Financial mathematics, or mathematical finance, is a set of mathematical tools allowing to express use cases on financial markets a way that can (or could) be solved using mathematics.
They are a lot of branches of financial maths:
- Modern Portfolio Theory is one of them: how to express portfolio management using mathematical formulas (and how does it leads to different solutions)? Think about Markowitz portfolios of course, but also about Black and Litterman ones too, or about Merton's intertemporal portfolios.
- Derivative pricing is another one: how to express the implementation cost of a contingent claim (and how it leads to the best way to replicate it)? This about Black and Scholes' option pricing of course.
- Optimal trading is a third one: how to write down the situation faced by a trader (or a market maker) having an inventory and trading with other market participants (and how it leads to optimal quotes or an optimal trading speed)? Algren-Chriss or Cartea-Jaimungal trade scheduling or Avellaneda-Stoikov+Guéant-Lehalle market making modeling are of this kind.
Financial mathematics do not tell you what is the appropriate model for a given practical situation, but tells you that "if you see the world this way, then an optimal decision process should drive you in this direction" (please take optimality in a very broad sense, financial maths do not tell you if agents are rational or not).
