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Rating: 
Summary: Good mix
Review: I have read this book... from a learning perspective of trying to learn what the theory behind options pricing is it is a great book. A lot of more recent topics are missing, but as a starter book for those who already price options/work in the industry without having learned all the theory (or in my case forgotten what they learned in school) it is a great read and a great reference.
Rating: 
Summary: Probabilistic approach to derivatives valuation
Review: The language of financial derivatives is, arguably, the language of the modern theory of martingale stochastic processes. In this approach pricing contingent claims is reduced to finding an "equivalent martingale measure". Practitioners would think in terms of risk adjusted or risk neutral valuation. To understant this topic from an abstract and rigorous point of view is a daunting task restricted to a relatively small elite. For those seeking to learn the mechanics of this discipline a good foundation is well provided by the texts from Hull, Options, Futures and Other Derivatives, as well as Jarrow & Turbull, Derivatives Securities. These books present the intuition behind the formulas and how to use them in practical situations, but they do not show where the formulas come from and much less the mathematics necessary to prove them. Before the book under review was published, this task was attempted by other authors with mixed degrees of success. Here we briefly mention three of them. Baxter and Rennie's Financial Calculus (233 pages) is written in an informal fashion about deep mathematics and one has the feeling that the essence of the topics covered can be grasped and understood from it. However, behind this innocent style there is a huge amount of sophisticated machinery that, in my opinion, should have in part been presented in more detail. An instructor is left with the feeling that it could have been much more profitable to work a bit harder on the students and give them a more complete picture of the theory. Next comes Neftci's Mathematics of Financial Derivatives (352 pages). Its language is more accessible than Baxter and it gives a more detailed and extensive description on most topics. Mathematically, though, it falls short of current usage and rigour. The book by Musiela & Rukowski, Martingale Methods in Financial Modelling (511 pages), is far more difficult than any of these and should be read and understood only by a few. It requires previous knowledge of stochastic processes at the level of, for example, Probability with Martingales by D. Williams. The book under review is an excellent text for courses and for individual readers with a modest background in probability. There is no compromise with mathematical language and concepts. They are presented precisely and illustrated by examples without the burden of more technical theorem-proving approach in advanced mathematical texts. After introducing the idea of derivatives and risk-neutral valuation, it gives a summary of modern probability theory including measure, integral, conditional expectation, modes of convergence, characteristic functions and the Central Limit Theorem. This sets the framework for the rest of the book. Stochastic processes and finance in discrete time are not pre-requites for the much more complicated continuous time but serve as a pedagogical preparation for it. The Third and Fourth Chapters are dedicated to the discrete case and key concepts are carefully analysed. Information and filtrations are discussed as well as the important random walk processes as a motivation for the Brownian Motion. The culmination of these efforts is the proof of the Fundamental Theorem of Asset Pricing: in an arbitrage-free complete market there exists a unique equivalent martingale measure. A very readable discussion on binomial trees is given, including the proof that in the limit of small time increments one recovers from it the usual Black-Scholes formula for a call option. Chapters Five and Six are dedicated to stochastic processes and finance in continuous time. This includes filtrations, a sketch for the construction of Brownian Motion, quadratic variation of Brownian Motion, stochastic integrals and Ito calculus, stochastic differential equations, etc. A continuous version of the Fundamental Theorem is discussed but not proven. The main formula for risk-neutral valuation in terms of expected values is proven. A general result about the relationship with other approaches is that solutions to partial differential equations have a stochastic representation in terms of expected values (Feynman-Kac Formula). On p. 211 a discussion is presented regarding our knowledge concerning continuous time securities market in comparison to the discrete case. If you are really interested in understanding the probabilistic foundations of modern financial derivatives theory, please consider seriously this book. Another reference, in the same spirit that I recommend is the excellent notes from Shreve, Stochastic Calculus and Finance, which is not yet in book form. After reading the text by Bingham and Kiesel you will gain a solid background well worth the effort and will be able to read profitably most of the contemporary texts and articles on this subject.
Rating:
Summary: Probabilistic approach to derivatives valuation
Review: The language of financial derivatives is, arguably, the language of the modern theory of martingale stochastic processes. In this approach pricing contingent claims is reduced to finding an "equivalent martingale measure". Practitioners would think in terms of risk adjusted or risk neutral valuation. To understant this topic from an abstract and rigorous point of view is a daunting task restricted to a relatively small elite. For those seeking to learn the mechanics of this discipline a good foundation is well provided by the texts from Hull, Options, Futures and Other Derivatives, as well as Jarrow & Turbull, Derivatives Securities. These books present the intuition behind the formulas and how to use them in practical situations, but they do not show where the formulas come from and much less the mathematics necessary to prove them. Before the book under review was published, this task was attempted by other authors with mixed degrees of success. Here we briefly mention three of them. Baxter and Rennie's Financial Calculus (233 pages) is written in an informal fashion about deep mathematics and one has the feeling that the essence of the topics covered can be grasped and understood from it. However, behind this innocent style there is a huge amount of sophisticated machinery that, in my opinion, should have in part been presented in more detail. An instructor is left with the feeling that it could have been much more profitable to work a bit harder on the students and give them a more complete picture of the theory. Next comes Neftci's Mathematics of Financial Derivatives (352 pages). Its language is more accessible than Baxter and it gives a more detailed and extensive description on most topics. Mathematically, though, it falls short of current usage and rigour. The book by Musiela & Rukowski, Martingale Methods in Financial Modelling (511 pages), is far more difficult than any of these and should be read and understood only by a few. It requires previous knowledge of stochastic processes at the level of, for example, Probability with Martingales by D. Williams. The book under review is an excellent text for courses and for individual readers with a modest background in probability. There is no compromise with mathematical language and concepts. They are presented precisely and illustrated by examples without the burden of more technical theorem-proving approach in advanced mathematical texts. After introducing the idea of derivatives and risk-neutral valuation, it gives a summary of modern probability theory including measure, integral, conditional expectation, modes of convergence, characteristic functions and the Central Limit Theorem. This sets the framework for the rest of the book. Stochastic processes and finance in discrete time are not pre-requites for the much more complicated continuous time but serve as a pedagogical preparation for it. The Third and Fourth Chapters are dedicated to the discrete case and key concepts are carefully analysed. Information and filtrations are discussed as well as the important random walk processes as a motivation for the Brownian Motion. The culmination of these efforts is the proof of the Fundamental Theorem of Asset Pricing: in an arbitrage-free complete market there exists a unique equivalent martingale measure. A very readable discussion on binomial trees is given, including the proof that in the limit of small time increments one recovers from it the usual Black-Scholes formula for a call option. Chapters Five and Six are dedicated to stochastic processes and finance in continuous time. This includes filtrations, a sketch for the construction of Brownian Motion, quadratic variation of Brownian Motion, stochastic integrals and Ito calculus, stochastic differential equations, etc. A continuous version of the Fundamental Theorem is discussed but not proven. The main formula for risk-neutral valuation in terms of expected values is proven. A general result about the relationship with other approaches is that solutions to partial differential equations have a stochastic representation in terms of expected values (Feynman-Kac Formula). On p. 211 a discussion is presented regarding our knowledge concerning continuous time securities market in comparison to the discrete case. If you are really interested in understanding the probabilistic foundations of modern financial derivatives theory, please consider seriously this book. Another reference, in the same spirit that I recommend is the excellent notes from Shreve, Stochastic Calculus and Finance, which is not yet in book form. After reading the text by Bingham and Kiesel you will gain a solid background well worth the effort and will be able to read profitably most of the contemporary texts and articles on this subject.
Rating:
Summary: Excellent and brief compendium of financial theory
Review: This book covers quite a few fields (axiomatic probability, stochastic processes, financial theory) to the extent that they relate to valuation of securities. Naturally, the scope of coverage in such a brief tome (< 300 p.) is limited. It is written clearly and with precision, with sufficient number of exercises provided at chapters' ends. I would say that it goes to greater depth than Neftci, and is far more rigorous than Wilmott. Incomparably easier to understand than Merton. The only shorcomings I can find are relative paucity of examples and inadequate Index.
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