Ergodicity of financial indices

We introduce the concept of the ensemble averaging for financial markets. We address the question of equality of ensemble and time averaging in their sequence and investigate if these averagings are equivalent for large amount of equity indices and branches. We start with the model of Gaussian-distributed returns, equal-weighted stocks in each index and absence of correlations within a single day and show that even this oversimplified model captures already the run of the corresponding index reasonably well due to its self-averaging properties. We introduce the concept of the instant cross-sectional volatility and discuss its relation to the ordinary time-resolved counterpart. The role of the cross-sectional volatility for the description of the corresponding index as well as the role of correlations between the single stocks and the role of non-Gaussianity of stock distributions is briefly discussed. Our model reveals quickly and efficiently some anomalies or bubbles in a particular financial market and gives an estimate of how large these effects can be and how quickly they disappear.