Falsifying Generalized Wiener Processes Melvin J. Hinich Applied Research Laboratories University of Texas at Austin

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Falsifying Generalized Wiener Processes Melvin J. Hinich Applied Research Laboratories University of Texas at Austin

The Black-Scholes-Merton model is a central tool for pricing & hedging options & other derivatives securities. Black-Scholes Model At present the derivatives business has become a more than 415 trillion US$ market. It has been implicated in the present world wide financial crisis. (portfolio.com/news-markets/national-news/portfolio/2008/02/19/) Wiener process The model employs a Wiener process

Generalized Wiener process (GWP) bandlimited filter non-gaussian i.i.d variates fixed sampling rate

Bandlimited Filter Nyquist sampling rate

linear The generalized Wiener process (GWP) is linear time reversible It is time reversible since the distribution of is the same as the distribution of

The Trispectrum The principal domain of the trispectrum in the cube of indices where

Normalized trispectrum is the spectrum of For the linear process GWP

Trispectrum Based Tests of Linearity & Time Reversibility The trispectrum version of the Hinich-Rothman (1998) time reversibility test & the Hinich (1982) linearity test can be used falsify the validity of the GWP model. DFT

Estimating the Trispectrum Estimated normalized trispectrum

Testing Linearity of a GWP Under this null hypothesis that the observed values of are generated by a GWP when 0 < c < 1/3 has a uniform (0,1) distribution Then

Testing Time Reversibility of a GWP has a uniform (0,1) distribution

Statistics of the 4 NYSE Returns MeanSigmaSkewKurtosisMaxMin BIR EOG FE IMN

MeanSigmaSkewKurtosisMaxMin AGL -0.17E AMC -0.45E ANZ 0.16E BHP 0.24E BIL 0.16E WOW 0.30E SGB 0.16E QBE -0.20E FXJ -0.49E PBL 0.14E CML 0.35E CCL 0.72E Statistics after 10% Trimming of the Australian Returns

Nonlinearity Test for NYSE Returns 73 Trifreqs Stocksp-values < % p-values < % p-values < BIR %37.0 % EOG %28.8 % FE %16.4 % IMN %0.0 %

Stocksp-v’s < 3. e-04 % p-v’s < 3.e-04% p-v’s < e-05 AGL %49.5 % AMC %52.3 % ANZ %10.5 % BHP %11.0 % BIL %54.7 % WOW %42.9 % SGB %46.4 % QBE %47.1 % FXJ %67.2 % PBL %53.3 % CML %45.7 % CCL %61.5 % Nonlinearity Test Results for AU Returns 2,920 Trifreqs

Time Reversibility Test Results All TR p-values are < e-05