I recall when I
was a graduate student studying math. A professor was trying to recruit me onto
a research project, and I was resisting. When I protested I knew nothing about
continuum mechanics, he walked into the corridor, poking his head into offices
asking "does anyone have a bad book on continuum mechanics?" What he was looking
for was a quick book that focused on results, downplaying rigor in favor of
rapid understanding. He wanted to find me the kind of book mathematicians hate
but engineers love. This book is that kind of book.
Author Navin
developed the book out of lectures he gave for programmers who needed to
implement financial engineering models. His was a mathematically sophisticated
audience with no background in finance.
Overall, the book
unfolds in a predictable manner: probability theory, stochastic calculus,
Black-Scholes, interest rate models, credit risk models, CDOs. All is covered
in just 136 pages (the balance of the book is exercises and appendices). As you
can imagine, the discussions are far from comprehensive. What is interesting is
the choice of topics at a page-by-page level. Downplaying overall theory, the
author shares with readers practical techniques and insights that financial
engineers use every day but more academic books don't pick up on.
Navin rearranges
the black-Scholes formula into a sort of reverse Kolmogorov equation to discover
three hidden symmetries. A few pages later, he is using Euler's relation to
disaggregate a derivative's price into a linear function of strike and spot
price. The result is handy for derivatives on multiple underliers or for
constructing optimal hedges. Later, there is an practical discussion
of finite difference methods.
Contents
1. Introduction to the Techniques of Derivative
Modeling
2. Preliminary Mathematical Tools
3. Stochastic Calculus
4. Applications of Stochastic Calculus to
Finance
5. From Stochastic Processes Formalism to
Differential Equation Formalism
6. Understanding the Black-Scholes Equation
7. Interest Rate Hedging
8. Interest Rate Derivatives: HJM Models
9. Differential Equations, Boundary Conditions,
and Solutions
10. Credit Spreads
11. Specific Models
Exercises
Solutions
A. Central Limit Theorem-Plausibility Argument
B. Solving for the Green’s Function of the
Black-Scholes Equation
C. Expanding the von Neumann Stability Mode for
the Discretized Black-Scholes Equation
D. Multiple Bond Survival Probabilities Given
Correlated Default Probability Rates
As you can
imagine, discussions tend to be minimalist, and are often somewhat cryptic. The
author assumes a lot of mathematical knowledge. Don't be intimidated. Much of
the math does not build upon itself. If you come across a concept that is
unfamiliar, you can make a mental note to later study it, and then move on. In
this regard, the book could motivate plenty of further study. Exercises and
solutions at the back of the book will be invaluable for any student.
No one is going to
learn financial engineering from this book. It is too short, too cryptic and too
idiosyncratic. On the other hand, anyone trying to learn financial engineering
will benefit from this as a supplementary text. It offers a unique perspective.
The exercises are excellent.
It is practical and, I think, quite motivational. [December 2, 2006]
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