Most areas of mathematics are very well documented. Financial mathematics is no exception.
However, like most maths books, most require at least some prerequisite knowledge,
often to the level of a Maths undergraduate,
or Engineering/Physics undergraduate.
I have read (parts of) quite a few books in this area by now, both specific financial
maths books and also other maths books.
Options, Futures and Other Derivatives - John Hull - This is the most famous book on this area.
A lot of it is reasonably simple, especially the earlier sections explaining futures and the
binomial lattice model for pricing options. However, after Chapter 11 it becomes more
complicating explaining Black-Scholes models. For this I found that it assumed quite a
bit of knowledge about calculus, particularly PDEs, something which I have not covered.
In any case you can avoid this area, if you are prepared to look at results (and not the proofs).
Most mathematics engineering books should be sufficient to cover this. Later on, it also has
sections about credit derivatives. There are also descriptions of numerical procedures for solving PDEs,
such as finite differences. Finally Derivagem, a software package is included
which is basically a set of Excel worksheets with macros that value lots of the derivatives covered in the book.
I found the software to be a good educational aid, and good for checking my paper valuations using the binomial tree model.
An Introduction to the Mathematics of Financial Derivatives -
Salih N. Neftci - This book unsurprisingly goes into the mathematics of derivatives quite quickly
with only a short section explaining the various types of derivatives (unlike Hull's book, which is
quite detailed here). However, it does explain the maths very well, for example explaining how to
classify PDEs, and how stochastic integration is different from normal integration etc. It often
gives proofs as well, which I found I read through with a bit of effort (it is going to be another
matter actually being able to reproduce them). To properly understand them, I shall need to go through
them on paper. It might still be necessary to buy an accompanying engineering mathematics book, if you
want to understand the mathematics in more detail. Later sections explain the more difficult models for
valuing interest rate derivatives, which I am hoping to get on to eventually. Once I finish the book,
this review will be updated.
A First Course In Probability - Sheldon Ross - This book contains virtually no finance! However,
it is a great book for getting a grounding in probability, which is vital for understanding financial
mathematics. It explains most of the important concepts such as moment generating functions, random
variables and expectations. It also describes most of the major distributions, proving certain properties
from first principles. There are many easy to follow worked examples as well. There are many exercises,
broken down into easier practical ones and more difficult theoretical proof based ones. There are also
later sections at the end, going into advanced areas such as simulation and modelling (which I didn't cover).
This book was recommended by lecturers for two introductory probability and statistics courses I attended.