Hi, I am looking for a good reference to start reading about Mathematical Finance/Financial Engineering. I have a good background in Math but have no idea at all in Finance.

4$\begingroup$ Is this a researchlevel question in mathematics? $\endgroup$ – Andrej Bauer Jan 24 '13 at 10:18

1$\begingroup$ To Andrej: strictly speaking, no. But this kind of question if very frequent and wellaccepted on mathoverflow: A researchlevel mathematician asks a question about where he should go to learn some other parts of mathematics/physics/otherscience. $\endgroup$ – Joël Jan 24 '13 at 13:54

$\begingroup$ You may also ask at quant.stackexchange.com $\endgroup$ – Alexander Chervov Jan 24 '13 at 19:12

$\begingroup$ Another really good site to get advice from is wilmott.com $\endgroup$ – Deane Yang Jan 24 '13 at 19:50

$\begingroup$ I've made this community wiki. $\endgroup$ – Scott Morrison♦ Jan 24 '13 at 22:31
If you have no idea at all of finance you should begin by reading a book of finance, with only elementary mathematics, which tells you what it is all about. A very standard, wellknown text book is "Options, Future, and other Derivatives" by John C. Hull. Now to begin with the mathematical treatment of finance, I recommend for example "Methods of Mathematical Finance" by I. Karatzas & S.E. Shrieve.

1$\begingroup$ I agree that Hull's book is a good place to start. If you ever decide you want to focus on bonds and fixed income, I also recommend Bruce Tuckman's book, Fixed Income Securities: Tools for Today's Markets. The advantage of these two books is that they focus on real world finance and use math only when needed. $\endgroup$ – Deane Yang Jan 24 '13 at 2:24
I definitely recommand "Introduction to Stochastic Calculus Applied to Finance" by Lamberton and Lapeyre. It is concise, precise, introduce all the mathematics you want by constructing the Ito calculus, the BlackScholes model, formulation of the pricing via martingales and PDE, some interest rate theory and then introducing jumps and finishing by the algorithmic side. Then you can read "The Concepts and Practice of Mathematical Finance " by Mark Joshi where you can gain more insight on the financial side and the technics used by the practionners. I think this could be a good start.
There is also the excellent lectures by Emmanuel Derman that you can find here http://www.ederman.com/new/docs/laughter.html. You can read theme even before the Joshi book. It goes from the basics to localstochastic volatility in a physicist spirit with lot of intuition
Some expert (physicist, working partly in finance) recommended me the book:
JeanPhilippe Bouchaud, Marc Potters (2003). Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management Amazon
It is econophysics approach to analysis of financial markets. It uses quite advanced mathematics including random matrices, stable distributions and so on. One can also look for the papers by these authors in arxiv.
Another expert recommended me the following site: http://www.opentradingsystem.com/quantNotes/main.html about quantative finances,
and the book " HighFrequency Trading. A Practical Guide to Algorithmic Strategies and Trading Systems" IRENE ALDRIDGE
The standard reference for derivative pricing and the role of Ito calculus are still the books by Shreve called Stochastic Calculus I (discrete) and Stochastic Calculus II (continuous). The whole theory is developed from a mathematical viewpoint with definitions and theorems and proofs; so if you appreciate the standard math textbook approach to life then you will find the presentation pleasantly familiar.
The material is developed slowly (I think most people who have taken a course in probability even at the undergraduate level can safely skip Shreve I) but by the end one starts to see and use nontrivial results.
These books are used in the Intro math finance classes at Chicago and Rutgers, both of which have wellreputed mathfinance programs.
I agree with Vel Nias' answer that Shreve's books are good. I would like to add that if you have literally no idea at all in finance, as I did when I first started reading about financial mathematics, it would be a very good idea to read a nonmathematics book about finance to learn what stocks are etc. Most authors of economics books try hard to show how clever they are, so their books can be quite difficult to read. On the other hand, children's books can sometimes be good, such as Growing Money by Karlitz and Honig. I also recommend The Great Game by John Steele Gordon, which is a relatively downtoearth history of Wall Street.
(By the way, it's almost impossible to find a book about finance which is unbiased in the sense that it doesn't try to offer some kind of investment advice.)