I am currently doing my masters studies in financial mathematics. However, I have had a good background in number theory and I don't feel like leaving it just like that. I am thus inquiring on any applications of algebraic number theory in financial mathematics. When I finish this degree, can I be allowed to enroll for another masters in algebraic number theory? Can I marry the two and apply number theoretic concepts in finance? Please advise me.



Algebraic Number Theory in Financial Mathematics
I do not know any applications of algebraic number theory in financial mathematics. However, there are attempts to use string theory inspired approaches to financial markets: <http://arxiv.org/abs/1109.0435> (The string prediction models as an invariants of time series in forex market, by R. Pincak and M. Repasan). On the other hand there is a deep and somewhat mysterious connection between number theory and string theory: <https://www.quantamagazine.org/20150312-mathematicians-chase-moonshines-shadow/> (Mathematicians Chase Moonshine’s Shadow, by Erica Klarreich). Therefore I would not exclude that you can finally find some applications of number theory in financial mathematics.


Other possible directions to search alleged applications of number theoretic methods perhaps is path integral and gauge theory approaches to financial modeling. For applications of the path integral see, for example, <http://deepblue.lib.umich.edu/handle/2027.42/44345> (The Path Integral Approach to Financial Modeling and Options Pricing, by Vadim Linetsky), <http://arxiv.org/abs/1410.1611> (Path Integral and Asset Pricing, by Zurab Kakushadze) and the classic book of Hagen Kleinert: <http://www.worldscientific.com/worldscibooks/10.1142/7305> (Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets). As for the applications of gauge theory, see
<http://www.ingentaconnect.com/content/iop/jphysa/2000/00000033/00000001/art00102> (Gauge geometry of financial markets, by K. Ilinski) and <http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1458886> (Gauge Invariance, Geometry and Arbitrage, by Samuel Vazquez and Simone Farinelli).


Although this is not directly related to your question, I'd like to mention that some real world applications of number theory is considered in the book
<http://www.springer.com/physics/theoretical,+mathematical+%26+computational+physics/book/978-3-540-85297-1> (Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, by Manfred Schroeder).


