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Why the world has no clue about volatility: difference between heat and fluid equations

Heat equation governs Brownian spreading of a drop of ink in water.  Still water.  The heat operator is $H_t = \partial_t - \Delta$.  It is an extraordinary operator whose sample paths are continuous and the modern mathematical infrastructure of finance is based on measures on the path space adapted to these called the risk neutral measure.  In no-arbitrage theories of valuation of financial derivatives, one calculates — assuming that heat equation governs sample paths of various fluctuating quantities — the expected value using this measure of discounted future transactions.

Heat equation is nice.  Too nice, in the author’s opinion, since the heat semigroup $P_t$ is not just smoothing but extremely smoothing.  It maps $L^2(X)$ to bounded functions.  The heat equation is associated with essentially the gas phase.

If we change the phase to liquid, things become nastier.  The Navier-Stokes equation is a highly nonlinear perturbation of the heat equation $H_t f = A(f)$ where the right hand side is like $f\nabla f$.  Now the heat semigroup being so smoothing implies that this equation is not going to have solutions anywhere as nice as the heat equation.

Intuitively the mathematics will reflect the difference between gas phase and the liquid phase, and intuitively fluid dynamics can have the kind of turbulence that cannot occur with the heat equation at all, which is borne out in physics and engineering.

When the world makes the error of modeling markets with heat equation when it behaves like a fluid there will naturally be a giant surprise during turbulence.  This has spawned Nassim Taleb on his campaign that produced ‘The Black Swan’.   Our solution is to start from scratch, consider volatility a global object and consider’s very carefully Onsager’s 1940’s results on vortex creation by negative temperature.