From: John Conover <john@email.johncon.com>

Subject: logistic non-linear extensions used in tsinvest

Date: 10 Nov 1999 05:32:13 -0000

On the home page of the tsinvest program, http://www.johncon.com/ntropix/, concerning the tsinvest and tsinvestsim programs, it is mentioned: "The programs use non-linear extensions to the random walk fractal model of equity prices. The extensions are similar to the discreet time "logistic" (parabolic) function. A derivation of the model is contained in the tsinvest documentation." To elaborate on the non-linear extensions, the program tsinvestsim was used to generate a time series for the tsinvest program using the data file "tsinvest/tests/optimal.equal.data" from the tsinvest regression test suite in the source tree, (which represents "typical" data for stocks on the US equity markets.) The -j option was used on tsinvest, (which means to use the non-default method of computing the index of all stocks in the simulation-similar to the method the equity exchanges use to calculate the broad market indices,) and the index of the 300 stocks, (all identical, with a Shannon probability, P, ie., a likelihood of an increase, of 0.51, optimal growth, ie., a root mean square of the marginal returns of 0.02,) specified in the data file analyzed. The frequency distribution of the marginal returns for the simulated index is:

and agrees favorably with the frequency distribution of the marginal returns for the indices of the US equity markets presented at: http://www.johncon.com/john/correspondence/981229233103.31169.html Of interest is the kurtosis in both graphs. (Note that in the simulation, the kurtosis was not the result of persistence, ie., a Hurst exponent greater than 0.5-it is the result of the non-linearity of the algorithm used by tsinvestsim, which used a random number generator, with independent increments, to make the index's increments a binomial distribution as an approximation to a Gaussian/normal distribution.) The Hurst exponent for the simulated index is:

and, also agrees favorably with the Hurst exponent for the NYSE Composite index presented at: http://www.johncon.com/john/correspondence/980807151309.11811.html Likewise, of interest is that the Hurst exponent is greater than 0.5. (Note that in the simulation, the Hurst exponent was not the result of persistence, it is the result of the non-linearity of the algorithm used by tsinvestsim, which used a random number generator, with independent increments, to make the index's increments a binomial distribution as an approximation to a Gaussian/normal distribution.) John -- John Conover, john@email.johncon.com, http://www.johncon.com/

Copyright © 1999 John Conover, john@email.johncon.com. All Rights Reserved. Last modified: Fri Mar 26 18:52:45 PST 1999 $Id: 991109213225.19168.html,v 1.0 2001/11/17 23:05:50 conover Exp $