data allows the identification of profitable trading rules. This treatment is not free, it has a price. Moreover, as the relativistic reaction patterns become increasingly diversified, research and development efforts will have to increase in the future to keep up with the ever-changing nonlinear patterns.

12.5 DISCUSSION OF THE CONVENTIONAL DEFINITION

As the markets consist of a diversity of components, different relaxation times occur because of the underlying relativistic effects between different components. It follows that the weak form of efficiency coupled with the rational expectation model cannot be attained. Because of the presence of different time components with heterogeneous expectations, current market prices cannot reflect all available information. The price discovery mechanism follows rather a dynamic "error correction model" where the successive reactions to an event unfold in the price. Why, then, did this not show up more clearly in previous scientific investigations? Some of the several reasons include the following:

High-frequency data are a prerequisite for the empirical investigation of relativistic phenomena.

Extensive computing power is needed to show the predictability in financial markets. Access to reasonably priced computing power has become available only recently.

It is in the past few decades that an increasing awareness for dynamic and nonlinear processes has been gained. Such an awareness is crucial for the study of relativistic effects.

The presumption of conventional economics that forecasting is impossible per definition has had a powerful impact on the research on market efficiency. Economists have focused on structural studies that were hamstrung by a lack of high-frequency data and theoretical shortcomings. Little academic research has been invested in actually trying to predict shorter-term price movements and build successful trading models.

12.6 AN IMPROVED DEFINITION OF "EFFICIENT MARKETS"

Although the current definition of efficient markets has shortcomings, we do not think that this concept should be abandoned; rather, it should be adapted to the new findings. It is important to find a good measure of how well a market operates.

From a dynamic perspective, the notion of reduced friction should be central to the notion of efficiency. We consider an efficient market to be a market where all market information must be available to the decision makers and there must be participants with different time scales and heterogeneous expectations trading with each other to ensure a minimum of friction in the transaction costs.

A quantitative measure of efficiency might be derived from the bid-ask spreads (those between real bid and ask prices being more appropriate for such a measure

12.6 AN IMPROVED DEFINITION OF "EFFICIENT MARKETS"

than the nominal spreads quoted in information systems). Spreads are not only a measure of "friction," they also contain a risk component. The volatility or, more precisely, the probability of extreme returns within short time intervals should be considered together with the spread in the quantitative measure of market efficiency to be proposed. We are sure that in the years to come this definition will prevail and we shall find precise measures of efficiency as it is the case in thermodynamics and engineering.

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