This book focuses directly on foreign exchange. It aims to provide a comprehensive review of the relevant mathematical methods applicable to this market, with an emphasis on the use of such methods for the practical valuation of derivatives instruments commonly found in forex. Despite this, the techniques described here can also be useful to those working in the equity, and to some extent interest rate, markets too.

But even within the restrictions of the forex arena, the scope of the book is nevertheless extremely wide. From a fairly brief introduction to the market and the standard instruments, it goes on to cover basic mathematical concepts such as probability theory, time-series analysis and the common continuous-time processes. It then provides a useful section on discrete time models, which offers a simple way of introducing concepts such as risk-neutral valuation and replicating portfolios in a natural way, before going on, in the multi-period setting, to deal with other important concepts such as the use of predictable self-financing trading strategies. At this stage we are introduced to some of the exotic options such as lookback, Asian and barrier options for the first time, and some of the issues relating to non-stationary markets and volatility smiles are examined.

These first three sections, which cover the first third of the book, provide the mathematical basics and an introduction to some of the key concepts involved in derivatives valuation. There is nothing particularly new here, but as an overview of the most important ideas it provides a useful guide to those coming to the subject with little relevant knowledge.

The final section of the book, which covers the bulk of the content, covers continuous time models. This is where the author's knowledge of a wide range of mathematical methods comes into its own. Beginning with straightforward geometric Brownian motion models of the forex rate with deterministic interest rates, we move on to examine non-deterministic interest rate models, nonlinear diffusions, jump diffusion models and finally stochastic volatility models.

The Black-Scholes formula is then derived in the continuous setting, using various methods (replicating portfolios, the heat equation, Laplace transforms and Fourier transforms) before moving on to more complex multi-currency options such as basket, index and outperformance options. At this stage it becomes clear that one of the key aims of the book is to describe and demonstrate a variety of approaches to solving the valuation problem rather than simply giving the solution in each case.

A useful section on dealing with non-constant volatility then follows that comprehensively covers the outstanding literature. This section also covers some practical issues such as sticky strike versus sticky delta, dealing with very short-dated options, situations when the volatility of volatility is small or the foreign exchange rate trades in a band. Although analysis of such models as the constant elasticity of variance model and jump diffusion model is given, the section on stochastic volatility models is particularly detailed, and some useful and intuitive results are provided.

A relatively short section on American-style options follows, again offering various alternative valuation methods, before two very large sections on path-dependent options. Here Lipton seems most at home, not simply providing valuation formulas for nearly every path-dependent option you might want, but more importantly demonstrating again a wide range of mathematical methods for solving sometimes very complex problems. Indeed, he tackles head-on issues such as discrete barrier monitoring and discrete fixings, fitting prices to known volatility surfaces, the hedging of discontinuities at the barrier, pricing with non-constant volatility models and more. At this point we are also introduced to some new methods Lipton himself has developed, for example in pricing quadruple no-touch options.

One of the key claims here is that for the majority of such path-dependent claims, the partial differential equation method can be used efficiently for valuation as opposed to a numerical approach such as Monte Carlo. The author demonstrates how this can be done, developing an 'augmentation procedure' to produce a general system for a wide range of path-dependent options. He then applies this to lookback, Asian, timer, fader and passport options, among others. At this point, a useful section on dealing with variance and volatility swaps and options, and cliquets, is provided.

There follows a brief look at issues such as imperfect hedging, transaction costs, liquidity risk and default risk before some short conclusions and thoughts on further developments.