For budding
financial engineers, this is an outstanding introduction to the mathematics that
underlies derivatives pricing theory.
Written by a
mathematician and published by the American Mathematical Society, it is
definitely a math text, but it is also a financial engineering text. It focuses exclusively
on math that is directly
relevant to financial engineering and packages it with financial applications
leading up to a derivation of the Black-Scholes formula. What it does along the
way is teach readers measure theory, probability theory and basic stochastic
calculus.
If you have
struggled with the concepts of sigma algebras, filtrations, measurable functions
and stochastic integrals, this is the book to read. Many competing books provide
intuitive treatments of this material without actually teaching it. There are also
plenty of rigorous books that are inaccessible to most readers. Few books combine accessibility with rigor. Of these, Dineen is the most
elementary. It also offers plenty of exercises that will help you build mastery.
Contents
1. Money and markets
2. Fair games
3. Set theory
4. Measurable functions
5. Probability spaces
6. Expected values
7. Continuity and integrability
8. Conditional expectation
9. Martingales
10. The Black-Scholes formula
11. Stochastic integration
I think anyone
with a strong background in calculus and elementary probability theory will be able to read this
book. However, you may struggle if you don't know the difference between a
countable and uncountable set or related topics. Readers who have taken a course
in advanced calculus that covered the topology of the real numbers will be best
prepared for the book. It would also be helpful to have some intuitive
familiarity with financial engineering. Don't make this the first book you read
on financial engineering.
This isn't an easy
read, especially as you get into the later chapters, but this is inevitable with
any book that rigorously treats this material. A good
motivational book to read before Dineen is the (largely intuitive) Baxter and
Rennie (1996). A good
supplementary math book is Bartle (1966).
Once you have completed Dineen, you will have a good foundation in basic
stochastic calculus and be ready for a more serious financial engineering text
like Bingham and Kiesel (2004).
[2/5/06]
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