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A Decomposition of Empirical Distributions with Applications to the Valuation of Derivative Assets

The selection of an appropriate parameterization of data is a fundamental step in a majority of empirical research effort. Likewise, detecting or estimating features of non-stationarities in data sequences is a critical point in conducting credible research that uses data for inference. In this spirit, this paper presents a simple decomposition of the empirical return distributions of financial assets into the sum of various normal distributions. The decomposition is motivated by the fact that market participants expect distributions to be drawn from two or three possible scenarios. It is also motivated by the recent applications of the EM algorithm to financial data. A parametric and a nonparametric approach are proposed and applied to the empirical distribution of the CAC 40 index traded in the Paris Bourse. We estimate the parameters of the mixture and propose a decomposition into three Gaussian distributions which essentially differ by their variances. The decomposition fits the observed distribution. An alternative approach, which consists in detecting these changes and estimating the distribution of the returns between two changes is developed. The results are obtained using a segmentation method, which is applied to financial data. One of the main findings in this paper is that the two approaches show the same results and give support to the proposed decomposition. There exists three kinds of regimes in the Paris Bourse and the series of the returns jump from a regime to another one at some random instants. This work might be applied to other data sets or other data generating conditions. It can used for the valuation of standard and exotic derivatives.