Implications of the Discreteness of Observed Stock Prices
研究股票价格只能以1/8为单位变动导致的偏差，发现收益率方差和高阶矩的自然估计量有偏，并推导了修正因子，而贝塔和市场组合方差的估计量近似无偏。
ABSTRACT Stock prices on the organized exchanges are restricted to be divisible by ⅛. Therefore, the “true” price usually differs from the observed price. This paper examines the biases resulting from the discreteness of observed stock prices. It is shown that the natural estimators of the variance and all of the higher order moments of the rate of returns are biased. An approximate set of correction factors is derived and a procedure is outlined to show how the correction can be made. The natural estimators of the “beta” and of the variance of the market portfolio, on the other hand, are “nearly” unbiased.