The behavior of large economic systems is by necessity unpredictable, for if it was, then the making of speculative profit would be impossible. The movement of markets is modeled as an example of Brownian motion, which is a consequence of the random motion of molecules. This is a complex process, where the only hope to predict the future is to apply statistical methods. Therefore, this book is largely a lesson in creating statistical models of random processes that allow for calculus methods to be used to analyze them.
The primary model is that of a discrete parameter martingale, which is where different price possibilities are computed based on probabilities that the parameter will have certain values. After years of teaching calculus, this is the first book that I have read where the concentration is on using calculus to model financial systems. Without question, I learned more new material from this book than I have in at least 90% of the math books that I have read. It was fascinating to see how a non-differentiable system is modeled so that it is then possible to use the continuous methods of calculus in working with it.
This book is perfect for advanced courses in the modeling of financial markets. The amount of calculus knowledge needed to understand the material is that of the standard three course sequence that is the start of nearly all undergraduate majors. A course in statistics based on calculus is also essential, and experience in differential equations would also be helpful, although not required.

The only reason that it does not get a fifth star is that there are no solutions to the exercises. I am a firm believer that solutions to at least 1/3 the problems should be included in any mathematics book.

Published in the recreational mathematics e-mail newsletter, reprinted with permission.

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