I have a bit of a conundrum on my hands, and I'm hoping that you can shed some light on it. I'm currently looking at PhD programs in mathematical finance & sorting out GREs. Most programs require the general GRE or GMAT, but a few also require a math subject GRE. My math background is fairly strong (completing econ & math undergrad, econ masters), except for a notable lack of abstract algebra. So, I have two basic options: work though abstract algebra on my own & take the subject GRE, or devote more time to research & take this requirement as a signal that those programs aren't my best option.A couple of comments on my thinking so far:First, the opportunity cost associated with devoting time to abstract algebra is quite high, so I feel like it would need to have value beyond the GRE for that option to make sense. Second, none of the programs that I'm looking at require (or recommend) further study in abstract algebra, so I'm quite uncertain about its value.Here's my question: Is abstract algebra (group theory, rings, modules, etc.) useful for mathematical finance? If so, how? Do you think it would be better just to take this requirement as a signal?Thanks.

a) all entrance requirements are signalsb) rings and fields do come up A LOT in the study of stochastic calculus as filtrations and various probability structures have rings and field structures. when you take 1st year stoch calc and advance prob, you will become intimately familiar with these structures.

QuoteOriginally posted by: wolf87I have a bit of a conundrum on my hands, and I'm hoping that you can shed some light on it. I'm currently looking at PhD programs in Here's my question: Is abstract algebra (group theory, rings, modules, etc.) useful for mathematical finance? If so, how? Do you think it would be better just to take this requirement as a signal?Thanks.These subjects will be ridiculously useful. The entire field of all of mathematics, not just mathematical finance is based on these structures. They are how you build probability, and neat things like that. Like Kack said, it will probably take a year of fairly intense study (depending on how smart you are) but that seems to be the consensus. I self-taught and my book on measure theory/integration said that it would take a full year for the first 6 chapters or so.. I crammed it in about 6 months ( the parts I need), and I still use it as a reference on a daily basis and probabily will so for many years to come. sigma-algebras are the most useful in finance and a lot more.

Interesting. Thanks for the info. I will start on algebra as soon as comps are over, although I think I will also focus less on those programs that look for it. It seems to be a signal that they want more of a pure math background than I have, and, given how interdisciplinary mathematical finance research is, I think that I should have a chance to pursue similar research & coursework in other departments in the same universities.

wolf,It is true that it is good to know the basic structures in abstract algebra, i.e. know what a group is, a group action, cosets, rings, fields and some examples of these structures. This can be picked up by reading the first few chapters of an introductory book on the subject.Most of what comes after that is mainly mental masturbation. There are some applications (notably in cryptography) but they are few and far between. I say don't waste your time -- unless it really fascinates you.

QuoteOriginally posted by: MrMartingalewolf,It is true that it is good to know the basic structures in abstract algebra, i.e. know what a group is, a group action, cosets, rings, fields and some examples of these structures. This can be picked up by reading the first few chapters of an introductory book on the subject.Most of what comes after that is mainly mental masturbation. There are some applications (notably in cryptography) but they are few and far between. I say don't waste your time -- unless it really fascinates you.Oh? "abstract data types" are kind of "universal algebras", "free algebras" are kind of "interfaces" (describe structure, without "implementation"). The vision of "generic programming" is a kind of "functor programming" (operations on structure-keeping functions?). mental m.?Derivatives have unnamed structures (you can think of bases, linear combinations,...). Not really elaborated, but a way of thinking?I am not saying, you need a deep knowledge in "near rings", ...

QuoteOriginally posted by: exneratunriskQuoteOriginally posted by: MrMartingalewolf,It is true that it is good to know the basic structures in abstract algebra, i.e. know what a group is, a group action, cosets, rings, fields and some examples of these structures. This can be picked up by reading the first few chapters of an introductory book on the subject.Most of what comes after that is mainly mental masturbation. There are some applications (notably in cryptography) but they are few and far between. I say don't waste your time -- unless it really fascinates you.Oh? "abstract data types" are kind of "universal algebras", "free algebras" are kind of "interfaces" (describe structure, without "implementation"). The vision of "generic programming" is a kind of "functor programming" (operations on structure-keeping functions?). mental m.?Derivatives have unnamed structures (you can think of bases, linear combinations,...). Not really elaborated, but a way of thinking?I am not saying, you need a deep knowledge in "near rings", ...Indeed, it is a universal language, I'm just at some "transition point" where the algebra is starting to flow together easily and it scares me really, being able transition between all sorts of spaces and mappings, knowing all the properties of all of these things by the efficient(logically optimal?) use of notation and language...I like almost periodic rings myself. It is only mental masturbation for those who have different measures of the value of abstract algebra. As far as elaboration, that is much harder than the abstract manipulation of symbols (which is easy to do and make mistakes if you arent careful). But doing the transition from abstract to advanced calculus it becomes trivial to get down to actually solving things concretely(i mean it just becomes gruntwork because all of the abstract algebra provides a map to "fill in"), because the "structure" makes the 'code' just sort of automatic and natural. At least, in this "universal algebra" wherever and whenever it exists.

The sigma-algebras and their lot have nothing to do with abstract algebra at all, only the word It's the theory of integration (real analysis) that should be mastered were you to study stochastic calculus, but fields, groups, ... are absolutely unnecesary (or probably arise in some trivial situations I can't even think of). However it could be that I don't understand what is hidden under the "abstact algebra" label (which is highly unlikely). Some examples of abstract algebra in financial math would be very interesting for me :-/ . That is not to say that it is somehow "bad" - it's fairly interesting if you find it interesting (and many people do).

QuoteOriginally posted by: wolf87I'm currently looking at PhD programs in mathematical finance & sorting out GREs. Most programs require the gene...Question: Do you have any real experience in finance and banking? If not, and you are considering a PhD program, then I watch out. Don't make the mistake of believing that a PhD is going to land you a great job.Also, a PhD in mathematical finance is extremely narrow. I would not do it, unless you already had experience and were established with a bank and they wanted you to do it.Do not forget that schools are businesses, and their programs are a product they sell. Their "advisors" are a kind of salesman for the school. YOU have to figure out their real value.

amiraliev, I believe you do not know what is hidden. You are correct, real(and more importantly complex) analysis are what is really needed to study stochastic processes. But once that is done, it turns out abstract algebra allows you to use the real and complex numbers as a 'base' for all sorts of extremely fascinating, deep and useful, results. (Integration on arbitrary seperable banach lattices)

Another question: Why are you considering a PhD program? Why aren't you, well, working?

QuoteOriginally posted by: amiralievThe sigma-algebras and their lot have nothing to do with abstract algebra at all, only the word It's the theory of integration (real analysis) that should be mastered were you to study stochastic calculus, but fields, groups, ... are absolutely unnecesary (or probably arise in some trivial situations I can't even think of). However it could be that I don't understand what is hidden under the "abstact algebra" label (which is highly unlikely). Some examples of abstract algebra in financial math would be very interesting for me :-/ . That is not to say that it is somehow "bad" - it's fairly interesting if you find it interesting (and many people do).amiraliev,It's refreshing to hear someone who knows what they are talking about. Stochastic calculus, by it's very definition, has no interesting structure. This reminds me of when a physics prof from Cornell associated Yang-Mills theory (a non-abelian gauge theory) with quantum mechanics. That idiot had no idea that Yang-Mills is a 'deterministic' gauge theory and there is nothing in common between it and QM (QM is obviously probablistic, a stochastic flow).Now, if you want to make money, skip stochastic calculus and go for abstract algebra (although I must admit the axiomatic approach in most texts give zero insight into utility).N

QuoteOriginally posted by: NNow, if you want to make money, skip stochastic calculus and go for abstract algebra (although I must admit the axiomatic approach in most texts give zero insight into utility).NWhat books or papers take the approach of focusing on utility, not the axiomatic approach?

QuoteOriginally posted by: CactusManQuoteOriginally posted by: NNow, if you want to make money, skip stochastic calculus and go for abstract algebra (although I must admit the axiomatic approach in most texts give zero insight into utility).NWhat books or papers take the approach of focusing on utility, not the axiomatic approach?You may want to start with group theory. A good text on the subject is 'Group Theory and its Applications to Physical Problems', by Hamermesh.

OK, I just ordered it.