Three Remarks On Asset Pricing

Abstract

We consider the time interval Δ during which the market trade time-series are averaged as the key factor of the consumption-based asset-pricing model that causes modification of the basic pricing equation. The duration of Δ determines Taylor series of investor’s utility over current and future values of consumption. We present consumption at current and future moments as sums of their mean values and perturbations during Δ of the price at current moment t and perturbations of the payoff at day t+1. For linear and quadratic Taylor series approximations of the basic equation we obtain new relations on mean price, mean payoff, their volatilities, skewness and amount of asset ξmax that delivers max to investor’s utility. The stochasticity of market trade time-series defines random properties of the asset price time-series during Δ. We introduce new market-based price probability measure entirely determined by frequency-based probability measures of the market trade value and volume. The conventional frequency-based price probability is a very special case of the market-based price probability measure when all trade volumes during Δ equal unit. Prediction of the market-based price probability at horizon T equals forecast of the market trade value and volume probabilities at same horizon. The similar Taylor series and probability measures alike to market-based price probability can be used as approximations of different versions of asset pricing, financial and economic models that describe relations between economic and financial variables averaged during some time interval Δ.