Market-Based Asset Price Probability

Abstract

The random values and volumes of consecutive trades made at the exchange with shares of security determine its mean, variance, and higher statistical moments. The volume weighted average price (VWAP) is the simplest example of such a dependence. We derive the dependence of the market-based variance and 3rd statistical moment of prices on the means, variances, covariances, and 3rd moments of the values and volumes of market trades. The usual frequency-based assessments of statistical moments of prices are the limited case of market-based statistical moments if we assume that all volumes of consecutive trades with security are constant during the averaging interval. To forecast market-based variance of price, one should predict the first two statistical moments and the correlation of values and volumes of consecutive trades at the same horizon. We explain how that limits the number of predicted statistical moments of prices by the first two and the accuracy of the forecasts of the price probability by the Gaussian distribution. This limitation also reduces the reliability of Value-at-Risk by Gaussian approximation. The accounting for the randomness of trade volumes and the use of VWAP results in zero price-volume correlations. To study the price-volume empirical statistical dependence, one should calculate correlations of prices and squares of trade volumes or correlations of squares of prices and volumes. To improve the accuracy and reliability of large macroeconomic and market models like those developed by BlackRock's Aladdin, JP Morgan, and the U.S. Fed., the developers should explicitly account for the impact of random trade volumes and use market-based statistical moments of asset prices.