Arbitrage Theory in Continuous Time (Oxford Finance S.) Review

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(More customer reviews)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.

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