Yet Another Book on Financial Mathematics
Yes, it was planned as a book on Finmath with emphasis on Finance but with a very thorough introduction to the underlying math, first of all, measure theory.
I did not expect many readers but hoped for a some kind of "kernel readers group", who were really interested and would help me improve the stuff and write more.
However, financial world turned out to be somewhat apathetic: professionals strives to applied stuff, lectures do not care whether the way they teach is digestible or not and students worry only about their grades
(hey, empoyers, these are YOU, who force them to prefer the good grade to the good knowledge ;) ).
There is at least one reader, who was grateful to me for(I cite him) "very clear introduction to the measure theory, much better than Shreve". This comparison with maestro Shreve is particularly flattering for me,
since I learnt by Shreve's textbook and it is really wonderful.
However, one grateful reader is not enough, that's why I pause writing for a while.
I am really short of time because I make MONEY at the stock market. And, guys and gals - believe it or not - without math knowledge I would make only losses.
However, what I have already written is a clear self-contained introduction to the measure theory.
That's why from now on(12.02.2010) the manuscript is [re]named
Yet another, yet very reader-friendly, introduction to the measure theory
Now my primary target audience is not the set of highly-motivated and proud students (unfortunately the measure of this set converges to zero) but the students who fear to fail their exam in Real Analysis.
So, guys and gals, if your Prof begins with an abstract definition of sigma-algebra, if he does not give examples (surely, he did not show an algebra which not a sigma-algebra), if the Caratheodory construction is what he starts from
very beginning with, so you came to the right website!
The 1st chapter is a teaser for financiers, here I quickly show that the math is valuable for financial modeling. Those, who are interested in Real Analysis only may skip it but I do recommend to have a look on it.
The 2nd chapter is the most important, here I slowly and thoroughly introduce the reader to the measure theory. I start with the measure on (0,1) and then on \mathbb{R}^1 (as Lebesgue himself did).
You will see, how semirings and sigma-algebras naturally arise and what advantages the Lebesgue measure has.
In the 3rd chapter we discuss the abstract measure theory (now, after chapter 2, your mind is mature for this). As a bonus (and this is my own "pedagogical invention") I demonstrate that a semiring as such, without
a sigma-additive measure on it, does not mean much. Additionally, I try to show an intuition behind the interplay of the measure theory and the theory of stochastic processes.
Chapter 4 is incomplete but it accentuates the important but frequently omitted idea that measurable functions can do without measures...
After you read Chapters 2, 3 and 4, your reference textbook on real analysis, which is recommended by your Prof, will become readable for you.
So my manuscript does not (so far!) substitute your textbook, my manuscript just gives you a "decryption key" to it.
Good luck and may the force be with you!
Chapter 1 is available online
Complete Chapter 2 is available online (some misprints corrected on 28.11.2008)
Complete Chapter 3 and
LaTeX source are available online (6.12.2008)
Some stuff of Chapter 4 and
LaTeX source are available online (last update: 8.01.2009)
See also: The Wedding Options (Notes on financing the wedding celebration with yield on American call options)
Remarks, comments, suggestions are welcome! 