The Confidence Limits of a Geometric Brownian Motion

This paper investigates whether the assumption of Brownian motion often used to describe commodity price movements is satisfied. Using historical data from 17 commodity futures contracts specific tests of fractional and ordinary Brownian motion are conducted. The analyses are conducted under the null hypothesis of ordinary Brownian motion against the alternative of persistent or ergodic fractional Brownian motion. Tests for fractional Brownian motion are based on a variance ratio test. However, standard errors based on Monte Carlo simulations are quite high, meaning that the acceptance region for the null hypothesis is large. The results indicate that for the most part, the null hypothesis of ordinary Brownian motion cannot be rejected for 14 of 17 series. The three series that did not satisfy the tests were rejected because they violated the stationarity property of the random walk hypothesis.