A measure of multivariate kurtosis for the identification of the dynamics of a NN-dimensional market

• We apply a stochastic geometry technique to reconstruct geometrical spaces from empirical data (stock returns).
• The shape of the geometrical spaces change with the occurrence of financial crises.
• We use extensions of the concept of kurtosis to measure distortions in the shape of the geometrical spaces.
• Modifications in the value of multivariate kurtosis correspond to changes in the shape of the market spaces.
• In relevant periods, the markets differentiate themselves from a random walk type of behavior.

This paper investigates the common intuition suggesting that during crises the shape of the financial market clearly differentiates from that of random walk processes. In this sense, it challenges the traditional analysis of the nature of financial markets implicit in the most popular models. For this, a geometric approach is proposed in order to define the patterns of change of the market and a measure of multivariate kurtosis is used in order to test deviations from multinormality. The statistical difficulties of this approach are discussed and a new solution is proposed to the consideration of a large space of variables in an accurate measurement of the dynamics of the market. The emergence of crises can be measured in this framework, using all the available information about the returns of the stocks under consideration and not only a single index representing the market.