Dear Professor Donoho,

Around 2000 Wim Schouten has been actively advocating a Levy distribution, the Meixner distribution which has a closed form expression for both the density and the characteristic function, and he had reported reasonable fits for some indices — especially the tails (see http://www.eurandom.tue.nl/reports/2001/002-report.pdf for some nice quantile-quantile plots for DAX and other indices). So I thought I would look at how this distribution fits some specific stock returns. This is highly reminiscent of work I did at Biospect many years ago under your guidance. Attached figure shows the empirical distribution of daily returns with a fitted parametric Meixner distribution. I don’t think it worthwhile to consider comparison to Normal distribution at all since it is an extremely well-known fact that normal distributions do not fit the tails at all. This fit is very crude, using an L1 penalty but the attached code shows the fitting procedure (Nelder-Mead, four parameters for Meixner etc.) I hope to try to understand this specific family of Meixner distributions as a stepping stone to NONLINEAR modifications next, as my previous week’s work indicated that one might expect improvements in likelihood.

The return data is cleaner in this code: previously I had not accounted for outliers on the tails from data errors and a giant spike at zero resulting from missing returns etc. Now the quality of the return distributions are clearer.

Of course the flattening of daily returns is not best, since there are non-Markovian memory effects etc. but these specific levy distributions are a good base case that is relatively recent and not absorbed by the public yet.

3 Attachments

Preview attachment Screenshot 2015-08-25 20.32.49.png

Screenshot 2015-08-25 20.32.49.png

Preview attachment schoutens-2001-meixner-finance.pdf

schoutens-2001-meixner-finance.pdf

Preview attachment meixner-mle.py

meixner-mle.py

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