thesis topic

[yetanotherquant]

Hi, I am currently doing my masters in financial mathematics but would like to do my thesis in applications of pure mathematics to finance. In other words, I want to apply either algebra or number theory to finance

Though I cannot give you a concrete advice, this book might be a starting point for you. However, I find your idea generally suboptimal, since (almost) noone wants (over)complicated models in practice.
BTW, I like mathematics, in particular I have written this tutorial on measure theory.

 

[Daniel Duffy]

I want to apply either algebra or number theory to finance that is during my thesis or on my PhD that I am about to embark on.

Well, pure maths(numbers, algebra, measure) is not so useful in computational finance IMO.

It's a non-starter.

 

[Daniel Duffy]

Though I cannot give you a concrete advice, this book might be a starting point for you. However, I find your idea generally suboptimal, since (almost) noone wants (over)complicated models in practice.
BTW, I like mathematics, in particular I have written this tutorial on measure theory.

Out of curiosity, do you have a degree in (pure) maths?

 

[yetanotherquant]

Well, pure maths(numbers, algebra, measure) is not so useful in computational finance IMO.
It's a non-starter.

Number Theory. Well, if one can create a better pseudo-RNG then it will be useful, and not only in computational finance (please do not interpret this statement as recommendation to try )

Algebra. Little can I say. Pure algebra likely no, algebraic methods in functional analysis may be yes, at least indirectly.

Measure.
How about the switch to T-Forward measures? Not only it allows to justify LIBOR model but also greatly reduces computational complexity (after this switch one needs to calculate an expectation of a random variable, not the expectation of the quotient of random variables).

 

[yetanotherquant]

Out of curiosity, do you have a degree in (pure) maths?

Short answer, no.

A longer answer - in Germany it is (was) not possible to make Diplom/Master in pure math, you also needed Nebenfach (minor). In this sense, I learnt not less (and actually more) than a typical German math graduate. I do not do it anymore but I dwelt pretty deeply in measure theory and in particular in Gaussian measures on L2. I did not apply it in practice but Carmona and Tehranchi have written a very interesting book on it in context of term structure modeling.

 

[Daniel Duffy]

So how is it possible to understand measure theory.

Personally, MT was not my favorite but I did have 4 years of it + applications from a prof who was one of William Feller's PhD students at Princeton (Brian Murdoch). No rooky.

That aside, computational finance needs applied/numerical maths _much_ more than pure maths.

MT plays a cameo role.

 

[Daniel Duffy]

The problem with measure theory is that is effectively non-constructive and has few applications. One exception is using the Radon Nikodym theorem, MLE and parameter estimation (good candidate for a MSc degree IMO).

http://www.wilmott.com/messageview.cfm?catid=8&threadid=97764

 

[yetanotherquant]

measure theory is that is effectively non-constructive

Non-constructive?
Yes, that's how it is taught, not how it actually is.
An axiomatic approach (one begins with Kolmogorov's triple ($\Omega, \mathcal{F}, \mathbb{P}$)) is succinct. But succinct != clear.
Some lectures start with Caratheodory construction (NB! construction!), which is still not intuitive. That's why I in my tutorial reproduce the original Lebesgue's way, starting with a semiring on (0,1). It is intuitive, constructive and can be even graphically sketched to some extent.

and has few applications.

What has significantly more applications in financial engineering (except classical subjects like calculus, linear algebra and (elementary) probability/statistics)?

One exception is using the Radon Nikodym theorem

[/QUOTE]
IMO it alone is worthy enough to learn it!

 

[Eric Lim]

Hello, I'm looking into Andy's first project "Dynamic replication of an equity option in the binomial model". If anyone had experience or reference paper on this topic, please let me know! Thank you!!

 

[Daniel Duffy]

Non-constructive?
Yes, that's how it is taught, not how it actually is.
An axiomatic approach (one begins with Kolmogorov's triple ($\Omega, \mathcal{F}, \mathbb{P}$)) is succinct. But succinct != clear.
Some lectures start with Caratheodory construction (NB! construction!), which is still not intuitive. That's why I in my tutorial reproduce the original Lebesgue's way, starting with a semiring on (0,1). It is intuitive, constructive and can be even graphically sketched to some extent.


What has significantly more applications in financial engineering (except classical subjects like calculus, linear algebra and (elementary) probability/statistics)?

IMO it alone is worthy enough to learn it! [/QUOTE]

Not sure is we are on the same page, but I meant this
http://en.wikipedia.org/wiki/Constructivism_(mathematics)

 

[Daniel Duffy]

Quote:
Classical measure theory is fundamentally non-constructive, since the classical definition of Lebesgue measure does not describe any way to compute the measure of a set or the integral of a function. In fact, if one thinks of a function just as a rule which "inputs a real number and outputs a real number" then there cannot be any algorithm to compute the integral of a function, since any algorithm would only be able to call finitely many values of the function at a time, and finitely many values are not enough to compute the integral to any nontrivial accuracy. The solution to this conundrum, carried out first in Bishop's 1967 book, is to consider only functions which are written as the pointwise limit of continuous functions (with known modulus of continuity), with information about the rate convergence. An advantage of constructivizing measure theory is that if one can prove that a set is constructively of full measure, then there is an algorithm for finding a point in that set (again see Bishop's book). For example, this approach can be used to construct a real number which is normal to every base.

 

[Daniel Duffy]

http://www.wilmott.com/messageview.cfm?catid=8&threadid=95199&FTVAR_MSGDBTABLE=

 

[vertigo]

it would be a downright insult to measure theory and (specifically) that of the lebesgue integral if one were to criticise it for not being constructive. that is not really the point of it. rather, it is the properties of the lebesgue integral and various rules on what we can/cannot do with the integral -> dominated convergence, fatou, product spaces, etc, that make it so powerful. no longer does one blindly switch sum/integral and so on. one will always look to discretise a lebesgue integral via riemann sums, just as that of the ito integral or any other integral.. we accept that computer implementations are only finite and deal best with riemann sums. that is our toolkit - we must work with it.

but looking at measure theory and saying 'well it's not constructive, it's a problem because. XYZ..' is the wrong attitude to have -> why are you trying to solve XYZ using measure theory?

 

[Mcdonald]

I'll send you them on mail.

Can i have one too, currently at the thesis stage of my masters in financial engineering