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 !
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.
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?
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.
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.
.... 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
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. 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.
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 ?