The author uses Visual Basic and Excel spreadsheets to compute the relevant financial quantities, and given the popularity of spreadsheets in finance, this is appropriate. The numerical solution of partial differential equations is most efficiently done using C (or Fortran) and no doubt the author does recognize this, for he does mention translating existing code in C to Visual Basic.
My only major objection to the book is the lack of exercises, which were a major selling point to me in the author's earlier book on derivatives. Having such exercises is indispensable in understanding results of this nature.
The first few chapters of Volume 1 give an elementary introduction to the theory of derivatives and stochastic calculus. The author does remain concrete in his explanations, and he gives a fairly straightforward derivation of the Black-Scholes equation. This is followed by a very quick discussion of Green's function solutions of the equation and introduction to the Greeks. Generalizations of the Black-Scholes model are discussed later, in the context of dividends, foreign currency, and time-dependent parameters. The author does not give a critical analysis of the Black-Scholes equation in these chapters. This would have been useful to both the practitioner and a newcomer to the field. Also, the Black-Scholes can be derived in many different ways, and it would have been instructive to see some of these alternative derivations. There are derivations of the Black-Scholes equations based on concepts from information theory, and these shed light on the limitations of this equation. All of the concepts in these chapters can be found in the author's earlier book on derivatives. The second half of the first volume is an overview of the mathematical techniques used to deal with path-dependent and "exotic" options. Consultation of the references is mandatory for a complete understanding of the ideas in these chapters, for the author is a little lacking on details. In addition, more discussion is needed on case history validation of the many formulas given in these chapters: are these formulas useful in practice? The author also introduces some new concepts in this volume that are not in the derivatives book, one being stochastic control. Also, the author introduces a similarity reduction technique for partial differential equations that is very much like the techniques used in neutron reactor physics. Physicists-turned-financial-engineers will see the similarity between these two approaches.
The last part of the first volume deals with extending Black-Scholes. The author discusses the problems with Black-Scholes but his treatment is too hurried. A better approach might have been to give (historical) examples of what might happen, from an investment/risk management perspective, if the assumptions of Black-Scholes are followed to the letter. He does give references though for a more in-depth discussion. Volatility surfaces, viz a viz the Fokker-Planck equation, are discussed here, and effectively. Again, the physicist reader will pick up on the dialog immediately. Information-theoretic techniques, via entropy minimization, are used, interestingly. It is refreshing to see in this part that the author gets down to an empirical analysis of some important issues (volatility for example).
The second volume is somewhat more specialized that the first and outlines in the first chapters fixed income products, swaps, and interest rate derivatives. Phase plane analysis is employed in the discussion on multi-factor interest rate modeling. The treatment here is too curt and needs considerable expansion. The theory of stability of fixed points under the influence of noise is non-trivial and requires careful consideration. A departure from the framework of partial differential equations occurs in the discussion of the Heath, Jarrow, and Morton model. Noting that this model is non-Markovian, he introduces Monte Carlo simulation as a technique to calculate the expected present values. He remarks that the simulation time to carry this out is very long. The sluggishness of Monte Carlo simulations in this model and others in financial engineering has motivated many researchers and start-up firms to devise techniques to speed up the simulations. Indeed, a whole industry has grown in recent years offering packages and algorithms to speed up Monte Carlo.
Risk and portfolio management are also discussed in this volume, beginning with modern portfolio theory. The most interesting and well-written part is on asset allocation in continuous time. Energy derivatives, an up-and-coming field are also discussed. The author is un-sure of himself in this chapter, but he does give a general but elementary introduction to the subject. This is an area that needs a lot more investigation and research given its importance.
The last part of the book addresses numerical methods, and there is some source code in Visual Basic. Monte Carlo simulation is discussed again, along with an introduction to low-discrepancy sequences. These sequences have been used extensively in recent years to improve the efficacy of Monte Carlo simulations. The author's treatment is very terse but he does give many references.
The author has done a fine job in these two volumes, and he spices up the reading with a litte humour, which does not detract at all from the seriousness of the topics, but instead makes for more enjoyable reading.