Mathematical Methods for Foreign Exchange: A Financial Engineer's Approach

Alex Lipton's Mathematical Methods for Foreign Exchange: A Financial Engineer's Approach is a comprehensive study of models used in practice for the pricing and hedging of derivatives. Despite the focus on foreign exchange, the methods detailed in the work go far beyond FX to apply to a number of other asset classes (e.g. equity, commodity, and some fixed income and credit derivatives). The core of the book applies the theory of parabolic partial differential equations to solve a vast number of problems that arise in the pricing of European and path-dependent options. It is difficult to find all the methods covered in the book treated in one place, and many are not documented elsewhere. The emphasis is on a mathematical treatment that highlights the structure of the problems at hand, not on numerical solutions to these problems. Important though numerical solutions are in their own right, they are far less illuminating in understanding the problem itself, and can easily be obtained with the appropriate mathematical tools, developed in this book, at one's disposal. The value of the work is further enhanced by the meticulous referencing and extensive bibliography that it provides. Having worked in the industry as a quant in foreign exchange, and in academia, I can strongly recommend this book to anyone interested in a rigorous mathematical treatment of the problems arising in the pricing and hedging of derivatives.

I have been using this book since I was a graduate student and, now working as a quant, I always keep it handy. The various techniques I have learned from this book helped me to solve a lot of problems related to financial engineering which I have encountered in my both academic and professional career. The book covers a lot of advanced material, most of which is original and unique, starting from pricing path-dependent options in the discrete binomial model up to pricing path-dependent volatility products under stochastic volatility models. I will briefly go through the content of the book and make appropriate comments. Chapters 1- 4) The author introduces the sufficient background which is necessary to solve financial problems arising by pricing and risk-managing of derivative securities, in particular, the stochastic calculus and backward/forward partial differential equations (PDE). By itself these topics are nowadays extremely broad and cannot be fully developed even within a series of books. In this book, only important results are given (and references for more specific texts are provided); these results will further be applied throughout the book and they include: basic properties of Brownian motion, connection between the diffusion problems and solutions to backward and forward Kolmogoroff equations, Ito's lemma, solution of SDE-s by discretization, solving PDE-s with Laplace and Fourier transforms, eigenfunction expansion. In chapter 4, the author does tangentially mention about numerical solutions of PDE-s. Here, it is important to note that PDE numerical solution methods are outside of the scope of this book, although the author almost always presents the problem as a solution to certain PDE so a relevant PDE solver can always be applied, the main scope of this book are analytical methods for solving PDEs. However, the author does illustrate the Crank-Nicolson and Crayg-Sneyd ADI scheme, which are the methods of choice (whether robust or not) at the majority of Wall Street firms. Chapters 5-6) The author fully develops the binomial model and carefully explains no-arbitrage pricing principles. What is important and, perhaps, is unique in his treatment of the binomial model is that he describes extensions of the model to the so-called implied trees, and, which is very important for pricing exotic options in the binomial model, he describes the augmentation principle to price path-dependent options, including American, asian, barrier, and lookback options. Chapters 7-9) The author studies continuous time dynamics for FOREX evolution (although by no means is the treatment specific only to FOREX) and pricing principles for European options. He applies various techniques from applied mathematics to solve a variety of pricing problems, including multi-dimensional problems, in an efficient way. These highly useful techniques include non-dimensionalization, Laplace and Fourier transforms, approximations, Green's functions, or the state-pr

I own pretty much all of the books on quantitative financeand this one holds a cherished place on my bookshelf.Anybody either working as a quant or with aspirations to become one should dust off some space on their bookshelf as well.Anybody who does serious research in finance in either academia or industry already knows that it is somewhat rare for top researchers to pen books of any length. Time is at a premium and the payoff to publishing journal articles or to finishing off code is typically much greater than it is for writing books. This is what distinguishes this book from its competitors.The author is well known in financial circles as one of a handful of quants who is capable of meaningfully contributing new results to this fascinating field. The book contains many results which cannot be found elsewhere in the public domain.Although the book title suggests that the results apply only to foreign exchange, it is straightforward to adapt the results to equities, commodities, and many other underlyings.Wall Street is a very secretive place and it is not easy to get a glimpse of the kind of things that consume a quant's time.I suspect that the only reason that this book was able to come to light is due to the acquisition of Banker's Trust, the author's former employer. Banker's was well known to be a fertile training ground for the best derivative minds and this book should prove to be a lasting legacy to that view.

Alexander Lipton Lifschitz has brought his extensiveexperience and years of research in the most diverse areasof applied mathematics as well as his experience in the financial industry to bear in authoring thisvery interesting book.The range of this book is impressive.Although the author chose to focus on currency options, his book really is a treatise ona wide spectrum of problems and methodologieswhich any quant wishing to tackle thesophisticated world of option pricing at a high levelmust master.The author demonstrates his mastery ofthe arsenal of the classic applied mathematician,asymptotic analysis, self-similarity, Laplaceand Fourier transform, and uses these to give anincisive analysis of both standard topics such as American options and more exotic topics such as options on one currency with barriers on the other currency, passportoptions (for which he was a pioneer in developpingpricing tools) , asian options and much much more.No, this is not as easy a read as Willmott's books. Willmott's books were and remain an important contribution with their quick and intuitive explanation of a varietyof instruments. Lipton- Lifschitz'sbook is more challenging and the reader willhave to pull up his sleeves on occasions where the author, while dealingwith a case analogous to one just treatedsays " the details are left to the reader".But let's face it, if you work on Wall Streetor nearby, you'll have to tackle those detailsalone at some point and Lipton-Lifschitzgives you all you need to know to do pull this off.And. last but not least, let's not forget the price. At less than 50$ thisbook is a real bargain and for a firstprinting, unusually free of typos or otherserrors.I highly recommend you buy this book now before the publisherdoubles the price.

Most of the books on mathematical finance fall in one of the two domains. Some books are written for people who are new to the field and, as such, do not go deeply into the mathematical details that are crucial for implementing these methods in practice. The more advanced books are usually written by academic mathematicians and sometimes suffer from poor readability and lack of awareness of relevant problems. What many people are looking for is a detailed and readable description of how to apply the latest mathematical methods to solving the problems appearing in day-to-day work of derivatives desks. Such books are few and far between: that is why Alex Lipton's manuscript was so welcome.As a quant in one of the Wall Street investment banks, I found this book a very valuable resource. Though written on a fairly high level, this book remains a readable and consistent exposition of latest methods of foreign exchange modeling. I particularly appreciated that the author does not skip steps in his derivations and gives out all those little practical details that are so important to people planning to use these methods in their work. The range of topics covered is fairly wide, with main emphasis on derivative pricing. I found the two chapters on path-dependent options to be particularly interesting and extensive. Some of the results included in the book came out of author's original work at Deutsche Bank. I also had an impression that some of his latest work was not included in the book, which is a pity.All in all, an excellent book. Well worth the price.