# Mathematical Techniques in Finance : Review

Books on derivative pricing come in all shades and colors. Some books give a brief introduction of derivatives at a leisurely pace and then suddenly the content becomes very mathematical. There are some books that have theorems and lemmas all through. There are some books that talk about risk-neutral pricing giving very little intuition about the concept. In the gamut of books available, I think this book stands out for a couple of reasons. The fact that we all live in incomplete markets is addressed right from the beginning. This immediacy has the reader’s attention right away. If the markets are incomplete, i.e there are more state variables than the instruments, how does one hedge an exposure ? Can there be a perfect hedge ? If not, how does one compare between two or more hedging options? These are very practical questions for an options trader. An option trader intuitively knows that a perfect option hedge that is taught in a grad school is an idealistic scenario that holds good under a ton of assumptions. Real world is messy. Pick up any book where Black Scholes is derived; in 9 out 10 books, you will see measure theory as a prerequisite to understanding the content. This book though, does not to have a math heavy prerequisites as most of the book can be read with linear algebra, elementary calculus and probability knowledge. So, in a way, this book can be read by a wider audience. Even though this book uses many numerical simulations, the author also believes that

There are computations one can do with pen and paper that even the fastest computers cannot perform.

Even though the book has 13 chapters and organized logically, the author suggests that one can take several trails for first, second or third pass through the book. One can follow discrete finance trail by working through Chapters 1, 2, 5, 6; One can go over the continuous finance trail by working through Chapters 8, 9, 10, 11 and 12; One can go over risk management trail by reading through Chapters 3,4 and 13. If you are like me who likes to read stuff that combines intuitive arguments and math, you might want to go over the entire book cover to cover.

Link to a document that summarizes the main points of the book

**Takeaway**

This book is definitely a class apart from the usual books on math finance. It uses a practitioner’s view to explain many concepts that are tricky to understand at the first go. The use of matrix computations, Fourier transforms, optimization methods, etc. makes this book more appealing to a practitioner rather than a theoretician. Great reference for someone working as a desk quant .