# Quantitative Challenges I spent about two to three hours studying and then working through mathematical problems on quantitative finance today. Specifically, these were questions from Blyth’s An Introduction to Quantitative Finance and dealt with interest rate swaps. These are contracts where one party typically pays fixed payments (e.g. 5%) and receives floating payments, which are dependent on market rates (e.g. LIBOR + 1.5%), though float-float swaps exist too (e.g. across currencies). Swaps can be used to mitigate interest rate risk (or gain exposure!); they can also be mutually beneficial depending on companies’ borrowing characteristics.

If I had to classify the mathematics involved in the problems I did, beyond “applied” I’m not sure how else I could label it. For many of these problems, the first steps involved figuring out how to mathematically model the financial products involved. There then followed some elementary algebra, along with proofs that required some intuition to pick the right approach (for lack of a better word). The exercises I did this time around relied less on probability than normal; much of this probably stemmed from a result that forward interest rates (i.e. the interest rate you’d get from future time T1 to later future time T2) could be valued independent of the distribution of possible values.

I’ve struggled quite a fair bit with the book, both in terms of the reading material as well as the exercises. Today’s chapter was relatively easier, though that might have been because I was reading through the chapter for the second time. It was my first time doing many of the exercises, though it seems like they went relatively smoothly today.

Some of this might be because I work through the chapters at a very relaxed clip of about one per month. Like many other mathematical domains, there tend to be many dependencies on previous topics. The earlier result I mentioned on forward interest rates, for example, was from the previous chapter; yet, it was instrumental in computing the valuation of a swap. There are certain fundamental ideas that I learned way back in 422 (Computational Finance at Imperial). I also think my mathematical knowledge and logical intuition have (hopefully) mostly stayed with me. Furthermore, I like to think that I remember the concepts at a high level. However, many proofs require recognizing that expressions are in certain forms and can thus be rewritten; I’m still yet to develop that level of familiarity – or shall I say intimacy – with the content.

This might also be partially self-created, especially where the reading material is concerned. When I see theorems, I tend to try my hand at proving them on my own first. These often prove to be rather tricky endeavors; the aforementioned lack of keen familiarity with the material certainly doesn’t help. Typically, I can understand the proofs fairly easily when reading them. However, I usually expect myself to figure out the intuition behind the proof (including reproducing it, at least at a high level), which isn’t always so forthcoming. That actually reminds me of what I used to do at Imperial for certain modules, especially the (in my opinion) two hardest of the course: 438 Complexity and 493 Intelligent Data and Probabilistic Inference. I would make the effort to understand why many of the proofs in those courses worked. I’d also try to figure out how the author might have come up with the proof, or at least what the core intuitions might have been. This included relatively nasty ones (e.g. SAT being NP-complete via direct argument, or ELBO results in variational inference), and I think it paid off in terms of understanding.

It could be argued that finding the material difficult is expected, because the subject matter is itself complex. I tried to obtain a popular estimate of the complexity of the material covered by the book, but didn’t find much data; there were only a handful of reviews on Amazon, which offered a wide spectrum of views (from “[o]ne of the best introductory treatise (sic)” to “I would hardly call it an “Introduction” to quantitative finance”). I’m not sure how to start more simply, though; there is a fair bit of assumed mathematical knowledge, but this is at least partly spelled out in the introduction.

While the book has proved challenging at times, I’ve not found it too hard to follow. Discussing the problems with a friend has also helped a fair bit, especially since the book doesn’t have solutions!