Fractal-Like Price Paths, Brownian Motion, and Market Predictability

# Weierstrass function as market movement's attractor


# Weierstrass function as market movement's attractor












I suddenly realized that BTC/USD index (while being unbounded to anything, yet liquid) in times of high volatility turns itself into something really beautiful and well-structured. Stochastic movements of unstable market spontaneously organize itself into highly structured fractal-like function filled with axes of symmetries, much similar to Weierstrass function. Such movements can be easily observed in the last week, as well as in many moments of whole 2017. Other way it can be proven is by changing candlesticks timescale and still observing same patterns over different scales, which all combined will form Weierstrass-like fractal function.

Here is an illustration for the most recent BTC/USD 'Weierstrass-like' movement:

So, I've come up with plenty of questions related:

- Is it really some kind of well-known market movement? If so, any articles/books/papers to read on this topic?

- What's the main cause of it?

- Can it be observed outside of cryptocurrency market?

- Is this tendency somehow exploitable by specific trading strategy?

## Answer by RRL (score 4, accepted)

https://quant.stackexchange.com/a/37938

This is not a new phenomenon indiginous to cryptocurrency.

As far back as 1900 when Bachelier wrote his thesis The Theory of Speculation, stochastic processes (Brownian motion or variants) have been used to model the random nature of stock prices and other financial assets. The assumption that stock prices follow a geometric Brownian motion lead to the original development of option-pricing models by Black, Scholes and Merton (and Thorp to be perfectly fair).

The sample paths of Brownian motion exhibit similar characteristics to the Weierstrass function in that they are continuous and nowhere differentiable almost surely.

The "fractal-like" behavior is also relevant. The fractal (Hausdorff) dimension of 1D Brownian motion is $\frac{3}{2}$ (which happens to fall in between the coastline of Great Britain and Norway). Much more on fractals in financial markets was developed and discussed by Mandelbrodt.

What is the main cause? If asset prices reflect expectations about future cash flows in an uncertain world then, obviously,they should undergo random fluctuations. Otherwise everything returns a risk-free rate.

Whether or not markets are so efficient that all price behavior is that of a martingale or there is some degree of (transient) predictability (momentum or mean reversion) is too broad a topic to go into here. It does have a bearing on the final question of whether the observed tendency is exploitable.