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Seeking some master's thesis topics, as as a math majors

I'm searching some master's thesis topics related to quantitative finance, as a math major's student.

I have had a glance at what a master's student in quantitative finance learn, there are many. So I would like to have someone suggest me some possible master's thesis topic based on the courses I have taken so far, and by giving some reference books, point out the additional knowledge that I need, to be able to start to do research.

Because I have to contact a master's thesis advisor by next February at the latest, my time to learn the essential tools/knowledges is very limited. I want to show the advisor that I'm very serious & motivate for the PhD.

Thanks in advance !

For the background :

I'm currently a second-year master's student in probability theory, I also did a math major in college. My ideal goal is to do research in math as a teacher-researcher in a university, and because I want to see that the math I learned can be applied, I want to be able to find/solve math problems from the applied field (quantitative finance). For this transition, I don't want to spend another year or two completing another master's degree, I plan to write a master's thesis next semester then find a PhD.

Courses taken in the 1st year of master : differential geometry, Lie groups & Lie algebras, algebraic topology, Riemann surface, functional analysis, Galois theory, statistics
Courses taken in the this semester : stochastic calculus, Markov process, statistical learning, ergodic theory.
I also learned numerical probability, Python, OCaml, Algorithms, data structures in college.

The type of math I like is more analysis, like stochastic calculus and functional analysis. I don't code regularly.
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easy topic: pairs trading and everything related to it
hard topic: how to select the optimal distance from mid price when placing orders as a market maker
Here is a paper that came out of my master’s thesis work for an MA in mathematics: Hausdorff and Fourier dimension of graph of continuous additive processes . Feel free to DM me and I can send you my thesis version.

The topic is at the intersection of geometric measure theory/fractal geometry and probability. Topics like the one listed above are far too applied for a masters thesis in (pure) mathematics. You may also find something dealing with Malliavin calculus to be interesting.