Modeling With Changing Moments
By Steven J. Grisafi, PhD.
Open systems imply the inconstancy of the zeroth moment for an arbitrary stochastic distribution. That the zeroth moment is ever changing also implies the first moment is changing. If indeed financial markets obey a Gamma distribution, then the mean, variance, and the kurtosis are ever changing for the stochastic distribution of an open system. Now, one should ask: What are the elements of a stochastic financial open system? The elements may consist of two categories. As one may restrict all measurements to a fixed number of assets, such as for a stock exchange index, then the elements would consist only of some measure of the capitalization for each asset. Ordinarily, for a arbitrary financial market, new assets may be introduced, and old ones may be removed, such that the number of assets, as well as their capitalization, is ever changing. For the purpose of phenomenological modeling, a toy market is chosen such that the number of assets within it never changes and the system consists only of the price for each asset.
Consider now whether asset price motions within financial markets obey a Gaussian and not a Gamma distribution. An ever changing zeroth moment for an open system obeying a Gaussian distribution would not necessarily have an ever changing mean because the variation of the elements could distribute themselves symmetrically about the current mean such that it does not change. But could this happen without also changing the variance of the distribution? No, it could not, because the changes must always be random within a system that is truly stochastic. Furthermore, any such random variations that could maintain constancy of the mean would cause skewness of the distribution. A Gaussian distribution can never have a nonzero kurtosis. So variation of the elements within a Gaussian system possessing an ever changing zeroth moment cannot distribute themselves such that they have an invariant mean. Hence, the challenge of phenomenological modeling of an open financial system requires a different approach, where indistinguishability of ever changing moments is of no consequence.
Obviously, if it should ever be perceived that there exists any skewness about the mean of a sample of measurements taken of a financial market, then one may disregard the suitability of Gaussian distributions. Thus, it is a fairly well accepted fact that asset prices do not distribute along a Gaussian distribution since they are clearly asymmetric about the mean. However, there are many other possibilities for the probability distribution of asset prices besides Gaussian and Gamma distributions. It is believed that asset price speeds, but not the asset prices themselves, do appear to distribute themselves along a Gaussian distribution, or something similar to it. This observation strengthens the candidacy of the Gamma distribution since it possesses special cases that can mimic Gaussian behavior. This is, of course, the Chi-Square distribution, which is widely used to predict Gaussian behavior for sample sizes of a relatively small number of degrees of freedom. But it is a hopeless task of evaluating time series data taken from financial markets when one cannot assign either a variance, mean or normalization to the data without ambiguity of the distribution. There are simply to many possibilities. We may suppose the validity of the Gamma distribution because theory predicts it. But true phenomenology takes no such presuppositions. Therefore we seek a method that does not require us to assume a priori the validity of any known probability distribution.
The reader has just learned the motivation for the creation of finance rheology. While it may at times prove insightful, proving that an open system obeys one stochastic distribution instead of another is merely of academic value. To prove that asset prices within financial markets obey a stochastic distribution requires evaluating the parameters of the distribution and comparing the fit of that evaluation to all other possibilities. But these parameters are ever changing for stochastic distributions of all open systems. No one set of parameters holds greater significance than any other. Moreover, there is also the possibility that the stochastic distribution of an open system can transform from one standard type into another when different types of state are achieved. For example, if it were truly possible that a open system can reach an equilibrium state, one cannot disregard the possibility that a Gamma distribution would transform into a Gaussian distribution. However, it is more proper to speak of an open system achieving a non-equilibrium stationary state than of achieving an equilibrium state, since being open, the system experiences external flows that one can only expect to balance. If all external flows were to vanish, leaving only internal flows, such as a rotation of price evaluations among fixed assets, then the system would no longer be open.
Finance rheology seeks to render each set of measurements as a complete universe of values. It achieves this by casting a set of measurements as a surface within the state space of an open system. From each probability surface is evaluated a potential function for a given financial market that is cast upon standard boundaries to which all markets can be compared. Thus, the money potential serves as a standard measure of the driving forces within a given toy market that can be directly compared to any other properly designed toy market. No one set of measurements is assigned priority over any other and all serve to estimate the ever changing time parameters of a toy market. The ever changing trajectories of time parameters serve to indicate the speed to which a toy market responds to changes in its external flows. When properly interpreted, this can provide observers of financial markets with some indication of the nature of the changing market state.