science of volatility = solution to ‘failure of poisson model’
June 9, 2015 by zulfahmed
Zulfikar Ahmed <wile.e.coyote.006@gmail.com>


11:30 AM (4 minutes ago)




Sir,
The snaphot of Laskin’s paper tells us the physicists’ side of the story from anomalous diffusion theory such as explicated in MetzlerKlafter’s attached book. The attached residual to fit on volatility shock distribution led us to conjecture that the proper generalization is not the power law but a more general form for EN(t) of renewal theory: products of powers and exponentials.
The large issue facing us is to understand what replacements are necessary to produce equivalent of fractional equations currently in use by physicists as article by Saichev from 1997 gives details for.
I am fairly sure that the EN(t) in this class of distributions mixing powers and exponentials is right on. For power law, the physicists have basically constructed a framework based on fractional derivatives.
So what is really going on here? It’s not hard to understand conceptually I think, but may be quite wrong:
WAITING TIMES are telling us about the ‘fractionality’ of the DERIVATIVES according to physics people at least and this seems reasonable but because of the conceptual baggage that came with the older forms of derivatives which we know from pseudodifferential equation theory where the symbol can be arbitrary functions where polynomials are special for their familiarity, one intuitively would expect PSEUDODIFFERENTIAL equations. These are necessary anyway for nonlocal phenomena.
My rule of thumb here to guide myself is
LINEAR LOGWAITING TIME<>NORMAL DIFFERENTIAL EQUATIONS
EXPPOWER LOGWAITING TIME<> PSEUDODIFFERENTIAL EQUATIONS
This is a question of mathematical interest motivated by the delicate missingoftheboat by the FRACTIONAL POISSON PROCESSES on the volatility modeling problem.
Zulfikar Ahmed <wile.e.coyote.006@gmail.com>


11:34 AM (0 minutes ago)




This is the stateoftheart mathematically held by what the physicists know about FRACTIONAL DIFFERENTIAL EQUATIONS. Returning to the problem at hand, I will fit a function in powerexponential class for N(t) and then the real problem is to understand where the physicists have gotten with ‘fractional dynamics’. ‘Fractional’ is the wrong emphasis surely even though it is the dominant effect from whatever is going on behind anomalous diffusions and related phenomena clearer seen in financial volatility.
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