Three Remarks On Asset Pricing

Asset pricing crucially depends on an averaging time interval Δ of the market trade time-series. The choice of Δ changes the basic pricing equation and determines Taylor series of investor’s utility functions over current and future values of consumption. We present current and future values of random consumption as sums of the mean values during the interval Δ and perturbations determined by random variations of the price at current moment t and the payoff at day t+1. Linear and quadratic Taylor series’ approximations of the basic pricing equation describe mean price, mean payoff, their volatilities, skewness and the amount of asset ξmax that delivers max to investor’s utility. We believe that the stochasticity of the market trade time-series must define the random properties of the price and introduce the new price probability measure entirely determined by the probability measures of trading value and volume. We define the set of nth statistical moments of the price as ratio of the nth statistical moment of the value to nth statistical moment of the volume of the market trades performed during the averaging interval Δ. The set of price statistical moments determines the price characteristic function and its Fourier transform defines the new price probability measure. Prediction of the price probability measure requires forecasts of all statistical moments of the trades. Definition of the price probability expresses the catch phrase “You can’t beat the market”.