Stable distributions, futures prices, and the measurement of trading performance

Abstract

n recent years there has been a growing awareness that futures prices and I other economic price series are not well described by normal probability distributions. Accordingly, a number of different investigators (Mandelbrot, 1963; Roll, 1968,1971;Dusak, 1973;Mann and Heifner, 1976, and others who have worked outside the futures price area) have considered the determination of parameters to fit one or another members of the family of stable distributions to an underlying economic price series. Here, we have used stable distributions to fit futures price series (including foreign exchange price series) from US.-regulated exchanges and have compared the results obtained against those obtained from fitting the same data series with normal distributions. In total, daily price series from 18 different futures markets were considered. These included 12 different metals, livestock, agricultural and world commodities (duplicate gold and silver contracts from different exchanges were reviewed), and four different currencies. They represent the most comprehensive selection of different futures contracts yet analyzed in this way. In contrast to previous measurements of this type, symmetry of the underlying price distributions was not an a priori assumption. The data series selected were all of long duration, having time periods ranging from three to nine years with each containing between 807 and 2291 data points.

Searching for fractal structure in agricultural futures markets

“…More recently, Helms and Martell (1985), using data for all commodities traded on the Chicago Board of Trade, conclude that returns on futures prices, although they are not normally distributed, are closer to normal than to any other member of the family of Pareto distributions. Contrary to their results, Cornew, Town, and Crowson (1984) claim that the stable Lévy-Paretian distribution offers a better fit for futures returns of several contracts than the normal distribution. So (1987) confirms that currency futures and spot returns are stable Lé vy-Paretian, whereas Hall, Brorsen, and Irwin (1989) and Hudson, Leuthold, and Sarassoro (1987) claim that futures returns are not stable Lé vy-Paretian.…”

Calendar Spreads, Outright Futures Positions, and Risk

“…Hence we perform the analysis on daily changes in both the outright and spread price series. This approach is consistent with prior research (Cornew et al [1984]; Hudson et al [1987]; Kim and Leuthold [1997]; Poitras [1990]). 8 One popular non-parametric method chooses the optimal bandwidth to describe the data as generated from a mixed-normal distribution (Butler and Schachter [1996]).…”

“…In light of the growing body of evidence suggesting that futures price levels are not lognormal (see, e.g., Cornew et al [1984] and Hudson et al [1987]), some researchers explicitly assume that spread price changes are normally distributed, and then investigate the validity of this assumption. For example, Kim and Leuthold [1997] and Poitras [1990] document that daily price changes in futures calendar spreads experience significant departures from normality, in the form of both skewness and kurtosis.…”

“…5 See Cornew et al [1984] and Hudson et al [1987] for examples of studies that examine the empirical distribution of daily changes in futures prices, and investigate departures from normality. 6 We address this complication in our analysis by re-estimating the ratio of standard deviations, and rebalancing the VaR-adjusted spread position, over two time frames-on both a daily and a weekly basis.…”

The systematic risk of futures contracts

“…These briefly summarized conclusions are characteristic of a literature that is becoming voluminous. The following studies all agree that futures returns are not distributed normally: Stevenson and Bear (1970); Dusak (1 973); Clark ( 1973); Mann and Heifner ( 1976); Tauchen and Pitts ( 1983);Cornew, Town, and Crowson (1984); Helms and Martell (1985);Gordon (1985); Hudson, Leuthold, and Sarassoro (1987); So (1987);Hall Brorsen, and Irwin (1989);and Gribbin, Harris, and Lau (1992). In addition to the non-normality of futures returns, there may also be a problem with serial correlation in daily futures returns.…”