I did not want to let too much time pass since the promise i made to you.
So here we are. One of the “problems” we have in modelling asset prices is that they do not have the same behavior in high time resolution and in low time resolution. This fact is called granularity (for those to like putting a name on everything (I don’ t)). Granularity is a view of the world different than the fractal one or self-similarity. Hopefuly my friend who coauthors in this blog will post some of his fractal art/vizualization he used to work on, and perhaps still is, when we were younger. (the Brownian motion has a self-similarity structure, that is the problem adressed in this article. Lungs also have a self-similarity structure (to give an example taken from nature), and both of them share a common feature (in some sence), but I won’t go any further into this)
It is well known that people who work in finance use Brownian diffusions to model asset prices, and they’ re really good at modelling asset prices (both people who work in finance and Brownian diffusions XD), (example: the Black Scholes stochastic process, though this one is not very good at modelling asset prices on the long run) but in “high frequency” we have properties of the trajectories of asset prices that cannot be explained by a Brownian diffusion. Let’ s see this graph I took from… (well, you can see from where, it’s the best I could find), where the value of the empiric volatility estimator is calculated:
I assume you all know that if the trajectory of the asset price followed a Brownian diffusion then the graph would be constant (self-similarity). And there are other effects that cannot be explained in the Brownian diffusion context (see the reference).
So, we need another model to work with high frequency data, one is suggested in http://www.cmap.polytechnique.fr/~bacry/ftpPapers1.php?paper=BDHM1.pdf. It’ s a process introduced by Allan Hawkes and based on Poisson processes and it is used in seismology, criminology and terrorism modelizations (I’ m not kidding).
Have a nice weekend.
I’ d like to thank: M. Rosenbaum and E. Bacry for their teaching, by the way.