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## The Dark Age of Finance

We’re in the equivalent of the medieval dark ages of finance.  While it is true that statisticians have made enormous strides in finance where there are reams of never-ending data, and the services of an ever growing statistical analysis.  I remember years ago I interviewed at the stat arb group at Barclays in New York.  I didn’t get the job, was not quick enough on my feet perhaps so used they are to getting Harvard and Yale whippersnappers, and I was technically rusty but I had given a good performace on their ‘written’ test that gave me the interview.  I’m a mathematician in my training, not a statistician.  I have large gaps in my statistical training I am endeavouring to fill.  But I’ve been thinking seriously about science for many years.  I’m more of a pioneer type, but enough about me.  Let’s get to the point: finance is today, despite the media blitz and the talking heads and the armies of statisticians working, not a science.  It is not a science because it’s marketing and gossip and is crippled by corruption of money.  Rich people with vested interests cannot make a science out of finance.  They want to make a buck and they are both dull and egotistical, a devastating combination in science because they have control of money supply of the west and the world.  They are trillionaires with gold and money in Swiss banks.  They may not want a science that hurts their interests.  So there’s no science in finance.  There is only one solution that truly makes sense.  People who take a vow of poverty could establish the right science of finance.  I’m not doing this myself anytime soon, but this is the only logical expectation that people who do science of finance must be uncontaminated by the corrosive influence of that poison.  Money is poison and you need some protection from it while studying it.

So what is the object of study?  First, economists don’t know what they’re doing besides kissing Rockefeller and Rothschild and the rest of the ultrawealthy ass, and parroting, “The fundamentals remain strong,” apparently to keep the public mob panic at bay.  That is their training.

The mere idea that one can create a science from the sort of relations –statistical relations of various types–you know, I discovered one myself years ago when I was a quant at a commodities fund between futures prices and inflation, nopt a fantastic find but not bad either with a 0.45 R-squared.  Anyway, even if you figured out the relationship universe of all macroscopic time series you would not be able to construct a real science of finance out of the jumble.

Let’s take a look at an example of great science.  Lars Onsager discovered conditions under which 2D point vertex hydrodynamic model produces new vortices.  Very good science, good numbers on measurements.  We want this sort of good science as science of finance.  In finance there is one giant object, the volatility of the entire world.  That’s basically the object of interest in finance.  That is what actually is the interest of finance to pure science.  Whether someone is making money or not is a gambling casino that really irrelevant to finance–in a casino, remember, the House always wins, speculators like Bachelier are amateur scientists not because they did not do good work but because their interest in finance was not that of a pure scientist but as smart people who wanted to make a buck.  And so are the rich people.  They may not be literally amateurs but for a pure science of finance they are amateurs as the portfolio managers of big pension funds and their traders.  They are corrupted by the profit motive.  They are entrepreneurs perhaps or miners but they are not pure enough in their interests to match science of the type Onsager did.  Therefore, they cannot pretend even to be dispassionate scientists with protective gloves.

How can we fix the situation?  very simple.  We ignore their primitive chatter and deny their science any merit.  We look at the volatility beast across the globe and we study it without exogenous variables.  So consider a large table of data with dates in the first column and daily returns for $N$ assets one for every traded stock, commodity and bond exchange-traded fund in the entire world.  Then use the standard latent volatility model:
$r_t = \exp(h_t/2) e_t$

where $h_t$ is the volatility and the $e_t$ is white noise, independent normal shocks, and $r_t$ is the return vector of length $N$.  Then we use the Whittle approximation

$h_t \sim \log(r_t^2) + \mu_0 + \sigma_0 w_t$

where $w_t$ is white nose and $\mu_0,\sigma_0$ are universal constants the mean and variance of log(whitenoise^2).  We then optimally denoise to extract $h_t$ using soft thresholding of wavelet coefficients and recover $h_t$.  This is the central object of scientific enquiry into finance, the beast of global volatility extracted from noise.  This object would contain the sum total of the collective emotions that drive global financial turmoil and storms that affect billions of human beings on this planet, storms that left me without a home recently and these are natural storms that without a science of finance we cannot quell as a race of seven billion.  Just as our ancestors had worked hard to manage the actual sea storms and track them and protect us from these, financial storms cannot be allowed to run rampant and cause havoc on our people.

This object $h_t$ I shall posit should be studied just as Gauss posited for the study of geometric surfaces, intrinsically.  We must explore the dynamics of volatility on its own terms without reference to who did what when and what events may or may not have affected the volatility.  We should first ask for dynamics of this object completely intrinsically.  Whatever laws are followed by this intrinsically must be the first stab at a real science of finance.

What can such dynamics on volatility be–the first model could the great Lars Onsager’s turbulent hydrodynamics model for vortex creation because financial volatility bubbles are like vortices not just metaphorically but in terms of quantitative statistical features.  Here is a Nature paper on cascade effects in foreign exchange markets; recall that cascades are hydrodynamics phenomena.  Recall now the Navier-Stokes equations of hydrodynamics:
$(\partial_t - \Delta)u = (a \nabla)u$

It would be surprising if this exact equation held for volatility — although I have recently been exploring the spatial variables for the $N$ assets that form the market as graphs and David Donoho has kindly pointed me to the work of Patrick Wolfe’s graphons and nonparametric estimation.  A graph structure is quite natural to financial assets using correlation of prices to define distance.  Partial differential equations on graphs are a new object of study that has been established over the past decade or so and are still in development in details.  In other words, Navier-Stokes is not undefined or vague but can be made rigorously precise.  It would be a surprise if an actual Navier-Stokes equation held for volatility even appropriately interpreted as a vector field on a graph.  But assuming that Onsager’s model for vortices and his results carried over to this context, in particular the gem of production of new vortices, we can try to see if it can be carried out fully.  The heat operator $H_t = \partial_t - \Delta$ is classical with enormous insights by mathematicians including my mentor from many years ago, Daniel Stroock whose understanding of diffusions is extraordinary.  The heat operator appears in Navier-Stokes with a highly nonlinear term appearing on the other side.  This term, were it not so nonlinear could not produce any turbulence and in that sense could be said to be a necessary condition of turbulence which only occurs for high Reynold’s number, a quantity defined simply in the two-dimensional hydrodynamic point-vertex model of Onsager.  In this direction, I recommend studying carefully the heat operator applied to $h_t$ in order to understand it in comparison to the Navier-Stokes right hand side.  The level of this excess is possible a key determinant of whether turbulence is possible for the world financial system.

This is simply a first stab at laws of nature in world volatility that are intrinsic using approaches in science which have led to great advances in the actual science of hydrodynamics.