This is a little note to demonstrate an “impossibility result”. Quite simply, it is impossible to figure it all out; to put it all in a box; to predict the future. At its core, this is what trading in financial markets is all about: a guess—however well informed it may be—about the future. If one is permitted by the powers that be privileged information that is one matter. But no one can predict the future, only guess it. God doesn’t send saints and prophets to share hot investment tips.
Every price has an underlying data generator that is clearly complicated. We obtain information about the data generator by observing the output generated by the systems. Thus financial systems are statistical systems—they may show a great degree of dependence that violates the use of fundamental tools, but they are systems possessing statistical properties.
Statistical mechanics moves from systems of differential equations and proceeds to systems of non-interacting particles. In the financial setting, we show that the equations are not known or are too complicated to be tractable. It is common to claim such mechanical systems are ergodic; that is, their systems have a solution for all finite times and initial conditions, and all such solutions converge to a unique steady-state distribution. If one accepts the ergodic conjecture, then one may completely describe the equilibrium properties of a large system in terms of a phase space average called the partition function Z. The prediction-masters need to prove is that these functions, which cannot be evaluated exactly for nontrivial systems, have ergodic properties which lead in the limit to physically correct consequences for macroscopic systems.
Consider a market. The nature of the process is characterized by N interacting buyers and sellers, dimensionless constants can be expected to grow exponentially with N.
Let N be the number of dimensionless constants required to characterize a process. Then a proper simulation experiment must sample n1X, n2X, …,nXn N points in the N-dimensional space of characterization. The cost of the program P should be proportional to the number of sampling tests:
Hence, if we define X to be the average value of the logarithm of the number of observations for each dimensionless constant, as was suggested to be the case.
True prediction requires a genius that can cut costs and time by going directly to the correct regime of the dimensionless constant of interest without conducting model tests over a broad range. However, the probability of an individual's a genius that gets "directly to the point" in the development of a trading strategy that involves N connected dimensionless constants decreases exponentially with N.
Here’s why. The relevant dynamical equations are approximated by difference equations. Problems arise from the sensitivity of the nature of the solution of the rate equations to the numerical choice of rate constants (the coefficients of the dependent variables) in the differential equations. In the case of non-linearity even the smallest change in the rate constants may lead to qualitatively different solutions of the equations.
This non-linearity result in several possible consequences: 1) one may have discovered an important "instability" or "phase transition" in the phenomenon or 2) one may have the wrong interpretation of a calculation because the rate constants were not properly chosen. Note that even in a system of five linear equations the required twenty-five rate constants may be difficult to find if it is a sparse system.
Such genius is unlikely in a world like ours. The winners here are patient craftsmen more often right than wrong, like my Uncle Larry; and the lucky pot-heads in the right place at the right time who aren’t even wrong.