I am an undergraduate maths student and I am tasked with writing a 10 page essay on a mathematical topic outside of what we studied yet, so something new and perhaps more advanced. The whole point of the essay is to get a student to learn about something new on his/her own by consulting various books.
Since my future interest lies in mathematical finance and since I enjoy Partial Differential Equations; I thought about choosing a topic concerning PDE's in finance. I have taken ODE's and PDE's undergraduate courses.
Aim: I would like to explore Brownian Motion, Black-Scholes Equation and topics such as Option Pricing and if I am not wrong I would have to learn about Stochastic Partial Differential Equations. -How should I delve into this field? The issue I am having is that I do not really know which areas of maths I actually have to learn in order to explore the above. Book recommendations would be helpful.