1. Non-gaussian behaviour of some commodities prices
José Augusto M. de Andrade Jr
Tabajara Pimenta Jr (PhD)

2. Agenda Introduction Stylized facts Impacts over financial engineering
Data and methodology
Discussion of empirical results
Conclusions

3. Introduction Pareto law Bachelier(1900)
Mandelbrot(1963) & econophysics
Black & Scholes options pricing formula
Stylized facts

4. Stylized facts High kurtosis (>3) – heavy tails – Noah effect
Volatility clustering
Long-term memory (Hurst & Joseph Effect)
Scaling

5. Impacts over financial engineering
Options & stocks pricing (B&S formula)
Brownian motion – Wiener process
S(T) = S(0) exp ( [ r – 1/2 σ 2] T + σW(T) )
CAPM
Risk / Return – Covariance / Variance
Risk
Unconditional distribution
Markovitz portfolio optimization model
Variance / covariance

6. Data and methodology
Data

7. Methods Non-parametric tests for normality BDS test for i.i.d.
Kolgomorov-Smirnov
Chi-square
Shapiro-Francia
Shapiro-Wilkinson
Anderson-Darling
BDS test for i.i.d.

8. Discussion of results

10. Results

11. Conclusions The commodities log-returns analysed are not gaussian
They are not i.i.d. – so the underlaying process is nonlinear stochastic (non-Gaussian) or deterministic (chaos)
The financial engineering tools are based on gaussian assumptions
So, these tools cannot give reliable results as they are based on false assumptions

12. Further research Another distribution – stable paretian models
Petrobras paper – the results