Implications of the Discreteness of Observed Stock Prices

Stock prices on the organized exchanges are restricted to be divisible by 1/8. 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. THE BEHAVIOR OF STOCK PRICES has been an issue of interest to the financial economist for many years. This interest resulted in a growing number of empirical studies which attempt to estimate this behavior (e.g., Blattberg and Gonedes [2], Fama [6], Fama and Roll [7, 8], Barnea and Downes [1]). To date, stock price behavior is estimated under the assumption that the observed trading price is the "true" equilibrium price. However, observed stock prices and stock price changes on the organized exchanges are restricted to multiples of 1/8 of a dollar.1 Therefore, if the "true" distribution of stock prices is continuous, an observed trading price can be different from the "true" price. This paper examines the biases in estimating the moments of stock price changes caused by the discreteness of observed stock prices. The major focus of the paper is in noting this problem, providing a model which explains the source of these biases, and quantifying their size. Section I demonstrates that due to the discrete nature of observed stock prices the natural estimators for the variance and for the higher order moments of the rate of returns are biased upward. This bias is larger for stocks with lower prices and smaller standard deviation. For instance, assuming that the standard deviation, a, is 0.001, the stock price is one dollar, the "true" probability distribution of stock prices is lognormal, and the observed prices are as close as possible to the "true" prices, then the natural estimator of r has expectation 0.01400; hence, it is biased upward by 1300%. Significant biases have important implications in option pricing. We derive an approximate set of correction factors which can be applied to the