Start with Stochastic Calculus for Finance I & II by Shreve. It starts of with a binomial, discrete time, model and builds up to the BSM model in continuous time and more, but really what you need is the math it teaches, not necessarily any specific model. This is the first book you would study in mathematical finance. Here's a free textbook pdf on stochastic calculus/ SDEs, just to give you an example of what you're looking at http://math.berkeley.edu/~evans/SDE.course.pdf . I can give more recs if interested as I'm currently getting a masters in math, concentrating on mathematical finance.
From Buffon's needles to $1 million per day. Fascinating thought! Thanks for the link. Can we take you up on your offer for a few more recommendations?
Sure thing! Here are some I've used in addition to the above:
Lecture notes from Dr Kohn at NYU: http://www.math.nyu.edu/faculty/kohn/math-finance-lecture-no...
Options, Futures, and Other Derivatives by Hull (considered one of the best books in this field, but I like the next one better)
An Introduction to the Mathematics of Financial Derivatives by Neftci
Numerical Methods in Finance and Economics: A MATLAB-Based Introduction (Statistics in Practice) by Brandimarte
Also, we take a general C++ course. Coming into the program they wanted us to have completed math through calc 3, differential equations (first course), and linear algebra (first course). I was a finance undergrad not math, so from experience I can say I wish I would have also studied some real analysis (just the basics), and a high level undergrad course in probability (although we took a grad level intro course first semester). Also, the better you are at proofs, the easier time you will have (I had no training in this and have picked it up as we go along).
Also, some schools make a distinction between computational finance and mathematical finance, so that's something to think about too.
Thanks.