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Stochastic Calculus for Finance II: Continuous-Time Models Paperback – Oct 4 2013
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From the reviews of the first edition:
"Steven Shreve’s comprehensive two-volume Stochastic Calculus for Finance may well be the last word, at least for a while, in the flood of Master’s level books.... a detailed and authoritative reference for "quants” (formerly known as "rocket scientists”). The books are derived from lecture notes that have been available on the Web for years and that have developed a huge cult following among students, instructors, and practitioners. The key ideas presented in these works involve the mathematical theory of securities pricing based upon the ideas of classical finance.
...the beauty of mathematics is partly in the fact that it is self-contained and allows us to explore the logical implications of our hypotheses. The material of this volume of Shreve’s text is a wonderful display of the use of mathematical probability to derive a large set of results from a small set of assumptions.
In summary, this is a well-written text that treats the key classical models of finance through an applied probability approach. It is accessible to a broad audience and has been developed after years of teaching the subject. It should serve as an excellent introduction for anyone studying the mathematics of the classical theory of finance." (SIAM, 2005)
"The contents of the book have been used successfully with students whose mathematics background consists of calculus and calculus-based probability. The text gives both precise Statements of results, plausibility arguments, and even some proofs. But more importantly, intuitive explanations, developed and refine through classroom experience with this material are provided throughout the book." (Finanz Betrieb, 7:5, 2005)
"The origin of this two volume textbook are the well-known lecture notes on Stochastic Calculus … . The first volume contains the binomial asset pricing model. … The second volume covers continuous-time models … . This book continues the series of publications by Steven Shreve of highest quality on the one hand and accessibility on the other end. It is a must for anybody who wants to get into mathematical finance and a pleasure for experts … ." (www.mathfinance.de, 2004)
"This is the latter of the two-volume series evolving from the author’s mathematics courses in M.Sc. Computational Finance program at Carnegie Mellon University (USA). The content of this book is organized such as to give the reader precise statements of results, plausibility arguments, mathematical proofs and, more importantly, the intuitive explanations of the financial and economic phenomena. Each chapter concludes with summary of the discussed matter, bibliographic notes, and a set of really useful exercises." (Neculai Curteanu, Zentralblatt MATH, Vol. 1068, 2005)
From the Back Cover
Stochastic Calculus for Finance evolved from the first ten years of the Carnegie Mellon Professional Master's program in Computational Finance. The content of this book has been used successfully with students whose mathematics background consists of calculus and calculus-based probability. The text gives both precise statements of results, plausibility arguments, and even some proofs, but more importantly intuitive explanations developed and refine through classroom experience with this material are provided. The book includes a self-contained treatment of the probability theory needed for stochastic calculus, including Brownian motion and its properties. Advanced topics include foreign exchange models, forward measures, and jump-diffusion processes.
This book is being published in two volumes. This second volume develops stochastic calculus, martingales, risk-neutral pricing, exotic options and term structure models, all in continuous time.
Masters level students and researchers in mathematical finance and financial engineering will find this book useful.
Steven E. Shreve is Co-Founder of the Carnegie Mellon MS Program in Computational Finance and winner of the Carnegie Mellon Doherty Prize for sustained contributions to education.See all Product Description
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A background with measure theory helps. Also basic financial knowledge of products.
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The first three and part of the fourth chapter serve as the mathematical preparation for the book. Shreve reviews basic concepts from probability, introducing just enough measure-theoretic concepts to understand the motivation behind the concepts of a filtration and its relation to conditional expectation, martingales, and later in a brief chapter on American options, stopping times. Since the book's main emphasis is on the application of the Ito-Doeblin calculus in solving SDE generated by Brownian motion, Chapter 2 covers the necessary elements of conditional expectation for risk-neutral pricing. Chapter 3 covers Brownian motion, although not rigorously - he gives just enough properties of the canonical continuous stochastic process to know how to identify it and to understand its crucial properties. This chapter is important for the first part of Chapter 4, which uses the properties of Brownian motion to develop the notion of quadratic variation and its role in the calculation of the Ito Integral. After developing the Ito integral and demonstrating its key properties, such as the martingale property and the Ito isometry, Shreve has enough math to start developing the Black-Scholes-Merton framework for actual finance.
Chapters 5 covers risk-neutral pricing as a more general case of the BSM model, and in addition to demonstrating important results to finance such as Girsanov's theorem and its role in the Martingale Representation Theorem, Shreve also covers extensions such as the relationship between Forwards and Futures prices. In addition, he extends the classical BSM formula to include dividends, a generalization which plays a key role in the pricing of currency options in the Garman-Kohlhagen model.
Chapter 6 shows how, through the application of the Feynman-Kac formula to Markov processes, the probabilistic (here, the martingale) approach can be connected to the PDE approach whenever a problem is (or can be made) Markov. At the end of this chapter and in the next chapter on Exotic Options, Shreve shows how adding additional states can reduce the pricing problem of a path-dependent option, such as an Asian option, to the Markov case. The presentation is particularly nice and through playing with some of the exercises, the reader can build the ability to reduce a seemingly complicated payoff to a simpler case and see how it's just another case of the same general theme.
Chapters 7 and 8 cover Exotic and American options, respectively, although each are meant only as introductions. One can see through the pricing of various exotic options that the difficulty lies more in algebra and basic calculus than in actual abstraction; the idea emphasized here is that setting up the problem correctly is the hard (although certainly less tedious) part of the problem. Chapter 8 only touches on the important concepts of American options, namely that to price them one must know how to identify a stopping time, understand what it means in non-mathematical terms, and understand its application to pricing.
Chapter 9, a generalization to the chapter on Risk-Neutral pricing, covers change of measures. While this isn't terribly difficult to grasp, it is important not just for currency pricing problems but also for more advanced Market Models through the use of forward measures.
Chapter 10, one of the longer ones in the book, covers a full range of term structure models. Shreve covers the older class of models, which require only the use of previously developed SDE, as well as an introduction to the HJM framework and its application to Modern Market Models. This is a subject not just of importance to quants working in the vast universe of fixed income derivative pricing, but also for all students wanting to test the power of risk-neutral pricing in a modern setting. Shreve's presentation seems to be a natural extension of the risk-neutral framework and makes a relatively difficult concept easy to grasp. Despite the emphasis on the HJM framework and the use of forward measures, Shreve doesn't neglect the classical term structure models, covering many of them both in the text and giving their solutions and some of their statistical properties through exercises.
The final chapter comes with a warning: Jump processes aren't easy to understand. Shreve succeeds wildly in teaching a very difficult subject quite well, building up from Poisson processes to compound processes, and then extending the same change-of-measure techniques to show how the risk-neutral approach works in this case too. While the book would have been complete in a pedagogical sense without this chapter, its inclusion reflects the increasing importance of jumps in everything from credit models to the volatility smirk/smile. It's no secret Levy processes and generalized jump models will play an increasingly important role in financial modeling, and Shreve is trying to show how the first 10 chapters of the book in some way provides some of the general ideas useful for these extensions.
The problems in this book are excellent and range in difficulty, length, and purpose, although the harder ones have copious hints; this book is clearly meant to learn how to apply a few basic ideas to models through applications, not to provide deep abstractions on the subject. Nonetheless, they span a range of topics and in some cases fill blanks in areas not covered in the text, ranging from the construction of the volatility surface to the portfolio dynamics of an arbitrage strategy.
Sometimes we like books which are both terse and mathematically elegant. This isn't one of them, nor does it pretend to be either. It's a way for a hard-working student to get up to speed on the basic mathematical tools and concepts used in derivative pricing and in other areas of asset pricing in finance. The emphasis is on learning by doing, many of the problems are extensions of examples in the text, while others are very long problems with plenty of hints, meant to encourage the reader to learn by "filling in the blanks."
Again, Shreve deserves my thanks as well as those of anyone who learned from this great book (or its predecessor, the lecture notes...). For those who want to complement this book with a more rigorous treatment of the SDE given in the book, Oksendal's book is about a half a step higher in mathematical rigor and covers important concepts not covered in Shreve related to PDE and diffusions, as well as applications to optimal control and other subjects important outside (and in!) derivative pricing. If you feel comfortable with PDE and Real Analysis, complement Shreve with this text to get a fairly strong background in stochastic calculus and its applications.
Suffice it to say that there were a lot of abstract building blocks one had to erect first before one could actually do anything useful.
Stochastic calculus is not easy. It is less intuitive than ordinary calculus. The vast majority of textbooks launch into a wall of definitions that seem divorced from the motivation for them. I am always suspicious of authors who do that. It's fine if you are writing for a very specialised audience but I am with Richard Feynman who reckoned that if you can't provide a simple explanation you don't really understand what is going on. In that context read his PhD thesis - it is most readable and understandable.
What Shreve has done - and this is a significant achievement in my view - is to present something that is rigorous enough (and we all know that in this and other areas of mathematics one can go on and on with minute points of detail all in the name of rigour) yet grounds the concepts in something that is understandable.
The simple pedagogical fact of life with this type of material is that there is a large overhead in getting to a particular point and Shreve had done a very good job in getting readers to a good standard without destroying their will to go on!
When one looks at areas of mathematics with much longer pedigrees - and Fourier Theory is an example - there are some extremely good presentations of the theory at both mathematical and physical levels. Elias Stein, for instance, has done some marvellous work in the area. Stochastic calculus is really very young in terms of mainstream appeal. I can recall actuarial subjects I did in the early 1980s that had no stochastic calculus at all in them. All that has changed and I think Shreve's attempts in this area can be improved upon too but this will only happen over time.
My colleagues in quant like Shreve's books so I guess that says something too.
Kazartas & Shreves' classic text book is too rigorous and very demanding, it would give readers solid theoretical background, but I think only few readers can really master in those material.
That's why books like Oksendal's SDE come into the market, they are easier than Kazartas & Shreve but deeper than many undergrad Financial Mathmetics in theory. Oksendal is easy reading and good for self study; however, it's Finance part is relatively weak.
For those Finance or Financial Engineering people, Steele's book fits well, It is right at the level like Oksendal, and root at Finance application. It's story telling style makes it joyful in reading, but bad in reference.
Finally we have Shreve's new book. This book is at the level of demanding as Oksendal and Steele's books. You may still need some grad-level mathematics training to understand the stuff well. But unlike other stochastic calculus books, it is designed for Finance field. Finance guy nomatter practitioners or researchers can soon find help they need in this books. Also it is well-organized and with nice writing style. Although the first couple chapters are a little too condense, I still highly appreciate this book.
This is a good book and covers all the topics in a well rounded manner, he also has a very good little section in which he addresses his competitors, such as Steele etc etc,
IF you want a really ridiculous read and to show off to your mates then I recommend Musiela and Rutkowski, which I use to prop my door open in hot weather, this book has pretty much everything but is written in a very dense and inaccessible manner, you get nervous opening it, as you find something new you didn't know every time, I don't like abook to make me feel that dumb and its not really an sde book!!!!!!
In summary I am happy with my purchase of Shreve, many moons ago, I will use it again to teach an MSc course and the students will again complain that its too hard, until I give them a few refs and they will understand that you can't just waltz into the city and say I wanna be a quantitative analyst it takes hard work. Reading Shreve puts you on the right road and you can't say anything more highly than that.
As to the discussion by previous reviewers on the Ito-Doebin formula I suggest Karatzas and Shreve will answer you arguments.
Other pros of the book include:
+ Self-contained (Well-written intro's to background materials are presented in the first two chapters.)
+ Presents both theory and intuition
+ Carefully planned end-of-chapter exercise problems
+ Consistent mathematical notations
+ Each chapter ends with an excellent summary
+ Historical notes (on who discovered which theorem and how they came about) help to put each theorem into perspective
Overall, it is an ideal textbook for a "first course" in mathematical finance.
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