Price, Volatility and the Second-Order Economic Theory

We introduce the price probability measure {\eta}(p;t) that defines the mean price p(1;t), mean square price p(2;t), price volatility {\sigma}p2(t)and all price n-th statistical moments p(n;t) as ratio of sums of n-th degree values C(n;t) and volumes U(n;t) of market trades aggregated during certain time interval {\Delta}. The definition of the mean price p(1;t) coincides with definition of the volume weighted average price (VWAP) introduced at least 30 years ago. We show that price volatility {\sigma}p2(t) forecasting requires modeling evolution of the sums of second-degree values C(2;t) and volumes U(2;t). We call this model as second-order economic theory. We use numerical continuous risk ratings as ground for risk assessment of economic agents and distribute agents by risk ratings as coordinates. We introduce continuous economic media approximation of squares of values and volumes of agents trades and their flows aggregated during time interval {\Delta}. We take into account expectations that govern agents trades and introduce aggregated expectations alike to aggregated trades. We derive equations for continuous economic media approximation on the second-degree trades. In the linear approximation we derive mean square price p(2;t) and volatility {\sigma}p2(t) disturbances as functions of the first and second-degree trades disturbances. Description of each next n-th price statistical moment p(n;t) with respect to the unit price measure {\eta}(p;t) depends on sums of n-th degree values C(n;t) and volumes U(n;t) of market trades and hence requires development of the corresponding n-th order economic theory.