[5/50] #Financial Mathematics# Summary 1 for Calculus for Finance: Volume 2: Continuous-time Model

Today I want to spend the whole morning to do a summary for the book Stochastic Calculus for Finance Volume 2: Continuous-time model. I just want to provide some information about this book for those people who interest in the Financial Mathematics.

Initially, I want to introduce my background of Mathematics and Financial Mathematics. Following are part of books I have read about and related to this book. 

Real Analysis & Complex Analysis written by stein (Various concepts of real and complex analysis and some Heuristic examples for Functional Analysis).

PDE & SDE written by Evans (Most important book for learning PDE theory, elementary book for learning SDE).

A Course in Financial Calculus (The textbook for my Financial Mathematics course).

Next, I want to list some advantages for this book (Personal View).

A1: How to introduce the concept of probability space into Probability theory?

As we know, the most important concept in modern probability theory is to introduce the probability space, and consider about the probability as on kind of measure.

This kind of idea is wonderful, but still pretty tough for me to understand at the first time. So I need motivation. I love the understanding by Evans (I always love his books) in SDE, and I also love the motivation to define the probability spaces in this book.

（Not for citation, just for emphasize.）

In classical probability theory, we know how to compute the event A,

Pr(A) = Pr(w1) + Pr(w2) + .... + P(wn) for all wi belongs to A.

We note that this formula only useful along with the at most countable infinity sample spaces. Because at the uncountable infinity sample spaces, the probability of any simple outcome is zero. Alternatively, we cannot determine the Pr(A) by sum up the Pr(element belongs to A).

This problem motivated us to define the probability spaces (Sample space, sigma-algebra, probability measure) so as to compute the probability of complicated events in uncountable infinity sample spaces.

For more information and heuristic examples, please look at the book.

A2: How to define the Ito's integral?

I think the second advantage in this book is the process to define the Ito's integral. I recall the process to define the Lebesgue integral the Real Analysis,

Characteristic function -> Simple function -> Bounded and supported in the compact set -> non-negative function -> general function

And this process generated two convergence theorems. i.e. Monotone convergence theorem and Dominated convergence theorem. 

Likewise, Steven used the same argument to define the Ito's integral. He also started with the Ito's integral for simple stochastic processes, then used the convergence theorem to define the integral for general stochastic processes (take the limit for the simple processes). 

I love this way to define the Ito's integral, because it connects the Lebesgue integral, Approximation theory (convergence theorems). However, the approximation theory in this book is rough. Precisely, how to prove that a general process can be approximated by a sequence of simple processes is unclear.

A3: To generate the Black-Scholes Equation: pricing for option by Hedge

In this book, Steven derived the Black-Scholes Equation for price of an option on an asset modeled as a geometric Brownian motion. I love it! Some book just introduce this equation in discrete case (also in Volume 1), but I think the continuous-time model is much more interesting,

Hedge:

Evolution of Portfolio Value  =  Discounted Stock Price and Portfolio

Using this equation, we will get a PDE, then solve this PDE by Fourier transform.

However, how to transform the BS equation to a diffusion equation (heat equation) and solve this equation by Fourier transform is unclear in this book (but can be found very easy in Google). 

I just read a part of this book, and found this book is wonderful and powerful. I want to do the summary 2 when I am free. 

FYI.

Eric

Feb. 11th