This is the second volume of a two-volume
introduction to financial engineering.
Volume I covers concepts in discrete time. This second volume is the meat of
the subject, covering the mathematics and application of financial engineering
in continuous time. While not absolutely necessary, it will be helpful to read
Volume I before taking on Volume II. Volume II does occasionally refer to
material in Volume I. Technical discussions are occasionally skipped with just a
reference to the discrete-time equivalent in Volume I.
One difference between the two volumes is a
substantial jump in the level of mathematical sophistication. About the first
200 pages of Volume II introduce concepts from probability, measure theory and
stochastic calculus. This is very well presented. Concepts are motivated
intuitively, and much of the discussion is broached with financial examples.
However, I doubt anyone with knowledge of just calculus and elementary
probability theory will be able to keep up. Do you know what an uncountable set
is? Have you ever worked with moment generating functions? Are you familiar with
Lebesgue integration? Formally, the author doesn't assume such knowledge, but
you better have it. I recommend that you have some familiarity with analysis at the level of Apotsol (1974)
and measure theory at the level of Bartle (1966)
before attempting this book.
Next, the book has a chapter on risk neutral
pricing and another on the alternative partial differential equations (PDEs)
approach. It is nice that both are covered. Many books tend to emphasize just
one or the other. For much of the balance of the book, risk neutral methods are
emphasized (which is consistent with common practice in financial engineering
today) but I like the fact that the traditional PDEs perspective is covered.
These two chapters are more cryptic that the earlier ones. At this point, the
book becomes increasingly formal—more about theorems and proofs than explaining
finance.
Contents
1. General Probability Theory
2. Information and Conditioning
3.
Brownian Motion
4. Stochastic Calculus
5. Risk-Neutral Pricing
6. Connections
with Partial Differential Equations
7.
Exotic Options
8. American Derivative
Securities
9. Change of Numeraire
10. Term Structure Models
11.
Introduction to Jump Processes
The next four chapters apply the math, with
discussions of:
exotic
options,
American
options,
changes
of numeraire, and
term
structure models.
The book closes with a chapter on jump processes.
Topics of more recent relevance, such as stochastic volatility, incomplete
markets or pricing for credit risk are not discussed.
The book uses a formal definition-theorem-proof
format throughout, but it is not entirely rigorous. At points, examples are
substituted for derivations. References to discrete time results are substituted
for more challenging continuous-time treatments. The book is about theory. You
won't learn about markets, quoting conventions, day-counts, etc.
Each chapter closes with historical notes and
references. These tend to be highly informative and fun to read. There are also
end of chapter exercises.
I recommend this book for mathematically
sophisticated readers who have practical familiarity with financial engineering
but want to delve into the formal mathematics. It is a challenging read, but it
is one of the most accessible introductions to formal measure-theoretic
continuous-time finance available.
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exotic
options,