Modeling of Stock Returns in Continuous Vis-À-Vis Discrete Time Is Equivalent, Respectively, to the Conditioning of Stock Returns on a Random Walk Process for Trade Imbalances Vis-À-Vis a Random Walk Process for Evolution of Information

Abstract

Let [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] denote, respectively, the current stock price, the future stock price that is conditioned on information, the minimum stock market tick size and the realized future stock price. Formal theoretical proofs in this study show modeling of stock returns in continuous time induces stock returns that have parameterization as gambles over lotteries. Stock returns have parameterization as gambles because in the presence of fairness of formation of [Formula: see text], regardless arrival of liquidity and speculative trades feasibly induces [Formula: see text]. Evolution of ‘(conditional) trade imbalances’ as random walks is shown to be a necessary and sufficient condition for parameterization of stock returns as gambles over lotteries. Suppose, on the contrary, a resort to modeling of stock returns in discrete time. The formal theory arrives at two dichotomous sufficiency conditions, which predict directionality and sizes of price changes, and facilitate evolution of stock returns as random walks. In presence of the two dichotomous conditions, fairness of formation of [Formula: see text] necessarily induces, regardless of arrival of liquidity and speculative trades, [Formula: see text]. Risk is parameterized by [Formula: see text], because all else constant, an inversion of the perturbing conditionally positive trade imbalance induces [Formula: see text]. Whereas then, [Formula: see text] has parameterization as ‘materialization of risk’, always, it is [Formula: see text] that is statistic for risk; risk, as such is well parameterized, that is, does not coincide with its materialization (note that whereas volatility is statistic for risk, it is not a statistic for materialization of risk; a statistic for risk necessarily is robust to non-materialization of risk). Given modeling in continuous time does not facilitate either of the two sufficiency conditions, always, risk has parameterization as the probability that [Formula: see text]. Since risk, as such coincides qualitatively with its materialization, it is not well parameterized. Given study findings parameterize general equilibrium, formal theoretical predictions have characterization as axiomatic statements, as opposed to propositional (parameter-dependent) statements.