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.