# what would be the most parsimonous sequence of study?

As a 3rd year undergraduate Economics student, I want to write my undergraduate thesis on Risk Analysis of bank failures. I want to prepare my theoretical background in summer time before the starting my senior year.So, what would be the minimum relevant sequence of study in my case to write an undergraduate thesis about risk theory.

My relevant background, non-measure theoretic probability & statistics ( read Newbold's Statistics for Business and Economics + Wackerly's Mathematical Statistics) , undergraduate level Econometrics ( read Gujarati's Basic Econometrics + Wooldridge's Introductory Econometrics + first few chapters of Davidson & Mackinnon's Econometric Theory and Methods) + Undergraduate level Linear Algebra + Advanced Calculus course series which constitutes topics such as intro to Point set topology and real analysis + Riemann Integration Theory and Differential Calculus. I have not taken any Risk Theory or Actuarial Mathematics courses.

Please give some field & topics recommendation so that i smoothly go into the target topics. Additionally, give some self-study textbook recommendations of your field recommendation.

• When you say undergraduate level linear algebra, advanced calculus, etc., are these courses given by an economics department for undergraduate economists, or are they for undergraduate mathematicians/physicists/etc, as often the level at which these begin might be similar, but the level at which a mathematics department will finish a linear algebra course is a lot higher. Can you possibly elaborate with the amount of hours you have, and possibly give some example theorems/equations you know how to solve, etc. This should help us better assess your level. – oliversm Jun 11 '16 at 1:48
• It seems like an impossible question for someone (other than you) to answer. I would suggest instead you browse some recent papers in bank risk and financial regulation by authors such as Hyun Song Shin, Tobias Adrian, Markus Brunnermeier, Anat Admati. Find out what concepts and techniques they use that you don't understand and try to read up on those. Or at least get some sense of what kind of work they are doing, what the deficiencies were in the pre-crisis framework and how these people are trying to improve theory and practice of bank modeling and regulation. Need more finance than math. – Alex C Jun 11 '16 at 2:01
• Olivesm , I took all the relevant courses from math department(not economics department). 2 linear algebra courses + 1 single variable calculus+ 1 multivariable calculus + 2 advanced calculus courses ( a little bit of set theory , closedness opennes in $R^n$ , compactness, sequences, differentiability, continuity, double integral, triple integral, line integral, surface integral which I have learned Greens , Divergence and Stokes' Theorems) – Quantes Jun 11 '16 at 5:34
• Alex C, I looked at some papers on the field. Especially at phd theses, authors are using like time to faillure, survival function,loss function, hazard function, gamma/gompertz , exponential distrubution, concepts from stochastic processes stuff like that. But to learn these stuffs, I always face the suggestions that i should learn at least a basic measure theory. What do you think ? Is it relevant with what i wanna do? – Quantes Jun 11 '16 at 5:36
• @Quantes my comment was too long, to I have posted it below. Also when mentioning people in a comment use the @ symbol and the username as it appears, and this should notify them. – oliversm Jun 11 '16 at 8:27

There are two main roots to risk analysis (as I see it):

• Statistics
• Modelling

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.

• Can you please elaborate on how this will help the OP with his thesis on 'risk analysis of bank failures' – Kiwiakos Jun 11 '16 at 10:22
• @Κίουι I see your point. I have added some extra info about why I think this would be a good direction to head towards, which hopefully makes it more relevant to Quantes – oliversm Jun 11 '16 at 16:46
• @oliversm You said there are two roots of risk theory. Does the second root (stochastic calculus) contain enough statistics which will come up as i go along the way or i will just need also a graduate-level stats book such as 'casella-berger's Statistical Inference' . – Quantes Jun 11 '16 at 18:59
• @Quantes, In my experience stochastic calculus is a very different beast to regular statistical analysis. Think of statistics and answering the question of "ignoring everything I thought I know, what do these data tell me", whereas stochastic calculus takes the opposite approach, "ignoring all that these data have told me, how should the world behave". You need very little statistics (possibly none) to learn stochastic calculus. They are two quite different styles of analysing financial behaviour. – oliversm Jun 12 '16 at 0:50