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Arbitrage Theory in Continuous Time

Arbitrage Theory in Continuous Time (Oxford Finance Series) By Tomas Bjork * Publisher: Oxford University Press, USA * Number Of Pages: 488 * Publication Date: 2004-05-06 * ISBN-10 / ASIN: 0199271267 * ISBN-13 / EAN: 9780199271269 * Binding: Hardcover Product Description: The second edition of this popular introduction to the classical underpinnings of the mathematics behind finance continues to combine sounds mathematical principles with economic applications. Concentrating on the probabilistics theory of continuous arbitrage pricing of financial derivatives, including stochastic optimal control theory and Merton’s fund separation theory, the book is designed for graduate students and combines necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises and suggests further reading in each chapter. In this substantially extended new edition, Bjork has added separate and complete chapters on measure theory, probability theory, Girsanov transformations, LIBOR and swap market models, and martingale representations, providing two full treatments of arbitrage pricing: the classical delta-hedging and the modern martingales. More advanced areas of study are clearly marked to help students and teachers use the book as it suits their needs. Summary: Nicely Prepared Intermediate-Level Treatment Rating: 4 The author has put together an excellent text that will take readers of an elementary text like Hull’s Options, Futures, and Other Derivatives to the next level. In the author’s treatment, the power of stochastic calculus is brought to bear on the options pricing problem from the point of view of modern martingale theory, if not the complete mathematical rigor needed to establish all the results. The text contains 26 chapters and 3 appendices. There is simply too much here to give a blow-by-blow account. So I’ll try to hit the highlights. The author gives intuitive definitions of some of the more heavy concepts from measure theory/Lebesgue integration, measure-theoretic probability theory and basic stochastic analysis. For the rigor, one need only look to the appendices, but the treatment is intuitive enough that can still follow along with only the occasionally glance to the back of the book. Readers of Hull’s text will find the first couple of chapters quite familiar, but starting in Chapter 4, stochastic integrals are (somewhat) formally introduced, along with the multi-dimensional version of Ito’s change of variable rule. This is not overkill as the development of multi-factor term structure models later in the book benefits from this early development. We note that these formulas are stated without proof, although they are motivated intuitively. In the next chapter, stochastic differential equations are introduced and the Feynman-Kac representation is established as a nice application of Ito’s rule. The chapter winds up with an intuitive treatment of Kolmogorov’s forward & backward equations. For the remainder of the first half of the text, readers of Hull will feel themselves in quite familiar territory, as the author develops the solution for the options pricing problem, studies the Greek letters and establishes parity using the now classical approach. The second half of the text delves into martingale methods for mathematical finance. As a consequence, the sophistication level jumps considerably. The reader is well-advised to get the basic analytical toolkit in hand before delving too far into the second half of the book. I recommend Rudin’s Real and Complex Analysis. Heavy machinery is pulled in from functional analysis to establish the first and second fundamental theorems of mathematical finance. Without some basic understanding of Hilbert and Banach space theory, the reader will understand very little of this treatment. A good reference for this is Rudin’s Functional Analysis The next highlight is the Girsanov Theorem. The author actual provides a proof in the scalar case, and presents (without proof) the Novikov condition to test when the Girsanov transformation is indeed a martingale (so the theorem holds). As a nice application, the Black-Scholes theory is revisted and re-established via these martingale results. Another highlight is the study of the Hamilton-Jacobi-Bellman model for stochastic control, along with a small catalogue of cases under which the HJB equations can be solved. As a nice application, Merton’s mutual fund theorem is established. The last several chapters of the book deal with martingale methods for term structure models. There is a nice survey and study of the 1-factor short rate models before loading up and doing the k-factor model framework of Heath-Jarrow-Morton. The martingale setting makes for a very rigorous treatment. The book ends with a really nice treatment of the Libor Market and Swap Market Models. Pure finance students may feel that the mathematics at the end unnecessarily overwhelms the intuition, but students of mathematical finance will appreciate the analytical treatment and may even feel inspired to implement their own LMM. There are a ton of terrific exercises at the end of each chapter. The exercises really solidify the understanding of the presentation and they make great technical interview questions as well. Summary: intuitive introduction to option pricing Rating: 5 I agree with several reviewers above that the book is written in a style very helpful for students to understand the material. It doesn’t contain a lot of small details of financial markets like Hull’s book, but the approach is very systematic. The derivations of formula for Barrier options is a nice example, Hull only lists a set of formula. The focus is on the theory, not on the practice. (No numerical method in the book). Bjork’s book is very valuable for a student with very good math skills but want to learn the reasoning style for option pricing. It is a quick and enjoyable read. A huge plus side of the book is to describe strategy before writing down all the proofs. This helps greatly. It can be contrasted with Duffie’s book "Dynamic Asset Pricing Theory", which is written like a dry math book (well, I have to admit that Duffie’s book is not an intro book) Only thing I can think of that can be improved is typo in the book, too many wrong formula, especially in the second half of the book, luckily enough, they are obviously wrong so that one can still understand the topics. I also find that using SEK and mentioning street name of Britain are amusing for a student in U.S. Summary: Hell, I should have rated it 5 stars! Rating: 4 If you’re going to be introduced to Derivatives pricing and Quantitative finance in continuous time, you need some basics in probability theory, an elementary introduction to stochastic calculus and you need "bjork". It tells you the equation and how to understand it. It’s the best source for a complete understanding of the basics of arbitrage free pricing in continuous time; whether it’s in complete or incomplete markets. The best feature of this book is how the author invariably provides an "intuitive interpretation or explanation" to convey critical concepts. {Things like market price of risk in the context of interest rate modelling, change of measure etc...} Why I rated the book 4 instead of 5? I will not forgive "Tomas bjork" not to have covered the Libor Market Model; it’s "THE" model and therefore should be covered in great details by any book of this calibre. A new edition of this book with the libor market model is needed. Having said that, the coverage he gives to the popular short rate models is worth every read! Guy, Msc Financial Engineering at ISMA Center, Reading - UK. Summary: Good introductory book Rating: 4 It is a good book to read as an introduction to the field. The author is successful in conveying the intuition behind the models instead of striving for complete mathematical rigor. I recommend this book if you want to quickly get acquainted with derivatives pricing but are a bit afraid of the higher math seen in other books. Summary: An FE Bible Rating: 5 The central text for IOE 552(financial Engineering I) at the University of Michigan. Halfway through the course and I really understand the application of Ito’s Lemma and the Feynman-Kac stochastic representation theorem. This book has just the right mixture of narative story telling, and mathematical rigor. The derivations are accessible to those with a semester of advanced calculus and a semester of probability. Over and over, Bjork shows that the secret of success in Financial Engineering is "RAIL" which stands for the "Relentless Application of Ito’s Lemma".

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