Introduction to
Stochastic Calculus Applied to Finance
Expertly translated from the original French by N.
Rabeau and F. Mantion, this was one of the first advanced books to fully embraced the "stochastic calculus" approach to financial
engineering. It doesn't ignore the "differential equations" approach. It
actually has an excellent chapter on that perspective that is well integrated
into the rest of the book.
Adopting a theorem-proof format, this is one of
those books that brilliantly develops mathematical concepts with little or no
financial or intuitive motivation. If you already have the math skills and some
familiarity with markets and financial engineering, you will love this book!
Stated simply, it covers rigorously the material that Baxter and Rennie (1996)
and Neftci (2000) cover intuitively.
The book opens with two chapters that introduce
probabilistic and financial concepts in discrete time. The second of these
focuses on optimal stopping times, anticipating the use of such concepts for
pricing American options. Both chapters are nice for familiarizing you with
notation and the martingale approach to modeling markets.
Chapter 3 introduces stochastic calculus and Ito's
lemma. The material is very formal, so prior knowledge will be invaluable.
Chapter 4 and 5 are the meat of the book. They formalize option pricing theory,
with Chapter 4 focusing primarily on the "stochastic calculus" approach and
Chapter 5 focusing on the "differential equations" approach. Chapter 5 also introduces
the method of finite differences for finding numerical solutions.
Contents
Introduction
1 Discrete-time models
2 Optimal stopping problem and American options
3 Brownian motion and stochastic differential
equations
4 The Black-Scholes model
5 Option pricing and partial differential
equations
6 Interest rate models
7 Asset models with jumps
8 Simulation and algorithms for financial models
App. A.1 Normal random variables
App. A.2 Conditional expectation
App. A.3 Separation of convex sets
References
Index
Chapter 6 introduces interest rate models, focusing
primarily on the Vasicek, Cox-Ingersol-Ross, and Heath-Jarrow-Morton models.
This is a field that has developed significantly since the book was published,
but the chapter remains a useful introduction.
Chapter 7 covers jump diffusions, and is largely
consistent with Merton's early work on the subject.
Chapter 8 introduces the Monte Carlo method, but is
too cryptic in my opinion. See Glasserman (2003)
for a better treatment.
I highly recommend this book for mathematically
sophisticated readers who want a rigorous founding in the mathematics of
financial engineering. You should be familiar with measure theory, as covered by
Bartle (1966),
and measure-theoretic probability, as covered by Resnick (1999).
While the book introduces stochastic calculus, prior familiarity will be
invaluable. See one of my
recommended books on the subject. Finally, prior familiarity with markets
and financial engineering will provide an intuitive perspective that this book
doesn't attempt to communicate. See, for example, Chriss (1997).
[review and table of contents based on the first edition]
