This book introduces stochastic calculus in a very intuitive sense not burdening the reader with heavy math from measure theory and functional analysis. The author in the preface says that the reader is deemed to have passed an examination based on the book if he/she can understand the derivation of Black-Scholes that is present in the last chapter of the book. If you pick up any standard text book that proves Black-Scholes , one needs to go through a ton of math before understanding Ito’s lemma that is one of the major tools in deriving the price of a call options. So, the author’s ambitious objective (as stated in the preface) is that the book should teach Ito’s lemma to anyone with a basic understanding of probability and calculus. Does he meet the objective ? Well, you got to poll a large group of readers to get a fair estimate to that question. My opinion is that the author has done a fantastic job of showing the appropriate concepts and tools that one needs to get a hang of, before understanding Black-Scholes PDE. How does the author mange it?
Chapter 1 : Preliminaries
Well, the book starts off with the basic introduction to probability and then quickly moves on to the definition of Stochastic processes and explains some of the ways to categorize them. Subsequently the book goes straight in to the definition of Brownian motion and its properties. Through various visuals, the reader is made aware of the fact that Brownian sample path is a nowhere differentiable function and has unbounded variation on any interval. This makes the standard Riemann integral useless for working with Brownian motion. Along the way the chapter explains the principle behind simulating Brownian motion and some derived processes from BM.
One can’t understand anything in stochastic processes without proper knowledge of Conditional expectation. Presenting the concept of conditional expectation in an intuitive way and at the same time touching upon all the properties and rules is a very tough act. The author starts off with a discrete conditional expectation variable and then takes a big jump by extending similar properties to a continuous conditional expectation variable. Something that is peculiar about conditional expectation random variable is that it has to be guessed!. You need to guess the right form based on some constraints. Since knowing its particular form is not always feasible, one needs rules to work with the variable. The chapter introduces about 7 most commonly used rules of conditional expectation.
The final section of the chapter is on Martingales. Well, Martingales are at the heart of math finance. They constitute an important class of stochastic processes. The author introduces the concept of filtration and gives definitions for continuous time martingales and discrete time martingales. The key aspect of Martingale transformation of a process by a previsible process is defined and explained via an analogy to a gambling fair game.
An interesting historical snippet is mentioned in this chapter. It’s about Karl Weierstrass , who was the first mathematician to come up with nowhere differentiable function. At that time it was considered a mathematical curiosity. But think of Brownian motion, a function that is a nowhere differentiable function. It has moved from the real of pure math to applied math as is now being used in a whole lot of scientific disciplines.
Chapter 2 : The Ito Integral
The author starts from scratch by defining Riemann integral , Riemann –Stieltjes integral, the latter being used to integrate a function with respect to another function. Subsequently he poses a question about the possibility of using Riemann-Stieltjes integral to integrate a function with respect to Brownian motion. The chapter then shows through various established theorems that Riemann-Stieltjes cannot be used. All the while, there is no formal proof for most of the statements here, as that is not the purpose of the book. It is supposed to be non-rigorous and it is so. The chapter then introduces Ito’s integral. For those who always associate value of an integral with a value, it might come as a surprise that Ito’s integral is a probabilistic average. Since one cannot integrate with respect to every Brownian path, all one gets out of Ito’s integral is a probabilistic average.
Now how do you define an Ito’s integral? Mathematicians love building stuff ground up. So, in order to define Ito’s integral, it is first defined for a simple step function process using Riemann-Stieltjes integration framework so that you get a probabilistic average of the integral. Then you define Ito’s integral of a general integrand as a limit of Ito’s integral of the simple processes. This is the thing that you see everywhere in math. If it is difficult to define a complicated object, you define it for a simple object, then use those simple objects to create a general object. For some one who is familiar with Measure theory, this is standard stuff. You define a Lebesgue integral of a non-negative function using Lebesgue integral for simple measurable functions.
Since the Ito’s integral itself is a random process, one can compute the expectation function and covariance function of the process . The chapter shows the beautiful the connection between Ito’s integral and Martingale theory. By using Riemann-Stieltjes integral in a specific way, i.e., the integrand evaluated at the left end point of the time interval, the resulting Ito’s integral becomes a Martingale, thus enabling one to use the rich theory of martingales. One downside is that the classic chain rule and produce rule of calculus no longer applies to Ito’s integral.
One needs tools to solve ODE, PDEs. Same is the case with Ito’s SDE. Ito’s lemma, the most useful tool in solving SDE’s is introduced in the chapter. Various forms of Ito’s lemma are introduced based depending on the case whether the function is driven by Brownian motion, Ito process or a combination of Ito processes. The chapter ends with the introduction of Stratonovich integral that evaluates the integrand in a different way. Despite the integral not being a Martingale, Stratonovich integral retains the classic chain rule and product rule properties and hence its utility.
Thus this chapter connects the concepts of Riemann-Stietljes integral, Ito’s integral, Stratonovich integral Martingales, Martingale transform of a process in such a way that the reader gets a fair idea of the Ito process and its properties. In all the cases, one needs to get used to the fact that the integral values are mentioned in the context of mean-square convergence. Integrals are probabilistic averages is one of the resounding take always from this chapter.
Chapter 3 : Ito’s Stochastic Differential Equations
The chapter starts off by defining a random differential equation and stochastic differential equation and explains the difference between the two using visuals. The first thing to notice is that the Stochastic differential equation is actually in the integral form. Since differentiation for a Brownian path is meaningless, the SDE is represented in the integral form, though one can write it in the differential form as a symbolic version. The author systematically builds up examples, visuals to explain ways to crack an Ito’s SDE.
Obviously there is no single way to crack it. It depends on the type of SDE. Straight forward application of Ito’s lemma to a hypothesized solution and then solving PDEs might work for some cases. Transformations might work for some cases. In this context, the chapter also has an interesting section of using Stratonovich integral. By working through this chapter, a reader is fairly equipped with techniques to solve some elementary to medium level difficult SDEs. The chapter ends with a discussion of numerical analysis of the solution to SDEs. Most of the times one want to get a handle on the distribution function for the solution and in this context the author shows two popular numerical analysis techniques, i.e. Euler’s approximation and Milstein approximation.
Chapter 4: Application of Stochastic Calculus in Finance
If a reader has worked through the first three chapters , i.e. around 170 pages of the book, the author assumes that the section on Black-Scholes derivation should be easy on eyes. My guess is, if a reader puts in some decent effort in going through 170 odd pages, he/she is more or less certain to understand the derivation of Black-Scholes PDE.
Well, even though the author seems to focusing on “Understanding Black-Scholes SDE” as a single test of measuring reader’s understanding, I think this book will open up many paths to a curious reader. May be you will like the way Martingales are introduced and you might decide to spend some time on understanding Martingales. May be you will appreciate that solving a SDE is more like trying out various techniques and you will venture out to understand and master those techniques. May be you will want to think about how to solve a SDE numerically, etc..Well, the list can go on. So, I guess this book will be a pleasure to read for anyone who is interested in knowing about the world of stochastic calculus and wants a pretty non-rigorous introduction. Again non-rigorous is a relative term. Something that is non-rigorous according to the author might be fairly challenging for someone. It goes without saying that this is a must-read for any math-fin student as it provides a superb intuition to formulating and solving SDEs.