Recursive computation of piecewise constant volatilities

Returns of risky assets are often modelled as the product of a volatility function and standard Gaussian white noise. Long range data cannot be adequately approximated by simple parametric models. The choice is between retaining simple models and segmenting the data, or to use a non-parametric approach. There is not always a clear dividing line between the two approaches. In particular, modelling the volatility as a piecewise constant function can be interpreted either as segmentation based on the simple model of constant volatility, or as an approximation to the observed volatility by a simple function. A precise concept of local approximation is introduced and it is shown that the sparsity problem of minimizing the number of intervals of constancy under constraints can be solved using dynamic programming. The method is applied to the daily returns of the German DAX index. In a short simulation study it is shown that the method can accurately estimate the number of breaks for simulated data without prior knowledge of this number.