this one differs from the many mainstream treatments … . The style of the book creates the atmosphere of a lively lecture … . Each chapter ends with a section of carefully chosen exercises, preceded by some motivating remarks. … I really liked the book." (R. Grübel, Statistics & Decisions, Vol. 20 (4), 2002)

"This book gives an introduction to stochastic calculus … with applications in mathematical finance. … As the preface says, ‘This is a text with an attitude, and it is designed to reflect, wherever possible and appropriate, a prejudice for the concrete over the abstract’. This is also reflected in the style of writing which is unusually lively for a mathematics book. … on the whole, the results are presented carefully and thoroughly … ." (Martin Schweizer, Zentralblatt MATH, Vol. 962, 2001)

"This is a book on stochastic calculus of a different flavour. Intuition is not sacrificed for rigour nor rigour for intuition. The main results are reinforced with simple special cases … . This is one of the most interesting and easiest reads in the discipline; a gem of a book." (D. L. McLeish, Short Book Reviews, Vol. 21 (1), 2001)

Synopsis

This book is designed for students who want to develop professional skills in stochastic calculus and its application to problems in finance. 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 Ito integral and aims to provide a development that is honest and complete without being pedantic.With the Ito 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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1 von 2 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Riskfree profit ! 9. März 2003
Format:Gebundene Ausgabe
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.
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3.0 von 5 Sternen Would be better without the ideology 27. Januar 2004
Format:Gebundene Ausgabe
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.

Note added 2/25/07: The martingales presented as elementary examples are very instructive, if you derive them yourself (in some cases, some thought is required to figure out why this initial condition or that boundary condition). The Black-Scholes solution as martingale for arbitrary drift and diffusion coefficients is presented as a Feynman-Katz functional integral on pg. 272, but with unnecessarily complicated notation. However, eqns. (15.25) and (15.27) are inconsistent and (15.25) is wrong: you cannot freely shift coordinate origins because the x-dependent drift and diffusion coefficients break translational invariance. For a more complete treatment, see cond-mat/0702517 on xxx.lanl.gov.
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Amazon.com: 3.9 von 5 Sternen  18 Rezensionen
41 von 43 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen A Beautiful MATH Book 20. Juni 2006
Von longhorn24 - Veröffentlicht auf Amazon.com
Format:Gebundene Ausgabe
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.
24 von 24 Kunden fanden die folgende Rezension hilfreich
4.0 von 5 Sternen nice treatment of a difficult subject in probability 22. Januar 2008
Von Michael R. Chernick - Veröffentlicht auf Amazon.com
Format:Gebundene Ausgabe
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.
24 von 25 Kunden fanden die folgende Rezension hilfreich
4.0 von 5 Sternen Very good intro to stochastic calculus and applications 15. Oktober 2001
Von Mr. Nikolay K. Kolev - Veröffentlicht auf Amazon.com
Format:Gebundene Ausgabe
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.
39 von 47 Kunden fanden die folgende Rezension hilfreich
2.0 von 5 Sternen I Hate It When Books Lie About Mathematical Requriements 2. Mai 2003
Von Ein Kunde - Veröffentlicht auf Amazon.com
Format:Gebundene Ausgabe
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.
16 von 19 Kunden fanden die folgende Rezension hilfreich
5.0 von 5 Sternen Riskfree profit !! 9. März 2003
Von Palle E T Jorgensen - Veröffentlicht auf Amazon.com
Format:Gebundene Ausgabe
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.
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