糖心TV

Speaker: Ole Peters (Department of Mathematics Imperial College London)

Title: From ergodicity to stochastic market efficiency

Abstract: 

Stochastic systems are ergodic if the frequencies at which states are visited approach a unique probability distribution in the long-time limit. Here, time averages of fluctuating quantities can be computed as ensemble averages where states are weighted according to the unique probability distribution. If the probability distribution is not unique, time averages will depend on initial conditions; if the probability distribution does not exist, for instance in non-stationary processes, it clearly cannot be used to weight ensemble members. In this case, if ensemble averages can be defined at all, they need not be identical to time averages. This is the case in geometric Brownian motion (GBM), a non-stationary model commonly used in financial economics. In GBM the time average of the growth rate differs from the ensemble average. Using ensemble averages, which are agnostic to fluctuations, it is impossible to optimize risk-taking. Using time averages, risk-taking can be optimized in a meaningful objective way. Thus we can compute the optimal leverage for an investment in an asset with given stochastic properties. This leads to the concept of stochastic market efficiency: A stochastically efficient market cannot be beaten by leveraging or deleveraging, which places strong constraints on the stochastic properties of markets. The predictions of this theory are strongly confirmed by empirical studies. In contrast to the ordinary "price efficiency" of markets, stochastic efficiency is a fully accountable theory, in the sense of the word introduced by Popper: The theory predicts its own imprecision and is thus meaningfully testable.