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Topic Title: Arbitrage and Holonomies
Created On Sun May 19, 02 01:46 AM
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IAmEric
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Thu May 23, 02 09:37 AM
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<< Have you found a geometric phase yet? >>



Of course! It is set to appear in Scientific Financial American,




Observations of Berry Phase in the SU(N) Gauge Theory of Merger Arbitrage

It has been observed that risk arbitageurs experience a pi phase shift upon completion of a successful merger.

"I don't know what happened. On the day the merger was scheduled to go through, I found myself showing up for work at 9pm!"

This widespread phenomenon has been attributed by researchers to SU(N) market holonomies.





Ugh... I think I need more sleep
 
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scholar
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Eric, I agree with your addition to my list. Also, thanks for the compliment

Barry phase rules !
 
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Onuk
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Eric >> Berry Phase

Eric, what is Berry phase? I'm now impatiently waiting to receive my "The Geometry of Physics: An Introduction", but can I have a sneak preview or is it too involved? I vaguely remember hearing it mentioned before, but never covered it.
 
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filthy
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berry phase refers to the phase a quantum system picks up when its parameters are cycled
and returned to its original state. ie the system "rembers" its history.

it could have been noticed in 1926 but somehow escaped discovery until 1984. it explains a lot
of "anomalies", among them the aharnov-bohm effect and the pauli exclusion priciple.
 
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scholar
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<< berry phase refers to the phase a quantum system picks up when its parameters are cycled
and returned to its original state. ie the system "rembers" its history.

it could have been noticed in 1926 but somehow escaped discovery until 1984. it explains a lot
of "anomalies", among them the aharnov-bohm effect and the pauli exclusion priciple.
>>



How does Berry phase explain the Pauli exclusion principle ?
 
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filthy
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Fri May 24, 02 10:58 PM
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Scholar,

you might want to look at this paper

Berry, M V and Robbins, J M, 1997, Proc.Roy.Soc.Lond A453 1771-1790 ?Indistinguishability for quantum particles: spin, statistics and the geometric phase?

available as a pdf file

http://www.phy.bris.ac.uk/research/theory/Berry/the_papers/Berry286.pdf

Not light reading...

F
 
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IAmEric
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This is a truly lovely little book and is fun to read:

FLATLAND: A Romance of Many Dimensions
by E. Abbott

The concept of holonomies is nicely illustrated by thinking about living in "Sphereland"

Read a bit of Flatland, and then start thinking about what happens if they lived on "Sphereland" instead.

Sphereland is like Flatland, except if you were to continue walking in any given direction in a "straight" line, you'd end up right back where you started from.

If you were to walk around a loop on Sphereland holding a spear in your hand trying desparately to keep the spear pointed in some fixed direction as you move around, i.e. as you turn left you compensate by rotating the spear to the right so it remains in the original direction, then when you return to your starting point, you will have found that the spear has actually rotated by a total amount proportional to the area of the loop you covered.

The exact amount is given by

delta(theta) = Area/a^2

where a is the radius of the sphere. For example, if you walk around a great circle, then you split the sphere into two equal halves of area 2*pi*a^2 each so that

delta(theta) = 2*pi

and the spear is not rotated at all when you return. If you trace out an octant of the sphere, i.e. a spherical triangle with each angle pi/2, then the area of the loop is 1/8*(4*pi*a^2) so that the amount rotated is

delta(theta) = pi/2.

Etc etc.

There is a nice java applet demonstrating this at:

Parallel Transport on a Sphere

Another thing to note is that in the limit that a -> infinity, then delta(theta) -> 0 for any loop of finite area.

In other words, in the limit that Sphereland becomes Flatland, then you will not observe any of this "rotation" business.

This means (among other things) that when you transport something around a closed loop, if it returns "rotated", then that means there was some "curvature" floating around somewhere.

The curvature is obtained from a connection. The connection is basically a rule that tells you how to "transport" stuff.

In the case of electromagnetic theory, there is a U(1)-connection. The group U(1) is just the group of elements of the form exp(i*t) for some real t, i.e. U(1) is the group of "phases". Hence, EM theory is invariant under changes of "phase", i.e. gauge transformations.

If you tilt your head sideways a bit and think really hard, you can picture the AB effect as transporting a SINGLE electron around a closed loop. When the electron gets back to its point of origin, it is "rotated." In this case the rotation group is U(1) so that a rotation means a "change of phase."

Now, like a said, if transporting around a closed loop results in a "rotation" that means there is some curvature somewhere. In the case of EM, the magnetic flux density B plays the role of the curvature. So if you turn off the B-field, you have no curvature -> no rotation. If you turn on the B-field you have curvature -> you have rotation = change in phase.

Now, in a feeble attempt to relate this to math finance, arbitrage is like transporting a portfolio around a closed loop (as in my very first post in this thread). So if there is no arbitrage opportunity (curvature), then no matter what you do, i.e. no matter how you carry your portfolio around in loops, you will never make any money (i.e. rotate your portfolio). So speculators and arbitaguers have the effect of eliminating curvature in the market

The analogies to geometry are pretty striking. I'm looking forward to learning more about it.

Cheers,
Eric

Edited: Sat May 25, 02 at 05:41 AM by IAmEric
 
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IAmEric
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WOOHOO!

My copy of Ilinski arrived (from inter-library loan) today

This idea of arbitrage and holonomies is precisely what Ilinski is talking about. Strange minds think alike

Has anybody here bothered forging their way further into this book? Any thoughts on non-equilibrium pricing in general? Seems pretty interesting.

Eric
 
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scholar
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Someone on this forum mentioned an interview with Ilinsky who admitted that he himself does not use his electrodynamical model in his present capacity as as front desk quant
 
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IAmEric
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Hi,

Well, that would make sense. The whole book is about "non-equilibrium" pricing. He mentions that most of the time, equilibrium models suffice. I guess when the market gets shaken up so bad that non-equilibrium effects become important, then the last thing people are going to be wanting to do is scramble for Ilinski's book. They'll probably have other things on their mind

I'm interested in the material enough to pursue it, at least part time. I'm still young (in finance years) and impressionable, so we'll see what comes of it. It's interesting to comtemplate the effect that a model has on the market. It would be difficult to convince me that Black-Scholes hasn't had a nonlinear impact on the very market that it is trying to model.

Eric

 
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filthy
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<< Someone on this forum mentioned an interview with Ilinsky who admitted that he himself does not use his electrodynamical model in his present capacity as as front desk quant <img src="i/expressions/face-icon-small-smile.gif" border="0"> >>



that may have been me you are referring to. i spoke with the person who hired
him. he said that they didn't really use the geometric stuff.

i am still interested to know if these methods have given us any new results.
at this point it seems like a classic case of someone coming to finance with a
bag of tools and being determined to use them, rather than finding a problem that really
needs solving and attacking it with the best methods.
 
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Omar
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Sat Jun 08, 02 10:26 AM
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"I spoke with the person who hired him. he said that they didn't really use the geometric stuff."

Were these his exact words? Did he say "We don't really use the geometric stuff", as if they used it a little bit?
Because, I would say, they don't use it at all.



Edited: Sat Jun 08, 02 at 10:29 AM by Omar
 
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filthy
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<<

Were these his exact words? Did he say "We don't <u>really</u> use the geometric stuff", as if they used it a little bit?
Because, I would say, they don't use it <u>at all</u>.
>>



no. those were not his exact words. i can't remember his exact words. the conversation
went something like...

Me: I've seen this Illinski stuff. On this point i agree with taleb. i think it is hogwash.

Him: really? he is a smart guy. we just hired him.

Me:To do that stuff!?

Him: No. To work as a quant.

None of these are quotes but i understood this to mean they don't use it at all.
 
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Omar
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Sounds more like it and I'm sure Illinski is a smart guy.
 
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