Mathematical Techniques In Finance
Tools for Incomplete Markets

A focus of considerable financial engineering research during the 2000s is incomplete markets. The notion that markets are complete was a convenient lie that facilitated much of financial engineering during decades past. It allowed financial engineers to dispel risk preferences and utilities from asset pricing. The price of a derivative was just the cost of maintaining a replicating portfolio. Absent complete markets, replicating portfolios don't always exist.

 

Complications such as stochastic volatility and volatility smiles have strained the simplifying assumption of complete markets. Today, growth in credit derivatives markets has forced financial engineers to, in many cases, abandon it completely.

Author Cerny has written an excellent introductory text on financial engineering that embraces incomplete markets right from the start. It explores complete and incomplete markets in simple discrete contexts using Arrow-Debreu securities and gradually adds sophistication. By the end of the book, you are reading about optimal continuous-time hedging strategies in incomplete markets.

Contents

1. The Simplest Model of Financial Markets

2. Arbitrage and Pricing in the One-Period Model

3. Risk and Return in the One-Period Model

4. Numerical Techniques for Optimal Portfolio Selection in Incomplete Markets

5. Pricing in Dynamically Complete Markets

6. Towards Continuous Time

7. Fast Fourier Transform

8. Information Management

9. Martingales and Change of Measure in Finance

10. Brownian Motion and Ito Formulae

11. Continuous-Time Finance

12. Dynamic Option Hedging and Pricing in Incomplete Markets

App. A Calculus

App. B Probability

The book's focuses on theory. There is little discussion of standard pricing models. Term structure models are not mentioned. Neither are credit derivatives, which would be an obvious application for much of what is presented. Despite all this, the book is very hands-on. There are plenty of worked examples, and readers are encouraged to perform simple exercises at every step of the way. The book promotes the use of Gauss analytic software for many tasks, and it presents Gauss code at points. If you are not familiar with Gauss, this will not be a significant impediment to using the book. 

Don't plan on learning much math here because you won't. Lots of elementary math is used, but it is all cookbookish—"here is some formula; here is how to plug some numbers into it." This is not entirely bad because it keeps the discussion moving at a reasonable clip. Familiarity with basic calculus, probability and linear algebra is all you need to get through most of the book. Cerny formally goes through the motions of explaining the basics of linear algebra, but I doubt readers who have never worked with linear algebra will follow this. Actually, that discussion is wonderful for readers with modest familiarity with linear algebra because it motivates concepts such as independence and span with relevant financial computations in multiple dimensions. A chapter on Fourier transforms will be incomprehensible to anyone who lacks familiarity with Fourier transforms, but it jut might motivate readers who lack such a background to pick up a good book on Fourier transforms—I recommend Bringham (1988). Cerny assumes no prior familiarity with stochastic calculus. He does an admirable job of explaining the basics. His transition from discrete time to continuous time is one of the best I have seen in an elementary financial engineering text.

I would say that the book's biggest shortcoming is that it lacks context. Its focus is on developing financial engineering for incomplete markets. It affords little sense for the vast body of techniques developed over thirty years for complete markets. For this reason, I do not recommend this as a first book on financial engineering. As a second or third book—perhaps following on the heals of Kolb (2007) and Chriss (1997)—it is exceptional.

For related books, see sections:

Financial Engineering - Basic Theory

Financial Engineering - Numerical Methods

Math - Financial Math

Math - Financial Programming

Finance - General

 

 

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