2. Falsifying Generalized Wiener Processes Melvin J. Hinich hinich@austin.utexas.edu Applied Research Laboratories University of Texas at Austin http://web.austin.utexas.edu/hinich/

3. 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

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

5. Bandlimited Filter Nyquist sampling rate

6. 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

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

8. Normalized trispectrum is the spectrum of For the linear process GWP

9. 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

10. Estimating the Trispectrum Estimated normalized trispectrum

11. 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

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

13. Statistics of the 4 NYSE Returns MeanSigmaSkewKurtosisMaxMin BIR -0.00761.390-0.34934.421.21 -27.60 EOG 0.00590.5210.69714.69.28 -6.23 FE -0.000200.311-0.75833.74.26 -8.49 IMN -0.000540.467-2.51102.06.06 -16.10

22. MeanSigmaSkewKurtosisMaxMin AGL -0.17E-030.031-0.0591.930.075-0.077 AMC -0.45E-040.036-0.0132.010.090-0.090 ANZ 0.16E-030.034-0.0040.7860.077-0.077 BHP 0.24E-040.0340.0060.5640.076-0.075 BIL 0.16E-030.0350.0202.010.087-0.086 WOW 0.30E-030.0350.0531.810.087-0.085 SGB 0.16E-030.0290.0271.900.071-0.070 QBE -0.20E-040.036-0.0312.100.090-0.091 FXJ -0.49E-040.038-0.0283.710.103-0.104 PBL 0.14E-030.0320.0222.300.080-0.080 CML 0.35E-040.039-0.0071.700.094-0.094 CCL 0.72E-040.041-0.0032.500.105-0.105 Statistics after 10% Trimming of the Australian Returns

23. Nonlinearity Test for NYSE Returns 73 Trifreqs Stocksp-values < 0.0137 % p-values < 0.0137 % p-values < BIR 5271.2 %37.0 % EOG 2939.7 %28.8 % FE 4054.8 %16.4 % IMN3649.3 %0.0 %

24. Stocksp-v’s < 3. e-04 % p-v’s < 3.e-04% p-v’s < e-05 AGL 163956.1 %49.5 % AMC 171658.8 %52.3 % ANZ 60920.9 %10.5 % BHP60820.8 %11.0 % BIL176060.3 %54.7 % WOW148951.0 %42.9 % SGB154452.9 %46.4 % QBE160555.0 %47.1 % FXJ202869.5 %67.2 % PBL170458.4 %53.3 % CML159554.6 %45.7 % CCL190665.3 %61.5 % Nonlinearity Test Results for AU Returns 2,920 Trifreqs

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