Think of this as a thank-you letter to Shreve for helping to teach me applied quantitative finance. This is a truly wonderful book and a great place to start learning the subject, regardless of your previous exposure to the subject or mathematical maturity, and has plentiful opportunities in the exercises to practice important results.
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.