Karl Malterling Ruderfors: The Cont-Bouchaud Percolation Model Can Realistically Model the Return Distribution: Derivation of the Explicit Distribution and How it Corresponds to Financial Data

Abstract.

In a 2000 paper, Rama Cont and Jean-Philippe Bouchaud presented a model that explains the high kurtosis usually observed in time series of asset returns as a consequence of herd behaviour. The herd behaviour in the model is based on a random graph, where market participants are vertices and where links between them — indicating a similar trading behaviour — are formed according to some law of probability, analogous to percolation models in statistical mechanics. In this master's thesis, we build upon the original paper by Cont and Bouchaud and analytically derive an explicit form for the distribution of returns. We also prove that it is equivalent to Cont and Bouchaud's implicit distribution and provide a scheme for calculating it numerically. This has allowed us to fit the distribution to various datasets of real returns in order to asses the model's suitability and accuracy. In many cases, the correspondence between model and data is very promising. We use two methods to fit the model to data. A method of moments-like approach uses the kurtosis predicted by the model to determine model parameters. Alternatively, we find the numerically most optimal regression with respect to the goodness of fit measure. The latter is almost always gives a better fit, and the discrepancies between the two can usually not be explained as a stochastic error in the kurtosis estimate. We conclude that the kurtosis predicted by the model is unreliable, but still qualitatively correct. Disregarding this, the optimal fit to data can still be accurate, and its shortcomings in comparison to other distributions is arguably compensated for by the fact that the Cont-Bouchaud model offers a micro-economic interpretations for its parameter, which is interesting on its own.