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 non-trivial results.

These books are used in the Intro math finance classes at Chicago and Rutgers, both of which have well-reputed math-finance programs.