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Mathematical Methods for Financial Markets succeeds to be both an excellent finance textbook *and* an excellent maths textbook. Contrary to what the profane may believe, it is therefore not just a textbook in financial mathematics or in mathematical finance. The enlighten reader shall be able to find in this book essential elements to understand options markets and in particular exotic options. He shall also discover an integrated vision of risk and of the valuation of credit derivatives. Beyond this, the incompleteness of financial markets lurks. As far as probabilistic aspects are concerned, the reader shall also not be disappointed, though the academic credit of the authors left no room for a doubt on this matter. Girsanov, Ito, Bessel, Lévy, Feynman, Kac, Azema, Tanaka become familiar names upon mastering of this book, where the clarity and elegance of the mathematical proofs is not incomparable to the deepness of the scientific content and goals, which are clearly to understand finance. Modern finance can not be understood without a minimum quantity of formalism. “No one enters here lest he be a geometer” would have said Plato, if not in practice, at least in spirit.

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The book is divided into two parts. The first one is dedicated to continuous finance – essentially the finance of Brownian motion. The second one is the one of discontinuous finance, that is to say of Poisson processes, marked point processes, Lévy processes, semimartingales… The first part is made of six chapters, when the second one is constituted of five chapters. We shall now detail the contents of this book chapter by chapter.

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The first chapter of this book exhibits, a little bit in the manner of the book by Jacod and Shiryaev, but in such a way that the layman does not get lost, the main probabilistic tools that are used in the sequel, at least in the first part of the book. To be noted: a very clear exposition of Markov processes, of stopping times and of continuous local martingales. The also very clear presentation of the oblique bracket permits then to display Ito’s calculus in a synthetic fashion. Finally, probability changes, very simple to put into practise by Master students, but seldom understood by them, are presented along with numerous illustrations, which is unfortunately not often the case in the literature.

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The second chapter exposes the roots of finance, with a clear preference for options theory. The story starts with self-financed portfolios and is pursued with PDEs in complete markets in presence of deterministic interest rates. The Black and Scholes formula and the treatment of Greeks are a natural sequel. A very important, and original, section on Numéraire changes then turns up. By the way, that so few finance textbooks concentrate on Numéraire changes is highly prejudicial, when this object is at the heart of the understanding of quantitative finance and is ubiquitous in dealing rooms. Next, some path-dependent products are introduced, like quantile options for instance. These products could have as well been introduced later on, in the fourth chapter, together with Parisian options. The second chapter is ended with the clear treatment of the Ornstein-Uhlenbeck framework for interest rates, and of the valuation of forex options, and in particular of quantos - a traditionally tricky subject for students.

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The third chapter is devoted to hitting times. It displays a presentation of the law of a hitting time, and of the joint law of Brownian motions (including arithmetic and geometric ones) and of their maxima or minima, which will remind the scrupulous reader of some developments in the book by Dana and Jeanblanc. Clearly, on this topic, it is difficult to be plainly original, and it is rather the clarity of exhibition that matters. Building on this, it is then possible to realize an extensive treatment of (simple or double) barrier options, binary options, lookbacks, and related products. The chapter then moves on to a fundamental application of hitting times: the structural theory of default. This is followed by a simple exhibition of American options. The chapter is concluded by an important application of options in the economic field: real options.

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The fourth chapter deals with objects that push away the limits of aesthetism: local times and the Tanaka formula, to start with. Dupire’s formula and local volatility (together with a bunch of derivatives) are hinted at by this exposition. Then, meanders, Brownian bridges and excursions pave the way of Parisian options, introduced in the literature by the very authors of this book. These options are second generation barrier options, in the sense that they allow to take into account the time spent beyond critical levels. As with barrier options, the applications in the finance sphere are vastly multiple. To be noted: the chapter end displays a subsection on American Parisian options.

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The fifth chapter shows nice complements in the theory of stochastic processes, for instance an interesting section on Kolmogorov and Fokker-Planck equations that can only but captivate the reader with a minimum background in physics. In this chapter, a nice treatment of Pitman’s theorem can also be found. It links, amongst other things, the law of the maximum of a Brownian motion to the law of the minimum of a Bessel process. These developments could as well have been displayed later on, because Bessel processes are explained in this book only in the sixth chapter. Finally, the fifth chapter is concluded by a first exposition of filtration enlargement.

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The sixth chapter (last one of the first part) is dedicated to Bessel and Bessel square processes. In fact, the CIR process (and its generalizations like the CEV process), together with its applications, is hinted at by this chapter. Important subfields of finance like stochastic interest rate models, stochastic volatility models… can be reduced to FFTs or numerical inverse Laplace transforms, because the Fourier and Laplace transforms of the CIR process are known. This chapter contains a section on Asian options, where the law of the (integral) sum of lognormal random variables is considered via Bessel processes, and where a Laplace transform formula for these options can be ultimately obtained. See also the seminal article by Geman and Yor on this subject.

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The seventh chapter is dedicated to credit risk. This is for sure an arduous piece of work. To the unprepared layman, we can only but advice to arm oneself with solid equipment, or even to ask for the help of one of the most experienced Sherpas of mathematical finance, before rambling the mountains of abstraction. When does a stopping time or a martingale remain a stopping time or a martingale when information is enlarged or diminished? Such are the questions that are examined in this chapter, and they aim at allowing the reader to price credit derivatives like CDSs. Finally, intensities and copulas are part of the trip, even if they obviously do not constitute the heart of the chapter.

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The eighth chapter is devoted to general Poisson processes. It starts by scrutinizing simple Poisson processes and their stochastic calculus, before following the same lines with inhomogeneous Poisson processes. Equipped with this, the reader is then ready to tackle Cox processes, that is to say Poisson processes admitting stochastic intensities, and to embark on a trip towards default risk. Next, the chapter examines compound Poisson processes, and the application of these processes in a concrete context: the ruin theory of insurance companies. Finally, an exhibition of marked point processes and of Poisson point processes concludes the chapter. The reader can view this chapter as the applied side of the seventh chapter, because it gives the practical tools to compute the conditional expectations obtained following the methods of the seventh chapter.

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The ninth chapter presents very general stochastic processes: semimartingales, together with their related stochastic calculus. The tenth and eleventh chapters are concentrated on particular processes admitting both a continuous and a discontinuous part. The tenth chapter displays the merits of mixed processes (according to their terminology), in other words of stochastic integrals built with respect to both a Brownian motion and a compensated inhomogeneous Poisson process. Jump-diffusion affine processes are then studied, before a deep treatment of (in)completeness. The eleventh chapter ends the book by a nice exposition on Lévy processes, so on processes admitting stationary and independent increments. Note that the continuous component of these processes can only be an arithmetic Brownian motion, whilst their jump component permits to go beyond Poisson processes by taking into account a potentially infinite arrival rate of small jumps. The valuation of options whose underlying is modelled by a geometric Lévy process is detailed in the book. It is important though to underline that many applications of Lévy processes have been developed in other subfields of finance in the past decade. These processes offer a nice balance in terms of complexity between compound Poisson processes and semimartingales, most calculations being easy to perform using FFTs.

To conclude, the work examined here is an excellent reading, going well beyond the Hull, that should be advised to all serious students in quantitative finance, and perhaps to a few colleagues who would want to enlarge their filtration about this topic. This is a prodigious encyclopaedia designed by the best authors in the field.

Olivier Le Courtois

Professor of Finance and Insurance at EM Lyon Business School

- R.-A. Dana and M. Jeanblanc (2007), « Financial Markets in Continuous Time: Valuation and Equilibrium », Springer Finance, 2
^{nd}Edition. - H. Geman and M. Yor (1993), « Bessel Processes, Asian Options and Perpetuities », Mathematical Finance.
- J. Jacod and A. N. Shiryaev, « Limit Theorems for Stochastic Processes », Springer, 2
^{nd}Edition, 2003 - J. Hull (2008), « Options, Futures, and Other Derivatives », Prentice-Hall, 7
^{th}Edition.

* *Jeanblanc Monique, * *Yor Marc, Chesney Marc, « Mathematical Methods for Financial Markets », *Finance*, 1/2010 (Vol. 31), p. 81-85.

URL : http://www.cairn.info/revue-finance-2010-1-page-81.htm

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