Feeds:
Posts

## Is financial volatility a space-time deterministic space-time fractional differential equation of Navier-Stokes type?

Since Louis Bachelier the speculator student of Poincare had looked at returns as a Brownian motion in 1900, the dogma in financial modeling has not only been Gaussian-tailed, heavily attacked in the past decades by Nassim Taleb who is a friend of Benoit Mandelbrot who had advocated fractional Brownian motion to model returns.  We have taken a different tack to the problem of a true real genuine science of finance that does not get all starry eyed about money which is the curse of this damn age, a curse that had taken away much of my life not because I was all that greedy by nature but I really do hate morons mistreating me because I’m poor.  Well, so here is my revenge.  I will build a real science of finance, an exact deterministic science, out of love for my true love Talia de Varsgaard, for a non-profit purpose of studying volatility without being swamped in the quicksand of the poison of money and save the world.  The first piece of the puzzle will be the really spectacular fit of the power law model of fractional Poisson model, a stochastic model, on data.  You can see that the residuals suggest model misspecification.

That’s what would be a straight line if the physicists and the fractionophiles were exactly right.  But the residuals of a line fit is this

These graphs show that the fractional models are not exact but there is an exact correction to be found.  Next is the question of which models are best to use for volatility on a complex network.  Our view is that fractionality cannot be evaded obviously but we should seek deterministic laws for a science of finance rather than stochastic models if we can.  Therefore we propose a deterministic volatility model of the following form:

$((d/dt)^{\alpha} - \Delta) = F(u)$

with due regard to the hydrodynamic Navier-Stokes equation which are in a form without any pressure term where $F(u) = u\nabla u$ where the great Lars Onsager who was probably really a mathematician but he was a good scientist too and made the great discovery of vortex formation in hydrodynamics.  We want to get exact fractional dynamics models for exact models with white noise and apply the Donoho-Johnstone wavelet threshold with theoretical guarantee of minimax recovery of Besov signals but we have theory of fractional differential equations and their solutions putting their regularity to these classes anyway.  Now we also have sample path Besov-ness for fractional Brownian paths so the denoising method is quite powerful for volatility.