This paper considers price volatility as the reason for description of the second-degree economic variables, trades and expectations } } aggregated during certain time interval {\Delta}. We call it - the second-order economic theory. The n-th degree products of costs and volumes of trades, performed by economic agents during interval { \Delta } determine price n-th statistical moments. First two price statistical moments define volatility . To model volatility one needs description of the squares of trades aggregated during interval \Delta%DIFDELCMD < }%%% . To describe price probability one needs all n-th statistical moments of price but that is almost impossible. We define squares of agent's trades and macro expectations those approve the second-degree trades aggregated during interval {\Delta}. We believe that agents perform trades under action of multiple expectations . We derive equations  on the second-degree trades and expectations in economic space. As economic space we regard numerical continuous risk grades. Numerical risk grades are discussed at least for 80 years. We propose that econometrics permit accomplish risk assessment for almost all economic agents. Agents risk ratings distribute agents by economic space and define densities of macro second-degree trades and expectations. In the linear approximation we derive mean square price and volatility } disturbances as functions of the first and second-degree trades disturbances. In simple approximation numerous expectations and their perturbations can cause small harmonic oscillations of the second-degree trades disturbances and induce harmonic oscillations of price and volatility perturbations } .
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 } %DIFDELCMD < }%%% 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 .
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This paper considers price volatility as the reason for description of the second-degree economic variables, trades and expectations
We introduce the price probability measure
A
\eta
A
(p;t) that defines the mean price p(1;t), mean square price p(2;t), price volatility
A
\sigma
A
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
R
We call it - the second-order economic theory. The n-th degree products of costs and volumes of trades, performed by economic agents during interval
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
R
\Delta
\sigma
D
determine price n-th statistical moments. First two price statistical moments define volatility . To model volatility one needs description of the squares of trades aggregated during interval
D
\Delta
R
. To describe price probability one needs all n-th statistical moments of price but that is almost impossible. We define squares of agent's trades and macro expectations those approve the second-degree trades aggregated during
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
R
believe that agents perform trades under action of multiple expectations
take into account expectations that govern agents trades and introduce aggregated expectations alike to aggregated trades
A
for continuous economic media approximation
R
and expectations in economic space. As economic space we regard numerical continuous risk grades. Numerical risk grades are discussed at least for 80 years. We propose that econometrics permit accomplish risk assessment for almost all economic agents. Agents risk ratings distribute agents by economic space and define densities of macro second-degree trades and expectations.
.
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and volatility
p(2;t) and volatility
A
\sigma
A
p2(t)
R
In simple approximation numerous expectations and their perturbations can cause small harmonic oscillations of the second-degree trades disturbances and induce harmonic oscillations of price and volatility perturbations
Description of each next n-th price statistical moment p(n;t) with respect to the unit price measure
A
\eta
A
(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