Book Recommendation for mathematical finance Does anyone know a book which covers topics on:
Brownian Motion 
Martingales 
Stochastic Calculus
Stochastic Differential Equations 
Options pricing. Black-Scholes model
Fundamental Theorems. Interest Rates 
Random Walk 
Applications in Insurance 
Simulations. Convergence 
Simulation methods 
I would like something in-depth, but at an undergraduate level.
 A: A recommended book which covers most of your topics is Options, Futures and Other Derivatives -  John Hull. Regarding simulation methods, I would suggest Monte Carlo Methods in Financial Engineering - Paul Glasserman..
Both books are a good starting point.
A: Oksendal, Stochastic Differential Equations is also a very good book to learn stochastic calculus.
A: Consider An Informal Introduction to Stochastic Calculus with Applications - Ovidiu Calin!
This book would cover the following topics (that were mentioned in the question): Brownian motion, Martingales, Stochastic Processes, SDEs. For mathematical finance topics, you may download the earlier version lecture notes from the author's homepage webpage
In my opinion, this is the easiest book for an undergraduate wanting to learn stochastic calculus. Do NOT regard this book as inaccurate or wrong just because of the word "Informal". Having worked through 4 chapters of the book, it is definitely not inaccurate/ wrong; neither will the contents be a walk in the park.
I feel that the pre-requisites would be 


*

*A basic undergraduate course in probability (Sheldon's Ross A First Course in Probability would be enough)

*Some introductory mathematical analysis (Bartle or Rudin is sufficient) - minimally you should be familiar with supremum, infimum, riemann integrals 

*An undergraduate course in multivariable calculus (AKA Calculus III/IV)


NOTE: You do not need to know measure theory, stochastic processes beforehand, Calin develops them in the book. However, this book can still be very useful even if you know these topics beforehand. 
Because the author asks for so little from his audience, you should be aware that this book does not offer the fullest details of the mathematical theory. I do not see this as a setback for undergraduates since chances are a typical undergraduate would not be familiar with measure-theoretic probability. 
I am sure that after learning the contents from this book, you would be able to transit to graduate-level stochastic calculus texts.
I am by no means affiliated to the author; I am just very fortunate to have chanced upon this book. It is such a hidden gem!
A: This is answered on the Quantitative Finance S.E.
https://quant.stackexchange.com/q/38862
https://quant.stackexchange.com/q/2019
https://quant.stackexchange.com/q/15013
https://quant.stackexchange.com/q/1
https://quant.stackexchange.com/q/2391
