Does anyone know a book that covers topics on:
Brownian motion and random walks
Martingales
Stochastic Calculus and Stochastic Differential Equations (SDEs)
Options pricing, including the Black-Scholes model
Fundamental theorems
Interest rates
Applications in insurance
Simulations and simulation methods. Convergence
I would like something in-depth, but at an undergraduate level.
On
Oksendal, Stochastic Differential Equations is also a very good book to learn stochastic calculus.
On
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
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.