There are two main roots to risk analysis (as I see it):
where the first is more real-world and requires data analytics, whereas the second is more academic/theoretical. Personally I am more of a theorist and will advocate the second approach, but to have the maths for the second is a bit of a step up, and is built around stochastic calculus
Modelling using Stochastic Calculus:
@Quantes, It's good you took lectures from the maths department, and it seems like you have a sufficient grounding for normal calculus. As you mentioned though in order to understand how variance/risk adds and combines in a financial setting you will have to take a course in Stochastic Calculus, for which the starting point is normally an overview of measure theory. However, pick up any introductory book (many typically recommend Steele or Shreve) and skip ahead to Brownian motion, then martingales, then semi-martingales, Ito's lemma, and stochastic/Dolean exponential. The final two tools you will need are then Girsanov's Theorem and change of measure theory. Once you have learnt this you will be ample prepared for any course on mathematical finance!
That though just gives you the mathematical tools, as for an intuition into solving financial problems the only extra thing that I suspect would be well worth looking into is change of numeraire, (extension of change of measure).
The book which I personally would recommend, and think is both comprehensive and self contained enough to skip from chapter to chapter is:
- "Introduction to Stochastic Calculus with Applications" - 3rd Edition - Fima C Klebaner
albeit that is my own preference and maths level, and this will change between individuals.
With regards to your thesis: An Example topic
I can only make a guess about how what direction of risk analysis you would want to go into regarding your thesis, (you could learn all of the above and then look into many topics which don't use any of this). To have an idea about what sort of topics this will let you look into, here are a few examples.
- Basic derivatives pricing theory (e.g. The Black-Scholes framework).
- Volatility modelling.
- Interest rate modelling and forward curves for fixed income markets.
The last of these I think is generally the most interesting (my own bias), but undoubtedly could be linked into a discussion of banking failures surrounding the US subprime mortgage crisis. E.g. having a mathematical idea of how mortgage products are priced and sold is not too difficult an idea qualitatively, the tricky quantitative bit is to put numbers on the risk (e.g. VaR), and to model the risk and sensitivities correctly. E.g. the prices quotes on US mortgage pools (cf. TBA products in the [Agency] MBS market) can be found by most suppliers (Bloomberg), and a large focus is on the risk analysis and interconnections between prepayment and default risk, which to understand quantitatively requires both statistics and a good grasp of financial calculus.