By Steven J. Grisafi, PhD.
Empirically it is known that asset price changes distribute along a Gaussian distribution, i.e., a normal distribution. Asset price changes occur over some differential of time, therefore they may be taken as the magnitude of a price velocity. While there exists empirical evidence for the Gaussian distribution of price velocity, this is not empirical evidence for the Gaussian distribution of logarithmic price velocity. Since the logarithmic operation does not necessarily commute with the expectation operation, i.e., the taking of moments, it is not warranted. Conversely, should one be presented with empirical evidence for the Gaussian distribution of logarithmic price velocity, this does not imply that the price velocity will distribute normally. The log-normal distribution is not a Gaussian distribution. The logarithms of price changes that undergo Brownian motion do not themselves undergo Brownian motion. This is by definition of a Brownian motion process as one that is continually distributed Gaussian throughout the entire period of the process. Diffusion does not require a Gaussian distribution of the property that is diffusing. Yet, the ansatz of this entire modeling procedure is the Ito Calculus, which presupposes Brownian Motion.