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Cuchulainn
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Re: Is the type of mathematics found in finance limited?

July 11th, 2018, 6:59 pm

neauveq wrote:
The question I have is this: Having only possess a very limited background in maths, should I then strive to learn the 'limited' maths of finance well (which is no easy feat in itself) or should one learn as much maths as one can outside of the 'limited' scope of finance and continuously think of applications to existing problems in finance? I am struggling to see the trade-off here. Will more maths outside of the 'limited' scope be helpful? Will more time spent on 'other' maths help me understand the current maths of finance better? I guess I am still very early in my maths journey. I am sure for other mathematicians out there the answer seems clear and self-obvious.

And the answer is
When a student comes and asks, "Should I become a mathematician?" the answer should be no. If you have to ask, you shouldn't even ask.
Paul,Halmos
Amen.
 
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Alan
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Re: Is the type of mathematics found in finance limited?

July 11th, 2018, 7:19 pm

Paul wrote:
The questions means at least two things: Do we only see the same maths over and over again? If so, why?

Thoughts?

There is a fair amount of interesting math (at least to me) in my second volatility book. Some examples:

1. I took one course in general relativity as a senior at Caltech. But, I think I learned more Riemannian geometry in doing my chapter on "Advanced smile asymptotics" then I ever learned in that course. If non-linearity is your criterion for "interesting", you get that in the interplay between the eikonal and geodesic equations. 

2. I have a chapter on "Spectral theory for jump-diffusions". This is a relatively undeveloped area -- as the finance operators are typically non-self-adjoint. (99% of textbook stuff is for self-adjoint operators). I came to appreciate some parts of functional analysis much more in writing this chapter.

3. Continuous-time inference for diffusions yields some interesting puzzles, esp. when boundary behavior is involved;  for example, with slowly-reflecting diffusions or even just the square-root volatility process. I came to appreciate Girsanov theory more from writing that one. As part of that chapter, you may remember this nice thread (although many of the equations seem to be lost).
 
 
Last edited by Alan on July 11th, 2018, 7:52 pm
 
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neauveq
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Re: Is the type of mathematics found in finance limited?

July 11th, 2018, 7:31 pm

Cuchulainn wrote:
neauveq wrote:
The question I have is this: Having only possess a very limited background in maths, should I then strive to learn the 'limited' maths of finance well (which is no easy feat in itself) or should one learn as much maths as one can outside of the 'limited' scope of finance and continuously think of applications to existing problems in finance? I am struggling to see the trade-off here. Will more maths outside of the 'limited' scope be helpful? Will more time spent on 'other' maths help me understand the current maths of finance better? I guess I am still very early in my maths journey. I am sure for other mathematicians out there the answer seems clear and self-obvious.

And the answer is
When a student comes and asks, "Should I become a mathematician?" the answer should be no. If you have to ask, you shouldn't even ask.
Paul,Halmos
Amen.

You've made a very fair point. Am I right in thinking that this quote advocates the idea of 'maths for maths' sake'? As I understand this is a very pure approach to mathematics. Do pure mathematicians make better quants? I realise I can't add anything useful to this discussion and will raise more questions that can possibly be answered but thank you for your response, I am trying to learn from the more experienced members of this forum!
 
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Paul
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Re: Is the type of mathematics found in finance limited?

July 11th, 2018, 10:25 pm

Alan wrote:
Paul wrote:
The questions means at least two things: Do we only see the same maths over and over again? If so, why?

Thoughts?

There is a fair amount of interesting math (at least to me) in my second volatility book. Some examples:

1. I took one course in general relativity as a senior at Caltech. But, I think I learned more Riemannian geometry in doing my chapter on "Advanced smile asymptotics" then I ever learned in that course. If non-linearity is your criterion for "interesting", you get that in the interplay between the eikonal and geodesic equations. 

2. I have a chapter on "Spectral theory for jump-diffusions". This is a relatively undeveloped area -- as the finance operators are typically non-self-adjoint. (99% of textbook stuff is for self-adjoint operators). I came to appreciate some parts of functional analysis much more in writing this chapter.

3. Continuous-time inference for diffusions yields some interesting puzzles, esp. when boundary behavior is involved;  for example, with slowly-reflecting diffusions or even just the square-root volatility process. I came to appreciate Girsanov theory more from writing that one. As part of that chapter, you may remember this nice thread (although many of the equations seem to be lost).
 
 

For a moment I thought these were good examples. Who doesn’t like an eikonal equation?! But really these just come from applying different techniques to the usual type of governing equation. I don’t think fancy techniques applied to boring equations quite satisfies me, I’m afraid!
 
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katastrofa
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Re: Is the type of mathematics found in finance limited?

July 12th, 2018, 2:03 am

Paul wrote:
The questions means at least two things: Do we only see the same maths over and over again? If so, why?

Autism?
 
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katastrofa
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Re: Is the type of mathematics found in finance limited?

July 12th, 2018, 2:10 am

Gamal wrote:
Paul, so called Mathematical Biology was my subject before switching to finance, namely Kimura model for evolution. It's some time back but still remember issues, people and biologists' comments.

Unlike financial mathematics, mathematical biology has been progressing since the Kimura model, and at a fast pace.

Gamal wrote:
Paul wrote:
But the same is true of any field trying to apply maths, including finance. 99% of research is not used. 

In biology it's 100%. Mathematicians and biologists don't understand each other, so they don't talk and even if they are forced to talk, they don't listen to the counterparty.

Today biologists and mathematicians organise conferences together, e.g. http://lcfi.ac.uk/news-events/events/va ... onference/ (there's also a lot of computer science guys, but they try to squeeze in their AI into everything today).


Paul wrote:
Mathematics is like poetry, everyone hates poetry, everyone hates mathematics and mathematicians. In finance mathematicians are a bit useful, so they can't just so get rid of them. In biology maths is so far useless.

Mathematics has been present in biology since 1964, when WD Hamilton published his mathematically rigorous evolution model. Biologists fell in love with it. Was your own PhD work do useless?
 
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Gamal
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Re: Is the type of mathematics found in finance limited?

July 12th, 2018, 5:08 am

Hamilton model is a toy model.
 
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Paul
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Re: Is the type of mathematics found in finance limited?

July 12th, 2018, 5:58 am

katastrofa wrote:
Paul wrote:
The questions means at least two things: Do we only see the same maths over and over again? If so, why?

Autism?

On this forum this is known as the "arctan problem."
 
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Cuchulainn
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Re: Is the type of mathematics found in finance limited?

July 12th, 2018, 10:44 am

Can anyone tell me the mathematics of AI.
 
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Paul
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Re: Is the type of mathematics found in finance limited?

July 12th, 2018, 10:46 am

Curve fitting, optimization,...When I lecture on it I do have a moan about the lack of 'modelling.'
 
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Cuchulainn
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Re: Is the type of mathematics found in finance limited?

July 12th, 2018, 11:22 am

Paul wrote:
Curve fitting, optimization,...When I lecture on it I do have a moan about the lack of 'modelling.'

Indeed, it's kind of soul-destroying. The model is the 'problem' while the optimisation is the 'solution'. 
There seems to be an unwilligness(paralysis?) to experiment with different solutions.Principle of least action, in action?
Last edited by Cuchulainn on July 12th, 2018, 11:26 am
 
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Paul
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Re: Is the type of mathematics found in finance limited?

July 12th, 2018, 11:25 am

Cuchulainn wrote:
Indeed, it's kind of soil-destroying. 

Soil has very interesting mathematical properties.

Cuchulainn wrote:
Paul wrote:
Curve fitting, optimization,...When I lecture on it I do have a moan about the lack of 'modelling.'


Indeed, it's kind of soil-destroying. The model is the 'problem' while the optimisation is the 'solution'. 
There seems to be an unwilligness(paralysis?) to experiment with different solutions.

But it is fun! (Until you get run over by a driverless car, at least.)
 
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Cuchulainn
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Re: Is the type of mathematics found in finance limited?

July 12th, 2018, 11:30 am

The nice thing about software systems is that bugs bet fixed in  next release e.g.

5.0 major release, more functionality + more bugs
5.1 major bug fixes
..
5.5 small bug fixes

6.0 major release, even more functionality + more bugs

//
One example that intrigues me is when the big three (Google etc.)  came to a y junction in the road  they chose for discrete dynamical systems (grosso modo, GD) and not continuous ones (ODEs).
 
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Cuchulainn
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Re: Is the type of mathematics found in finance limited?

July 12th, 2018, 1:55 pm

  I came to appreciate some parts of functional analysis much more in writing this chapter.

Functional Analysis is so important as foundation for lots of things. Without FA it is very difficult to get a grasp of FEM, PDE, optimisation.

I did a lot of (theoretical+ applied) FA as undergrad. It was great. Teaches you think abstractly and and analytically.
 
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Alan
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Re: Is the type of mathematics found in finance limited?

July 12th, 2018, 3:56 pm

Paul wrote:
Alan wrote:
Paul wrote:
The questions means at least two things: Do we only see the same maths over and over again? If so, why?

Thoughts?

There is a fair amount of interesting math (at least to me) in my second volatility book. Some examples:

1. I took one course in general relativity as a senior at Caltech. But, I think I learned more Riemannian geometry in doing my chapter on "Advanced smile asymptotics" then I ever learned in that course. If non-linearity is your criterion for "interesting", you get that in the interplay between the eikonal and geodesic equations. 

2. I have a chapter on "Spectral theory for jump-diffusions". This is a relatively undeveloped area -- as the finance operators are typically non-self-adjoint. (99% of textbook stuff is for self-adjoint operators). I came to appreciate some parts of functional analysis much more in writing this chapter.

3. Continuous-time inference for diffusions yields some interesting puzzles, esp. when boundary behavior is involved;  for example, with slowly-reflecting diffusions or even just the square-root volatility process. I came to appreciate Girsanov theory more from writing that one. As part of that chapter, you may remember this nice thread (although many of the equations seem to be lost).
 
 

For a moment I thought these were good examples. Who doesn’t like an eikonal equation?! But really these just come from applying different techniques to the usual type of governing equation. I don’t think fancy techniques applied to boring equations quite satisfies me, I’m afraid!

Well, OK. By the same rationale, I guess one could say all of biology and chemistry is simply applying fancy techniques to the same boring equation (non-relativisitic Schrodinger equation). 
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