Ladies and Gentlemen,

Calm water diffusion models fit volatility univariate series relatively well but the question of whether a good fit is good enough is a delicate question. Calm water diffusion models, including the Gorenflo-Mainardi fractional diffusion models do not have the theoretical possibility for phase transition to turbulence. For the latter, fractional Navier-Stokes equations with exact solutions might be better. Attached find code and paper for the latter. Our plan is to prepare careful model checking after the computations are properly done in R.

I’d like to remind you that turbulence in volatility affects not only financial market participants but also millions of people at the mercy of a pre-scientific financial science that has not properly looked into global financial volatility as a fundamental scientific object of study since the profiteers hire the armies of quants not for science but to make the rich richer. This research is a preliminary stab deep into the heart of the quantitative features of volatility as a natural which we hope to show has a phase transition to turbulence just as the three-dimensional Navier-Stokes equation does based on Reynold’s number. It’s intuitively obvious that this should be the case also for fractional Navier-Stokes type equations.

Now consider the way in which we might decide whether a fractional diffusion or fractional hydrodynamics model is more appropriate for data.

v<-noisy.volatility

excess<-function(theta){

alpha<-theta[1]

beta<-theta[2]

ddt<-fractional.derivative(v,alpha)

ddx<-laplacian.term(v,beta)

ddt+ddx

}

excess.l1norm<-function(theta){

l1.norm(excess(theta))

}

theta0<-c(1.0,0.8)

fit<-optim(theta0,excess.l1norm)

theta0<-fit$par

x<-excess(theta0)

Now x will be the requisite ‘excess’ from the fractional diffusion that fits data the best. While it is difficult to imagine a fractional Navier-Stokes to hold directly on data, we can easily check whether the model fit is correct for diffusion. If it is indeed incorrect, we have some evidence that there is more to volatility than diffusion and the SIZE of this excess is going to tell us whether a phase transition possibility exists in data.

Now if the data tell us that hydrodynamics equations better explain the data then we have discovered that there are natural laws of volatility for which the phase transition to turbulence is a feature of the financial markets based on viscosity of money in the market. Intuitively Reynold’s number in hydrodynamics just tells us this. In particular global financial volatility studied scientifically would indicate that the turbulence in volatility is not due necessarily to malicious large traders but simply because there is a deep non-science of finance where the basic underlying models have not been able to capture some of the features of volatility series that explain phase transition possibilities.

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