This book gives a good general overview of financial engineering but only for those who have had a lot of prior exposure to the subject, at least from a theoretical or academic point of view, but have yet to get their feet wet in actual practice. For physicists with a background in quantum field theory, stochastic dynamical systems, or statistical mechanics, the mathematics in this book will be straightforward, and physicists will be intrigued that some of their ideas are being applied to finance. It is not a book for beginners though, as it will demand a lot of attention to details, as well as a considerable amount of outside reading. Space does not permit a detailed review of such a large book, and so only selected chapters will be reviewed.



In chapter 4, the author analyzes plain-vanilla equity options and discusses in particular the case of American options. The calculation of the probabilities of exercise at different future times involves the determination of the critical path followed by a Monte Carlo simulation to determine to the fraction of paths crossing the critical path in each interval of time. The hedges are then distributed in time as the delta times these probabilities of exercise. The author unfortunately does not give the details of how to obtain the critical path in this chapter, but these details can be found in later chapters on path integrals.



In chapter 5, foreign exchange options are discussed including how to hedge with the Greeks. The author shows how to price FX forwards and FX European options. He mentions that the Garman-Kohlhagen model is used to price the FX options, but he does not elaborate in any detail on the model. This model, which is the standard pricing convention in the FX market, is the analog of the Black-Scholes model, but where a foreign riskless interest rate is used as the payout on the underlying asset. Particularly interesting in this chapter is the author's discussion on the "two-country paradox". This paradox arises because the change of variables in foreign exchange instruments forces one to do a separate normalization of the drift of each variable, and does not arise for ordinary options. The drift after the change of variable is not consistent with interest-rate parity. Also discussed are the `volatility smiles' that are empirically observed in FX. As the author illustrates in a diagram, the smile corresponds to an upward-facing parabola, and he explains its occurrence by a "fear factor" (sometimes called "crash-o-phobia" in the equity option literature), which causes the implied volatilities of OTM puts to be bid up, thus putting a premium on this volatility relative to the ATM volume.



There are five chapters in the book that discuss the use of path integrals in finance, and these chapters include the formalism and how to calculate them numerically. The writing in these chapters is very lucid, and this no doubt reflects the author's background in physics and his consequent bias toward the use of functional integration in financial modeling. The discussion of the Black-Scholes in the context of functional integration is good motivation for later developments, and should convince readers as to the viability of this approach in finance. In addition, the author gives examples where the path integral approach does not merely reproduce the standard results in finance, one of these examples being the inclusion of dividends in options valuation. Including dividends can be done via the use of an "effective drift function", as the author shows in detail. He also shows that jumps in stock price can be studied in the same way as dividends in the context of path integration. Discrete-schedule Bermuda options are also tackled using path integral methods, as well as American options, and the author shows the reader how to calculate the critical path for these scenarios, following up on a promise in an earlier chapter. The chapter on numerical methods for the calculation of path integrals is interesting because it introduces some techniques and concepts that are no doubt new to many readers, such as "geometric volatility", which corresponds to an approximate volatility that would lead to a particular set of paths.



Perhaps the most interesting and "exotic" of the discussions in the book is included in chapter 46, and regards the application of `Reggeon field theory' (RFT) to financial engineering. Even for physicists working in quantum field theory, this type of field theory may be unknown to them, but the author does give a very brief review. He assumes background in scattering theory, the renormalization group, dimensional regularization, and other topics in field theory and high-energy physics, in order to read this chapter. RFT is presented as a theory to describe high-energy diffractive scattering, as a field theory for a particle called the `Pomeron'. The author's interest for the application of RFT to finance concern its ability to model nonlinearities and non-linear diffusion. He writes down the Lagrangian for RFT, which involves the nonlinear product of three fields, and when the interaction is switched off reduces to an ordinary diffusive model in imaginary time. One could apply ordinary perturbation theory to the case of weak interactions, but the author instead is interested in the non-perturbative region for the theory. This he tackles with the renormalization group, the object of which is to find the critical dimension, in order to test for the occurrence of a phase transition. Therefore the Gell-Mann Low beta function is to be calculated (using perturbation theory) and its zeros found. The author summarizes what is known for RFT from the research in the literature. The applications to finance consist of the ability of the RFT model to describe deviations from "square-root time", the latter of which arises from the standard Brownian motion assumption in financial theory. The RFT model reduces to the standard financial model when the interactions vanish. The nonlinear interactions are expected to produce interesting "fat-tail" jump events, but the author does not elaborate on this in any detail.