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exact model for volatility shock distribution is not fractional Poisson and it does seem to have an exact law

Zulfikar Ahmed<wile.e.coyote.006@gmail.com>

4:52 PM (2 minutes ago)

 to David, bcc: aimee
We fit a linear model

log(cumsum(log(shocks))) ~ a + b log(time steps)
and took the slope.  This fit with low p-value and R^2=0.7 but the residual graph shows this is an exact misspecified model.  Therefore it is imperative that we find the correct functional form and then rethink the possible underlying theory for volatility.  The above linear model is simply from the consideration of EN(t) = a t^a and taking log of this equation.  It’s not a bad fit but clearly missing the exact model.

PARAMETER CHECKING BY TRIAL-AND ERROR

> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**2)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**0.1)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**0.3)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**0.4)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**0.5)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**0.6)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**0.7)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**0.8)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**0.9)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**1.0)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**1.1)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**1.2)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**1.3)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**1.5)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**1.6)
> plot(log(x+1e-6),log(cumsum(log(shocks+1e-6))+100)**1.7)
> plot(sqrt(log(x+1e-6)),log(cumsum(log(shocks+1e-6))+100)**1.7)
> plot((log(x+1e-6))**2,log(cumsum(log(shocks+1e-6))+100)**1.7)
> plot((log(x+1e-6))**1.5,log(cumsum(log(shocks+1e-6))+100)**1.7)
I just observed the changes in these curves to visually see the geometry of the relations and whether any of them leads to a straight line. None of them look exact.  I am not yet sure whether power law is quite right and there is something to investigate for exact formula in this case because the power law for EN(t) seems to be close to the correct exact model.
Sir, an exact fit to the shock distribution to a large complex market network would be a significant scientific achievement in finance by any standard.  It is not exactly a fractional Poisson process sir, but close to one.
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