The Market-Based Probability of Stock Returns

We show how time-series of random market trade values and volumes completely describe stochasticity of stock returns. We derive equation that links up returns with current and past trade values and show how statistical moments of the trade values and volumes determine statistical moments of stock returns. We estimate statistical moments of the trade values and volumes by the conventional frequency-based probability. However we believe that frequencies of stock returns don’t define its probabilities as market and financial concepts. We present the market-based treatment of the probability of stock returns that defines average returns during “trading day” that completely match conventional notion of the weighted value return of the portfolio. We derive how statistical moments of the market trade values and volumes define approximations of the characteristic functions and probability density functions of stock returns. We derive volatility of stock returns, autocorrelations of stock returns, returns-volume and returns-price correlations through corresponding relations between statistical moments of the market trade values and volumes. The market-based probability of stock returns reveals direct dependence of statistical properties of stock returns on market trade randomness and economic uncertainty. Any reasonable forecasting of stock returns should be based on well-grounded predictions of the market trades and economic environment.