Fractals
tags: [ mathematics , finance ]
src: talk at Microsoft Research
Nature vs Culture
Scale-free, or scale-invariance. If you’re given a picture of some rock, then you can’t tell what scale it is at, since you have self-similarity across scaling. Hence you need that reference scale on the bottom right. I wonder if this relates to scale-invariance (à la [[effectiveness-of-normalized-quantities]]).
Roughness. Self-similarity. Now there’s a mathematical framework for thinking about these very human, imprecise notions.1 Curiously, this reminds me of how I think about social networks and how a lot of this messiness is due to social agents. Somehow with this framework it almost seems to abstract away the messiness, and focus on some curious scale-invariance.
Graphs of stock prices have a similar issue (definitely felt this first-hand), whereby if you’re not given time scales, you really can’t distinguish between day or month (besides the upward trend for, say, S&P500).
Independence vs Uncorrelated: if you look at stocks, and if there is a spike one day, then it’s pretty likely that in the next day, there’s also going to be spikes (volatility is high). The only thing is that you don’t know the sign of the activity. So you have expectation zero, but these two events not independent.
Classical take on stock markets is to treat it like brownian motion (continuous time gaussian process). But this doesn’t capture fat-tail-ness and long-range dependence. But with the fractal view, using one parameter (with two real-valued parts), you can basically describe the whole gamut of stock market fluctuations. You can elicit things like anti-persistence and persistence. Start with a trend (upward), add a generator, and then just keep repeating (i.e. fractal).2 Definitely feels like the way you can get brownian motion as a limit to finite random walks, except the key thing is that you’re not taking step sizes to zero, but something a little more self-similar.
- keep moments finite, because then you can take central limit theorems. so things can average out.
The fractal view of markets is descriptive, not prescriptive.
It’s sort of interesting that, with the advent of ML, what you want is over-parameterization. Like that’s sought after now. Whereas in the old days, it’s all about finding the simple thing that then produces complexity. So here Mandelbrot talks about how using multi-fractals, he ends up with only two parameters, and by varying only those two, is able to produce remarkably different scenarios. He contrasts this to other methods, that would have to imbue their model with various extensions in order to be able to capture all the various facets of the characteristics of the price movement.
In some sense, this is good: there seems to be some scale-free nature to price movements. So there is some pattern going on. I’m not too familiar with the math, but I feel like specifying the roughness is half the battle. Or you just say that the rest is just noise or perturbations.