Dear Professor Donoho,

I’ve had some chance finally to re-evaluate all my intuitive ideas based on empirical data for a complex network model of 1900 stocks seeking natural law for global financial volatility in terms of the thinking that arises from fractional diffusions. This is a class of models for transactions of a single financial instrument as the fractional differential equations has been applied by Scalas, Gorenflo and Mainardi to tick-by-tick transaction data. First, let me clarify what they are doing exactly and then we can see how our approach differs substantially without losing the value of their technical insights.

First, we’re grateful to Scalas-Gornflo-Mainardi for producing a very good scientific model of modeling of PRICE and WAITING TIME for tick-by-tick data by the approach of Continuous Random Walks, so they are assuming that these are the objects for their modeling universe. They model price as {x(t_i)} where prices are the size and the times are random from the renewal theory model. So they have a bivariate distribution in JUMPS (price differences) $\xi$ and WAITING TIMES $\tau$ with a Fourier-Laplace transform of the density following some fractional equation.

Attached papers show the main thrust of this approach from Montoll-Weiss (1965) to Hilfer-Anton and 2008 paper of Gorenflo and Mainardi on fractional differential equations.

So what is wrong with this approach?

(a) It is essentially a UNIVARIATE model that is meant to model something other than global financial volatility through daily data.

(b) It is unrealistic to expect the bias of tick-by-tick data which happens in very developed and funded markets to reflect sources of global financial volatility which has the conscious effort of banks like Goldman Sachs and Wells Fargo and Chase which have trillions of dollars in funds and other we do not know. Therefore, the type of models that these wonderful scientists have developed are excellent for some tasks such as studying regularity in individual markets, but not suitable as a fundamental empirically sound model of the global financial volatility, which is not mentioned at all in any of their models because there is no solid science of finance yet.

Sir, in order to emphasize a lack of science, consider the tradition on which Scalas-Gorenflo-Mainardi’s work stands, which is Samuelson school of finance from MIT which laid the ground for continuous-time finance encapsulated in Robert Merton’s book attached. Our model of volatility models for deterministic laws for global financial volatility suffers from the same bias perhaps but we could take their results as suggesting that WHEN there is an efficient market from the viewpoint of a market-maker certain classes of renewal type models fit tick-by-tick data. This can then be a preliminary observation for a MULTIVARIATE VOLATILITY model for global finance with the geometry of the market encapuated including the topology by a graph and a renewal-type setup as a first approximation.

CONTROL OF FRACTIONAL ORDER SYSTEMS

Sir, the control theory for fractional order systems is addressed for applied problems in hard sciences and engineering but we have the set of tools to apply these now to global volatility to undertand the state of the arts methods for controlling the volatility storms whether or not the POWER exists within the current world order or not the problem can be posed and resolved clearly.

ACTUAL IMPLEMENTATION

Implementation would probably be perfectly successful as a Non-Governmental Non-Profit with people with some serious commitment to the well-being of seven billion human beings as physicians of the human civilization as the best natural science in Europe and America’s history have done (physics was in the seventeenth century a hobby of the physicians rather than an independent discipline). I am myself not wealthy and believe that making money is the wrong focus for understanding volatility as a scientific object. The quantitative science of finance grew out of games of chance and is in shambles by the failure of Long Term Capital Management. An independent organization like weather predictions may an idea that people with the resources to implement could consider. In the next century, our children (I have none) and grandchildren may thank us for taking a risk toward quelling these storms. They are a civilizational thorn on the side of humanity.

## Leave a Reply