Cumulative distribution function values for symmetric standardized stable distributions

Abstract

There exists a significant body of empirical evidence indicating that the underlying distribution from which samples of stock returns have been drawn may reasonably be characterized as stable Paretian with characteristic exponent less than two. This paper employs the methodology of Woxsdale (1975) to calculate five-decimal-place cumulative distribution function (c.d.f.) values for standardized stable distributions with characteristic exponents alpha [α] between one and two. The standardization is that originally proposed by Fama and Roll (1968). In addition to extending the Fama and Roll four decimal c.d.f. table to five decimal places, the current paper demonstrates that the Worsdale procedure using numerical integration techniques readily overcomes the computer overflow problems found by Fama and Roll in employing the gamma expansions of Bergstrom (1952).

Keywords:

• Stable distributions
• simulation
• numerical integration