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Background

I had a very good and bright student with some decent exposure to markets and the standard college math curriculum who got overwhelmed during a mathematical finance course due to unnecessary off-topic formalisms. In particular, the curriculum of the class was a bit misleading because it promised to follow a fairly practical book (Tomas Bjork's Arbitrage Theory in Continuous Time) as its main text, while the actual class went far beyond that, largely ignoring the text. The course spent a considerable amount of time dealing with concepts such as Radon-Nikodym derivatives, Formal Measure Theory, Filtrations, Martingales (just to name a few) which - crucially - the book only mentions as an addendum in the appendix.

Of course, there is absolutely nothing wrong with any of the topics above. It is just that my student was expecting a largely more applied class in Finance, versus a purely theoretical one in Measure Theory and Martingales.

Questions

This leads me to some questions.

  1. What is the extent to which abstract topics in mathematical finance, such as Radon-Nikodym derivaties are actually relevant to markets in the real world? Do finance instructors spend any time in the markets or do they prefer to engage in a series of endless mental gymnastics often for their own sake? (Incidentally, I am not the first one asking these questions)

  2. Is this top-down approach and over-formalism necessary? Does one really need to know Radon-Nikodym derivatives (and all their intricate details) in an introductory Math of Finance class? Especially one without any prerequisites which uses a largely informal book as its main text and which is not designed for math majors?

EDITS

  • Post shortened, following the suggestions of @Noah Schweber (Thank you!)

  • There is also a link to a chat below (thank you @roddik!). Thank you all for being constructive and respectful.

Pellenthor
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    It is true, I think, that many texts (and, I suppose, lecturers) on the subject work at a level of generality which, while not incorrect, obscures the material. For a very advanced class, I think this can be justified...as the students ought to be able to supply their own examples and they'll need the technical details in order to move forward. But an introductory class? Well, I don't see the point. All that said, I don't think this is a good forum in which to criticize some specific professor or class. Let alone the concept of tenure and the structure of various degree programs. – lulu Oct 31 '20 at 20:32
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    In a class titled the Mathematics of Finance I would expect it to be mathematics, which is to say formal, proof based mathematics. I would certainly feel misled if it lacked rigor and abstraction. – CyclotomicField Oct 31 '20 at 20:35
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    About 8 years ago, I meant a Woman in a book store and we got to talking. She was getting a PhD in the Mathematics of Finance. She was hoping to invest money for other people. The subject of Exxon Mobil came up. She had not heard of the company. It seems to me that if we are teaching finance at the graduate level in the United States, students should know the name Exxon Mobil. I also found out that she knew nothing about debits and credits. She had no understanding of accounting. It seems to me she did not learn finance so I do not think it is right to use the word Finance in the title. – Bob Oct 31 '20 at 20:40
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    There are several questions here, not all of them appropriate for math.stackexchange, and the bulk of the question is a personal anecdote which could be drastically shortened (or even removed) without impacting the actual question. I think in its present form this question isn't appropriate for MSE (which isn't to say it's a bad question!). – Noah Schweber Oct 31 '20 at 20:42
  • @lulu Thank you for your comment. To be clear, I am not criticizing a "specific" professor. If I did, I would not maintain their anonymity. Their example is only given to provide context and raise a much bigger issue. – Pellenthor Oct 31 '20 at 20:43
  • @CyclotomicField Thank you. That makes sense and I totally agree. Would you also expect it to spend the bulk of its time on the appendix of the main textbook? – Pellenthor Oct 31 '20 at 20:46
  • Thank you for your comment @NoahSchweber, it is very helpful. I often tend to go overboard sometimes with details which is my fault. Would you care to elaborate on suggestions on which parts you would leave out? – Pellenthor Oct 31 '20 at 20:49
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    @Pellenthor First, as I said I think the lengthy anecdote can either be shortened to a single paragraph (along the lines of "I had a hardworking student with some exposure to markets and the standard college math curriculum who got lost in a mathematical finance course due to the abstract mathematics involved. This motivated me to ask:") or even removed entirely. As to the questions, only 1,2, and 6 seem appropriate for MSE, and while there's some overlap there really are at least two distinct questions there (relationship between abstract mathematical finance and the real world, and pedagogy) – Noah Schweber Oct 31 '20 at 20:55
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    which should be treated separately. I think a question along the lines of "What is the extent to which abstract topics in mathematical finance, such as [examples], are actually relevant to markets in the real world?" would be fine. – Noah Schweber Oct 31 '20 at 20:55
  • @Noah Schweber. This is very helpful thanks. I ll get to it. – Pellenthor Oct 31 '20 at 20:57
  • As for using the appendix of the main text I found that as I went into the higher level material the class notes became more important with the text being supplementary or referential. This is typical of the axiomatic approach as you want everyone to be proving and using the same theorems in a consistent fashion as the course progresses. Professors often have personal preferences that factor into the topics covered. – CyclotomicField Oct 31 '20 at 21:19
  • Thank you @CyclotomicField. I think my point is not yet clear. This is not about how people get to the right proofs, but rather whether they understand what is it they are proving. You are of course correct that a top-down approach like the one you are suggesting is more helpful towards the former but it completely lacks the motivation that the students need of what is it that they are doing, especially during an introductory course. Nobody learns how to cook by reading a cook-book. One can only learn how others learned how to cook by reading it. – Pellenthor Oct 31 '20 at 21:32
  • Are they PhD students ? –  Oct 31 '20 at 22:02
  • @YvesDaoust My student was not, although this was a graduate course. – Pellenthor Oct 31 '20 at 23:19
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    If it's a graduate course then they should have found the motivation during their time as an undergraduate. You shouldn't be covering rudimentary material in an advanced course. – CyclotomicField Nov 01 '20 at 02:18
  • @CyclotomicField What is rudimentary is largely relative. For math majors, sigma algebras, filtrations, RN derivatives are typically considered elementary (and yet he covered them all). This is still beyond the point. The issue is that the class headed in a completely different direction than advertised leading to confusion. If we knew the instructor would spend half the class going over Martingales I could have simply advised my student to take a class in Measure theory first to prepare. Regardless, I still think MT is unnecessary and so does Bjork. – Pellenthor Nov 01 '20 at 02:49
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    Radon-nikodym derivatives and related tools have applications in options pricing. The demand for options pricing experts has fallen after the financial crisis as banks have been shifting towards simpler products. A criticism I have heard is that the standard Math Finance curriculum is still too focused on areas primarily relevant for options pricing, despite this reduced demand. – fes Nov 01 '20 at 12:24
  • This is very helpful @fesman. Other than their use in justification of the (generalized) definition of conditional probabilities (which one can reasonably circumvent by understanding the discrete case well), I have never seem RN derivatives been applied on anything else non-theoretical. Can you give me an example of such real world application? – Pellenthor Nov 01 '20 at 16:28
  • Options and other derivatives are usually priced under the risk neutral measure defined via Girsanov theorem, which uses the RN derivative. Derivatives pricing modellers working for banks will likely be somewhat familiar with RN derivatives. But I believe this area has lost some importance. – fes Nov 01 '20 at 17:18
  • Thank you @fesman. You are correct. The only this is, Girsanov's theorem falls under what I would call a theoretical result. For example, it can be formally used to prove the Black-Scholes Model (BSM). However, it is also not "necessary", as one can give a pretty solid justification for the BSM (and the risk free measure) - using simple hedging ideas - without mentioning RN derivatives at all (see chapter 7 in Bjork). – Pellenthor Nov 01 '20 at 20:00
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    There are many tools related to math finance you can learn at an intuitive level, for example Ito calculus. Most of the time it works fine but then you might run into some models like CEV pricing model where Ito integrals have strange properties and here some more formal understanding of the math is useful. But I also agree that the average student would be better off focusing on more applied topics and this intuitive and practical understanding might actually be more important. – fes Nov 01 '20 at 20:56
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    This is a few days old question, but let me provide my own perspective as a lecturer at a university. When I teach graduate level classes I focus a lot on the technical aspects of the material. I might spend a little bit time motivating what we do here and there and connect it back to real world example, but that is not the focus. The reason for it is that it is very hard to acquire knowledge of technical aspects by self-studying. On the other hand, once one masters the technical aspects it is much easier to apply them. I mention some applications but often will not go into details. – Mdoc Nov 04 '20 at 02:21
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    If I had a lot of time and energy, I would do both. I would cover technical aspects, discuss why we need the abstraction, and cover applications. But the time available for teaching and my energy are finite. Thus, the hope is that the students are mature enough to be able to bridge the gap between theoretical work and applications along the dimensions I would discuss in the class. – Mdoc Nov 04 '20 at 02:24
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    @Mdoc. Great comments. The question then becomes: without applications where is the "finance" part in all of this? Why can't we simply call these something more appropriate (and frankly less unethical) like Stochastic Calculus. By analogy, should we start calling complex analysis, the mathematics of Quantum Mechanics? The issue is that not all people interested in the subject are math majors, yet they are understandably allured by the title. This is why I am raising the question of accessibility. Are these "intro" courses only truly available to people with hard-core math backgrounds? – Pellenthor Nov 04 '20 at 20:17
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    I have two commentaries. Someone might have already pointed out these.
    1. Mathematics is made by mathematicians. No one else will ever find natural what a mathematician does.
    2. When the average mathematician learns a tool, he uses it. My algebra professor would refer to this with the term "shooting a fly with a cannon" - don't know if there's a proper English equivalent for this. This is because unless you really master a subject, simplifying things is actually more difficult than to apply them. If everyone was Poincaré, I think we'd have much simpler courses.
     – rod Nov 04 '20 at 20:43
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    It seems to me that many professors just lose the bigger picture. This happens not only in interdisciplinary courses. It even happens in pure maths courses. Where the teacher will simply expose the theory, without actually ever explaining it. It look to me like there are two radically different ways of teaching. Even if you look afraid to say so, I'm not: one of them is stupid to me. – rod Nov 04 '20 at 20:48
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    This is a brilliant observation @roddik thanks! Especially the comment about "overkill" (Incidentally I believe an equivalent expression is "bringing a gun to knife fight"). The irony is (as I explained in the other version) that my student performed really well when we approached the concepts with the "big picture" in mind. He understood hedging, arbitrage, and even the more complex risk-free measures, SDEs and the BSM model. But then, how do I teach him all the details of Measure Theory, RN derivatives, Martingales all in a couple of weeks so he can catch up with the HW? – Pellenthor Nov 04 '20 at 22:15
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    Well, I think this is simply not possible. Although many courses pretend it to be. You can even get away with sigma algebras, martingales, filtrations, RN derivatives in relatively a short time. However, definitions and theorems little have to do with comprehension. Speaking for me, I need some "orienteering" about what I'm studying. In two ways: there has to be some purpose (i.e. topology should not be explained just as a piece of set theory) and some technical basis (Ito's calculus must follow measure theory and basic probability). – rod Nov 05 '20 at 00:16
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    The reason why some people remain blind to the problem you're arising is still a total mistery to me. While a tiny fraction of them might be geniuses with alien brains, I came to the conclusion that most of them are either stupid or careless.

    I had asked a different but quite correlated question, here on Stackexchange. Basically, I was taught tensor product via its algebraic construction ("gargantuous quotient"+universal property) three times. But I only started to grasp something when I approached differential geometry and Realtivity. And those definitions were pretty useless to the scope.

     – rod Nov 05 '20 at 00:21
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    Well said! GR is the way for tensors. Definitely not geniuses btw. They may occasionally fool students, but not fellow academics. Look at Terry Tao for example: he is one of the top mathematicians alive today and yet one of the most comprehensive scientists I have met. This is not a coincidence. True knowledge generates clarity. We run into trouble when we step into areas where we don't belong. We are academics and thus - for the most part - theorists. We have no time to be anything else and there's nothing wrong with that. The issue starts when we pretend we are more than that. – Pellenthor Nov 05 '20 at 00:41
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    Let us continue this discussion in chat. – rod Nov 05 '20 at 11:28
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    Sorry for bringing this back from the grave, but, as a mathematical finance student myself, I'd be interested to know if you arrived at a nice conclusion to this question. – Jose Avilez Aug 11 '21 at 15:47
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    No worries. I guess it would depend on what you would consider a "conclusion". I think the lack of a response so far, at the very least indicates the existence of an issue. I feel that as academic educators we tend to fail our students time and again, by being fairly detached from reality, while operating within the confined, self-serving and consequence-free space of tenure. This is inevitable within a system that does not allow to correct for "failure". That is particularly evident in finance related domains, where facing the negative consequences of mistakes is necessary for progress. – Pellenthor Aug 12 '21 at 05:11
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    @Pellenthor I feel that your question, although interesting, does not really belong in MSE. The site https://quant.stackexchange.com is dedicated to this kind of topics and you would probably get a better answer over there. – PatrickR Feb 15 '22 at 08:38
  • @ PatrickR Thank you for you comment. – Pellenthor Feb 15 '22 at 20:01

0 Answers0