For the reader well-versed in Mathematica and in economic theory, this book gives a fairly good overview of how Mathematica can be used to study mathematical economics and finance. It is also assumed in the articles in the book that the reader has a strong background in mathematics. Since the book was published in 1993, Mathematica has considerably expanded, with many new features that make some of the accompanying code in the book somewhat dated, but the notebooks can still be used beneficially.In addition, economic theory is currently making more use of symbolic programming, and financial analysis has exploded as an area which is now making heavy use of high-performance computing. Although Mathematica cannot compete from a performance standpoint with the needs of financial engineering, it still has an advantage from a didactic standpoint. I did not read all of the articles in the book, so my comments will be limited to the ones that I did.
The article on "Mathematica and Diffusions" is an overview of how to use Mathematica to do stochastic calculus. The Ito calculus is reviewed briefly, and the authors begin with constructing a Weiner process. The Mathematica package they employ and on the disk accompanying the book is not discussed in detail, but is merely used to simulate realizations of the process. Readers who want a more in-depth view will have to go over the code themselves. The authors use the package to generate realizations of Weiner processes that are correlated with each other, and show this correlation via Mathematica graphics. The Black-Scholes formula is derived using the standard self-financing trading strategy and ignoring transaction costs and dividends. The algebraic manipulations are done with Mathematica, and this obscures (a little) the underlying concepts behind the derivation of this important formula. Since data structures in Mathematica are essentially lists, the authors outline the construction of the data structure that could be used to represent a diffusion, namely a list consisting of five terms: the diffusion, Weiner process name, expression for the drift and dispersion, and the initial value. For the reader familiar with OO-programming, accessor functions are used to extract the components of this data structure. This is a nice move by the authors, for it is an example of how Mathematica can be used to emulate OO-programming.
The article "Itovsn3: Doing Stochastic Calculus with Mathematica" is an overview of how to use the Itovsn3 package that is on the disk to implement Ito calculus. It is assumed that the reader has a background in stochastic calculus, since the author does not give a review. However, semimartingales, so important to those working in financial engineering, are discussed and their statistical behavior described using Mathematica. The Ito formula is presented as a semimartingale-type decomposition for smooth function of Brownian motion and the author shows using Mathematica plots how the higher order terms in the second-order Taylor expansion vanish asymptotically. This article is not merely Mathematica code for Ito calculus, for the author gives an example of how to use the package in a hedging problem.
The article "Option Valuation" is a more detailed overview of how to use Mathematica in the context of the Black-Scholes model to perform options valuation and risk management. Heavy use is made of the graphics capability of Mathematica to illustrate how option values change as a function of stock price and time of expiration. The author also shows how Mathematica can be used as a OO-language to treat options as self-contained objects with accessor functions. He does however state that Mathematica does not live up to the OO toolkits available elsewhere, contrary to my experience. He closes the article with a consideration of how to use Mathematica to value options that can be exercised before expiry, the binomial model playing the central role in the discussion. It is here in particular that the performance of Mathematica is readily felt. The numerical number-crunching needed to do the calculations in these types of models cannot be done in Mathematica efficiently and profitably.
The article "Time Series Models and Mathematica" gives a general treatment on how Mathematica can be used to study ARIMA models for time series. Mathematica is used more interactively than the other articles and the visualization obtained is quite nice in giving the reader insight into such concepts as the moving average and the spectral density function. The author shows how to estimate the spectral density function and why periodogram techniques fall short in this estimation. I would have liked to see other techniques for studying time series discussed, such as neural networks and hidden Markov models, but the author does do a fairly good job with the ARIMA models.
