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

July 12th, 2018, 3:57 pm

Chemistry only.
 
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Paul
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Re: Is the type of mathematics found in finance limited?

July 12th, 2018, 4:04 pm

Alan wrote:
Paul wrote:
Alan wrote:
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). 

I can see that might be the case, in an extremely deep, and yet strangely meaningless, way!
 
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katastrofa
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Re: Is the type of mathematics found in finance limited?

July 12th, 2018, 5:06 pm

Alan, could you possibly post a link to your book?
 
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Alan
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Re: Is the type of mathematics found in finance limited?

July 12th, 2018, 5:32 pm

katastrofa wrote:
Alan, could you possibly post a link to your book?

Delighted to -- thank you. Two amazon links:
amazon.co.uk
amazon.com

(Also, the financepress link at the bottom has a lot of supplementary material not found at amazons: full table of contents, codes, etc).
 
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katastrofa
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Re: Is the type of mathematics found in finance limited?

July 13th, 2018, 12:46 am

Thanks! I have little experience with math finance, especially compared to you or Paul, but tend to side with Paul on this one (everything revolves around the same type of an equation). Still, such finance books I could actually read :-)

Regarding your remark about biology and chemistry ("applying fancy techniques to the same boring equation (non-relativisitic Schrodinger equation)"), large parts of the disciplines require the Dirac equation, and when it becomes too complicated to solve, they experiment with other ways of modelling (e.g. statistical models/ML).
 
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Cuchulainn
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Re: Is the type of mathematics found in finance limited?

July 13th, 2018, 3:31 pm

katastrofa wrote:
Thanks! I have little experience with math finance, especially compared to you or Paul, but tend to side with Paul on this one (everything revolves around the same type of an equation). Still, such finance books I could actually read :-)

Regarding your remark about biology and chemistry ("applying fancy techniques to the same boring equation (non-relativisitic Schrodinger equation)"), large parts of the disciplines require the Dirac equation, and when it becomes too complicated to solve, they experiment with other ways of modelling (e.g. statistical models/ML).

You are probably speaking as a physicist, who are less concerned with (or maybe do not know) the mathematical niceties involved with PDEs. Alan's book is the most rigorous to date in the PDE literature galaxy. Most leave out the meat which leaves you high and dry when you try to implement it. Alan goes to the quick..
Anyone can write down a PDE but it's the detail and this is what Alan had done for his PDEs. For the record, I reviewed several of Alan's chapters in detail.
 
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Cuchulainn
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Re: Is the type of mathematics found in finance limited?

July 13th, 2018, 3:37 pm

Look at the average PDE finance article. All juicy stuff is missing

1. Boundary condition? use linear BC
2. Crank Nicolson ..
3. Skimpy algo details/implementation
4. Impossible to check validity of the results
5. Working code

Alan tackles these issues.

And good PDE models for fixed income with embedded optionality.
 
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katastrofa
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Re: Is the type of mathematics found in finance limited?

July 14th, 2018, 12:24 am

As I said, Alan's books are not bad.
 
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Paul
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Re: Is the type of mathematics found in finance limited?

July 14th, 2018, 4:41 am

Alan’s books are a definite exception, which rather illustrates my point.
 
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katastrofa
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Re: Is the type of mathematics found in finance limited?

July 14th, 2018, 7:13 am

BTW, physics is very broad field. My humble PhD thesis managed to encompass three fields of mathematics. Don't forget physics is also Navier-Stokes equation. (Regarding the mathematical niceties, the stochasticity of your equations is not a complication but a simplification.) In Eastern research institutions physicists don't habitually ignore the mathematical foundations of their models.
 
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Alan
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Re: Is the type of mathematics found in finance limited?

July 14th, 2018, 7:36 pm

Thanks for all kind comments about my books from everybody.

Speaking of mathematical biology, the great probabilist William Feller (source for Feller sqrt model, aka CIR, aka Heston vol model) was very interested in all that:

 Feller's Contributions to Mathematical Biology

So, this supports Paul's point of that field as a good source of inspiration for modelling. 
 
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Cuchulainn
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Re: Is the type of mathematics found in finance limited?

July 14th, 2018, 8:10 pm

I have a few beehives. The beekeeper told me how it works. Based on that optimisation algos were developed.

http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf

Bees don't get stuck in local minima (in contrast to gradient descent and people); they abandon depleted food sources (aka neighbourhood search).


// my analysis prof Brian Murdoch at Trinity was a PhD student of Feller at Princeton. 
 
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TinMan
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Re: Is the type of mathematics found in finance limited?

July 15th, 2018, 10:32 am

Any concrete suggestions for alternative maths or just grumbling?
By the way, nobody cares about derivatives pricing any more, another PDE for fixed income.... yawn!
 
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Cuchulainn
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Re: Is the type of mathematics found in finance limited?

July 15th, 2018, 10:59 am

TinMan wrote:
Any concrete suggestions for alternative maths or just grumbling?
By the way, nobody cares about derivatives pricing any more, another PDE for fixed income.... yawn!

ML afficiandos claim they can solve 100-dimensional PDEs before breakfast.

Somehow Lagrange, Hamilton, Bellman and Nash have been short-selled?
Last edited by Cuchulainn on July 15th, 2018, 11:00 am
 
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Paul
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Re: Is the type of mathematics found in finance limited?

July 15th, 2018, 10:59 am

TinMan wrote:
Any concrete suggestions for alternative maths or just grumbling?
By the way, nobody cares about derivatives pricing any more, another PDE for fixed income.... yawn!

At least if it were a different PDE...but it's the same one every time. (There is the nonlinear wave equation model...Epstein & Wilmott...that's more interesting!)
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