The Stable Paretian Distribution, Subordinated Stochastic Processes, and Asymptotic Lognormality: An Empirical Investigation

Abstract

I. IntroductionLOGNORMALITY OF ASSET RETURNS is a popular assumption in investigations of investor behavior. Empirical observation, however, has disclosed that daily returns are consistently more leptokurtic (are more peaked and have fatter tails) than lognormality indicates. Two responses to these empirical findings have evolved. The first was that the stable paretian' (SP) class of distribution, with a-characteristic <2, is better suited to the description of asset returns [7,9,18,20,23]. More recently the description of asset returns as a subordinated stochastic process (SSP) has developed [4, 6, 19, 21, 22, 24].The use of a daily period in investigations of the nature of the return distribution contrasts sharply with the use of monthly [3], quarterly [5], and annual [4, 15] returns in investigations of investor behavior. Several authors [2, 12, 13, 16] have noted that the nature of the return distribution may change as the period length changes. Section II and the appendix discuss this disparity and point out that the asymptotic tendency under the SSP approach is lognormality. Alternatively, the SP distribution reproduces itself asymptotically. Thus, the observed distributions of asset returns over longer period lengths have relevance for the SP/SSP controversy.The present paper, then has two purposes. The first is a direct empirical investigation of the suitability of the lognormal assumptions for returns to common stocks over monthly, quarterly, and annual periods. The second is to comment on the SP and SSP approaches by comparing the empirical observations to the asymptotic implications. The tests and comparisons are reported in Section III. Our results support the use of the lognormal assumption for the period lengths studied. We also conclude that the asymptotic tendencies of the return distribution are in agreement with the implications of the SSP approach. II. Asymptotic Implication of SP and SSPThe traditional justification of lognormality is based on a multiplicative version * The authors would like to thank Peter K. Clark for his insightful comments. ' While the description "stable" is often used in the literature, the distribution referred to is actually ln-stable, since the variable investigated is usually the ln of returns (price changes). Unfortunately, the literature often fails to discriminate between the two variables. See, for instance, Officer [20] p. 811, note 13. 1031

Call Option Valuation for Discrete Normal Mixtures

“…To assess the impact of skewness on the call price, the parameters 1-' -1 and 1-' -2 are varied to provide positive skewness ranging from none to very high. Table 3 6This tendency of empirical returns for longer periods to converge to normality is shown by Upton and Shannon (1979) and others. Yet, a more realistic model of returns would allow for dependence between the Ai to account for such empirically observed tendencies as high volatility followed by high volatility.…”

Volume and Volatility Surrounding Quarterly Redesignation of the Lead S&P 500 Futures Contract

“…7 See Chung (1993, 1995), Koch (1993), and Lamoureaux and Lastrapes (1994). 8 Time series studies that support this result include Bessembinder and Seguin (1992), Bessembinder et al (1996), Clark (1973), Cornell (1981), Epps and Epps (1976), Foster (1995), Garcia et al (1986), Grammatikos and Saunders (1986), Jain and Joh (1988), Karpoff (1987), Kawaller et al (1990), Kawaller et al (1994), Rutledge (1979), Upton and Shannon (1979), and Wood et al (1985).…”

“…Coincidentally, the smaller population of liquidity providers associated with the nonlead contract fosters a diminished capacity to absorb these imbalances. 6 For example, see Admati and Pfleiderer (1988); Bessembinder and Seguin (1992); Bessembinder et al (1996); Clark (1973); Cornell (1981); Daigler and Wiley (1999); Lee (1993, 1995); Epps and Epps (1976);Foster (1995); Foster andViswanathan (1993, 1995); Garcia et al (1986); Grammatikos and Saunders (1986); Heimstra and Jones (1994); Lipson (1994a, 1994b); Karpoff (1987); Lamoureaux andLastrapes (1990, 1994); Lauterbach and Monroe (1989); Lee, Ready, and Seguin (1994); Merrick (1987); Mitchell and Mulherin (1994); Morgan (1976); Rutledge (1979); Upton and Shannon (1979); Tauchen and Pitts (1983); and Taylor (1985). 7 See Chung (1993, 1995), Koch (1993), and Lamoureaux and Lastrapes (1994).…”

The Valuation of FDIC Deposit Insurance Using Option-Pricing Estimates

“…Several authors (Fama 1976;Hagerman 1978;Upton and Shannon 1979;Oldfield and Rogalski 1980;McFarland, Pettit, and Sung 1982) find that as the trading interval increases from daily to monthly periods, the kurtosis of the distribution of returns lessens, contrary to the prediction of the (independent and identically distributed) stable-Paretian model. For time horizons greater than a few months, the normal distribution may be a good working approximation for security returns.…”