Lie Groups and underlying trading

[Martinghoul]

Quote from bundlemaker:
This is also why signal analysis (Fourier, wavelets, etc) won't get you far; at least not consistently.

Hmmmm, signal analysis has been consistently getting me pretty far, but then again maybe it's the mean-reverting nature of interest rates.

 

[nitro]

Sorry Vikana, I must be going blind - I missed your response.

We are in agreement that the metric is the key. But think on this and allow me an analogy. ..

No, angle is key, not metrics. Metrics are not unique.

 

[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.

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.

Correct! Although p-adic metric is probably false. Simply, Spinors that have embedded Lie groups in them.

The continuous groups are not translational, but rotational. This also explains why fractals apply to markets (although almost everyone applies it wrong). We are interested in similar triangles, and therefore distance is not correct.

 

[Sig]

Hmmmm, signal analysis has been consistently getting me pretty far, but then again maybe it's the mean-reverting nature of interest rates.

I know it's an old quote but interesting nonetheless. Not asking you to spill the beans, but are you talking high frequency or low, i.e. a transform of the past few minutes or the past few weeks? Also curious if your experience has continued to support that statement now six years on? Thanks.

 

[ironchef]

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.

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.

After reading through all the posts in this thread a few times, reading wike a few times on Lie, I still do not have any idea what it is and how to apply the concept.

Can you explain in simpler terms what you are talking about here and how can I use this for a profitable trade?

 

[nitro]

After reading through all the posts in this thread a few times, reading wike a few times on Lie, I still do not have any idea what it is and how to apply the concept.

Can you explain in simpler terms what you are talking about here and how can I use this for a profitable trade?

If you take anything from this thread, you should take away that "distance", e.g., correlation in price, is a poor way to implement trading systems. Phase should also play a very strong role.

Look at the two triangles above, the important thing is that they are similar. That one is bigger than the other is irrelevant if you scale one into the other using a simple scalar multiplication.

The reason I like to think in terms of Lie Group/Algebra is because all of my systems are in continuous matrix forms, and since there is a natural (although hidden) manifold in which price is embedded and evolves, Lie Theory applies.

 

[nitro]

This is great that you can see the COB for free:

http://www.cboe.com/cob/cob.aspx

 

[ironchef]

Thank you nitro for your reply.

Still sounded like Greek to me , but I think I understand correlation in price and similarity. I think I also appreciate price and price forecast are embedded information and am trying to find a way to exploit that without all the fancy mathematics.

Regards,