Lie Groups and underlying trading

[ASusilovic]

Quote from nitro:

On the contrary. This tries to have a higher resolution of cross risk accuracy. And this has been said a million times, at LTCM it was not the models that went bad, it was the immense leverage that did not allow them time to recover from a massively unexpected standard deviation event. The people that bailed them out made out like bandits when mean reversion of their positions occurred. The leverage used was not the fault of the model, but the emotion of the managers when they thought one hundred year floods came on average every one hundred years.

You guys continue to harp on one instance where a quantitative approach went bad, but ignore the hundreds if not thousands where money is made literally on a daily basis with scalpel-like precision using these quantitative tools.

It gets really monotonous and tiring actually.

Nitro,

as monotonous and tiring as LTCM went down, Amaranth Advisors, Nick Leeson, Jerome Kerviel and others !

You forget, IMHO, one very important factor in your all too "logic" approach :

manipulation and sabotage.

Parties I mentioned above went into trouble because counterparties have been very well informed about their positions...

Can you model manipulation and sabotage ? No !

 

[sjfan]

And... that's why you need to have counterparty risk groups and operational risk groups. What's up with ET's cultish idea that all strategies that have a quantitative component MUST encapsulate everything? Yet at the same time the so called mechanical systems that it touts are so completely simple minded. Amateur hour, yet again.

Quote from ASusilovic:


Can you model manipulation and sabotage ? No !

 

[rluser]

Dare I point out that Euclidean geometry represents a Lie group?

Is there a point here? p-adic numbering system seems a non-starter, but perhaps there exists some other Lie group more useful. What is the perceived problem with present representations? Is there a quick transform to get all my present useful tools into the new space?

 

[heech]

I thought the foundation of modern stochastic finance assumed continuous process, rather than discreet/jump models. This is what allows us to use Ito's.

 

[nutmeg]

So if we take this analogy to managing book risk with many different types of underlying (different "forces" affecting our risk), we see that in order for them to fit together under one big symmetry group, we need the equivalent "book risk" Lie group that unifies the forces affecting risk in that book. That is the basic motivation for this idea.
--------------------------------

This analogy brings to mind an automobile. All of the different parts (underlying/forces) a finished product presents one symmetry group. If you propose to unify this force which is a completed automobile working under normal conditions, one still has to deal with exterior uncontrollable forces anyways. (The driver, road conditions etc). How are you any further ahead?

Ps. I am out of my league and may have missed the point but your post is intriguing.

 

[sjfan]

.... that's why they call that branch continuous time finance. Ito is only applied in a fair small part of it - just a useful little technical lemma

Quote from heech:

I thought the foundation of modern stochastic finance assumed continuous process, rather than discreet/jump models. This is what allows us to use Ito's.

 

[nitro]

Quote from nitro:
A Lie group is a mathematical object of tremendous importance to modern mathematics.

http://en.wikipedia.org/wiki/Lie_group

I have not seen a lot of applications of them to finance though. Perhaps it is because Lie groups deal with continuous processes, as opposed to e.g. stock prices that are discreet and contain jumps.

Lie groups can be continuous and have discreet solutions. In fact, this is why Heisenberg originally thought that his matrix mechanics was superior to Schrodinger wave mechanics for explaining the spectrum of atoms, which are by quantum mechanics, discreet. Today we know more about the [Lie] group theory of differential equations.

But there is a way to salvage this. If instead we deal with probabilities in the same way that Quantum Mechanics treats position and momentum as probability [wave] functions, continuity is reintroduced and Lie theory can be brought to bear. Therefore, it seems that even for the underlying trader, a possibly more coherent place to do analysis is in the option domain.

I have always thought that the notion of distance in the price domain using the standard Euclidean distance was flawed when it comes to stock prices (correlation, etc). Instead, it seems that a more natural object is to create a complex Lie Group, i.e, the p-adic [Lie] group of probabilities. Now, distance is not the standard definition that one thinks of in terms of price nearness, but something else. In the options domain, it could be a constructed object completely unrelated to price distance. This would have obvious implications to portfolio theory, as then one can better delta-gamma-vega hedge a portfolio of many different instruments, imo. "Rotating" a position in one instrument into another would be trivial if you had the right Lie group, and hence you would know the risk of one in terms of the other.

It has also occurred to me that a currency pair is not a Lie group, but is the Lie Algebra of some Lie group. For those that understand this stuff, that may make quite a bit more sense. Then trading the options on them are simplified considerably, especially if what you see is interpreted in acceleration space, not velocity.

We note the obvious application to currency triplets and their commutators.

 

[MasterAtWork]

Quote from sjfan:

.... that's why they call that branch continuous time finance. Ito is only applied in a fair small part of it - just a useful little technical lemma

True. That's why Taylor expansions does the job most of the time.

 

[bundlemaker]

The issue with attempting to apply Lie groups is that the symmetries implied are both continuous AND linear. But, the markets are the farthest thing you can get from linear. Same thing with the notion of quantum logic. This is also why signal analysis (Fourier, wavelets, etc) won't get you far; at least not consistently.

Fractal is the correct model. The mistake every phd math guy in the markets that I've talked to makes is they're looking at the wrong data series to look for fractal patterns Hint: it's not in price.

 

[MasterAtWork]

Quote from bundlemaker:

The issue with attempting to apply Lie groups is that the symmetries implied are both continuous AND linear. But, the markets are the farthest thing you can get from linear. Same thing with the notion of quantum logic. This is also why signal analysis (Fourier, wavelets, etc) won't get you far; at least not consistently.

Fractal is the correct model. The mistake every phd math guy in the markets that I've talked to makes is they're looking at the wrong data series to look for fractal patterns Hint: it's not in price.

Well, for sure Laplace, Legendre, Fourier are just tools for pricing.
But I never heard something about real traders making real money with fractal theory. Some examples ?