Distribution of detrended stock market data

For stock market data, the empirical distribution of the return for stock price and the empirical distribution of the return for stock market index are well known. However, for the detrended data (defined as data divided by trend), which is a different fluctuating quantity compared to the return, only the distribution of detrended daily stock volume is known so far. In this paper, we show that for both stock price and stock market index, the detrended daily data is well fitted by a stable probability density with characteristic exponent parameter less than 2. The trend was modeled using either cubic smoothing spline or principal component analysis. The significance of our results for stock market modeling is discussed.