[.ca] Stochastic Calculus and Financial Applications (ISBN 0387950168)

graduate course taught at Wharton:
Mike Steele has used the material in this text to teach stochastic calculus to business students. The text presupposes knowledge of calculus and advanced probability. However the students are not expected to have had even a first course in stochastic processes. The book introduces the Ito calculus by first teaching about random walks and other discrete time processes. Steele uses a lecturing style and even brings in some humor and philosophy. He also presents results using more than one approach or proof. This can help the student get a deeper appreciation for the probabilitist concepts. The gambler's ruin problem is one of the first problems that Steele tackles and he uses recursive equations as his way to introduce it. Brownian Motion, Skorohod embedding and other advanced mathematics is introduced and emphasized. After motivating the stochastic calculus and developing martingales Steele covers arbitrage and stochastic differential equations leading up to the fundamental Black-Scholes theory that is important in financial applications. It is not fair to criticize this book for lack of applicability. It is strickly intended to develop a firm theoretical background for the students that will prepare them for a deep understanding of financial models important in applications. I am not enough of an expert in this area to know if Professor McCauley's criticism in another amazon review of this book is valid, but I do think he is a little too harsh in criticizing the ideology that Steele presents. The ideology is what makes Steele's lectures stimulating and interesting to the students.

Would be better without the ideology:
Good intro to stochastic calculus and sde's, can compare roughly with Baxter and Rennie in readability. However, the book unnecessarily propagates ideology. First, it makes excuses for the fact that the empirically wrong notion of utility ('maximizing behavior') is totally disconnected from the Black-Scholes model. Second, the text propagates Black and Schole's original mistaken claim that CAPM produces the same option pricing pde as does the delta hedge. A careful and correct calculation shows that this claim is wrong, that with the wrong assumption made by B-S the fractions invested in both the stock and the option are zero! For the correct result, including the difference in option pricing via delta hedge and CAPM, see my recent paper "An Empirical Model for Volatiliy of Returns and Option Pricing' with Gunaratne. A third criticism is that only Gaussian returns are discussed in this text, but the empiciral distribution is far from Gaussian and is approximately exponential, with nontrivial volatility.

Riskfree profit !!:
The book is at the interface of three areas, math, statistics, and finance. While connections between the first two have a long history, it was the connection to finance that caught my attention. Coming from math myself, I needed first to take a closer look at the book to orient myself. The mathematical subjects, smooth sailing, include stochastic differential equations (SDE) as they relate to PDEs; and the ideas from probability and statistics include Brownian motion, martingales, stochastic processes, and the Feynman-Kac connection. Browsing the chapters I found them to be a lovely presentation of ideas with which I am familiar. For me, it was chapter 10 that turned out to have stuff that I wasn't familiar with. That is the finance part, and it is based on a model for Option Pricing developed in 1973 by Fischer Black and Myron Scholes. An arbitrage opportunity \osimplified\c amounts to the simultaneous purchase and sale of related securities which is guaranteed to produce a *riskless* profit. It was after reading more in this chapter I understood why the book is used in a course at the Wharton School at the University of Pennsylvania. I am impressed with the level of math in this course. Part of the motivation in the applications to finance is that arbitrage enforces the price of most derivative securities. And I learned from ch 10 that the SDE of the Black-Scholes model governs the processes which represent the two variables S, the price of a stock, and B the price of a bond, both S and B representing stochastic variables depending of time t, i.e., both stochastic processes. In the model, S is a geometric Brownian motion, and B is a deterministic process with exponential growth. The two are determined as solutions to the SDE of Black-Scholes.

Review from a grad student not at Wharton:
Reading Steele's book without attending has classes at Wharton leaves the reader looking for explanations to equations. Ideas are not clearly explained and problems are not worked out in detail with a descriptive process of how to solve the problem. The brief explanations in this book intended for a reader with knowledge of calculus and probability but not having a background in Stochastic calculus do not provide a sufficient basis for the reader to learn the material.

I Hate It When Books Lie About Mathematical Requriements:
The book says that its only prerequisites are calculus and probability. This is not true. To be able to understand everything that's going on, you'll need to have a very good grasp of subjects like measure-theoretic probability, Hilbert spaces, and functional analysis. I quit reading the book in the early chapters, when Steele starts talking about things like "spans" and "denseness" for function spaces. I don't know where you went to school, but at my school, I didn't learn these subjects in my intro calculus and probability classes. To summarize, don't buy this book if you don't know measure theory. If you want to learn quant finance at an elementary level, Baxter and Rennie is much, much better. Moreover, if you're comfortable with measure theory,and you want to learn the math that's necessary for option pricing, you'd be better off buying Oksendal's excellent book, which is at least as rigorous as Steele's book but much more clear.