On Economic Space Notion ABSTRACT

On Economic Space Notion Victor Olkhov TVEL, Kashirskoe sh. 49, Moscow, 115409, Russia, [email protected]

ABSTRACT This paper introduces Economic Space notion to expand capacity for economic and financial modeling. Introduction of Economic Space allows define economic variables as functions of time and coordinates and opens the way for treating economic and financial relations similar to mathematical physics equations. Economic Space allows study economic models on discreet and continuous spaces with different dimensions. The number of risks measured simultaneously determines Economic Space dimension. We present examples of modeling on Economic Space: option pricing and derivation of Black-Scholes-Merton equation on ndimensional Economic Space; Markov processes and derivation of Fokker-Plank Equations. Usage of Economic Space allows construe approximations of Economics and Finance similar to physical Kinetics and Hydrodynamics and derive Wave Equations for Economic and Financial variables.

Keywords: Financial Modeling, Economic Space, Risk Ratings, Economic Wave Equations. JEL: C500, C520, C530, C600, G110, G130



This work was performed on my own account only. 1

1. Introduction Economics and finance are systems with extreme complexity similar to complexity of theoretical physics; nevertheless the phenomena’s of these disciplines are too different. During last decades Econophysics has delivered many contributions for understanding and modeling economic and financial features (Mantegna and Stanley, 2000; Roehner, 2002; Stanley, 2003; McCauley, 2006; Special Issue, 2008; Schinckus, 2013) and presented many applications of statistical physics methods to economics and finance. Methods and models developed by theoretical and statistical physics might be useful for description of economics and finance. These results present first steps for modeling economic processes with accuracy similar to and with understanding comparable to current description of physical processes. We treat Econophysics as a way to adopt current methods and schemes of theoretical and statistical physics to economic and financial modeling. Such adoption should follow economic and financial laws and these requirements change basement of most methods and models developed within theoretical and statistical physics. We study economic analogies of physical schemes that can be useful for economic and financial modeling. One of the most general and common physical notions is the spacetime issue. We state a simple question: is it possible to introduce certain analogy of spacetime notion for economic and financial modeling and what are the possible advantages? Further we mention such analogy as Economic Space. We introduce Economic Space notion as an extension of matter that is already used in economics for many years and thus we assume that Economic Space notion conforms economic phenomenology. At the same time Economic Space notion allows regard economic variables as functions of time and Economic Space coordinates. It allows treat economic and financial properties in a way similar to mathematical physics equations and applies methods of current theoretical physics to economic modeling. It is already well known (McCauley, 2006) and we also underline the statement that the lack of conservation laws and symmetries in economics and finance make differences between economic and physical systems vital. Theoretical physics methods should be changed to obey economic and financial phenomenology. Space-time determines the foundation of theoretical physics and description of Nature. Thus motivation of our study is based on assumption that introduction of Economic Space may give a new look on Economics and build solid foundation for broad usage of theoretical physics methods for economic and financial modeling. The plan of the paper is as follows. In Section 2 we argue Economic Space notion.

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Then we present examples of financial and economic modeling on Economic Space. In Section 3 we study option pricing on n-dimensional Economic Space and derive extension for Black-Scholes-Merton equation. In Section 4 we study Markov processes and derive FokkerPlank equations on Economic Space. In Section 5 we study approximations of macroeconomics that are similar to physical kinetics and hydrodynamics and derive wave equations for economic variables on Economic Space. The conclusions are in Section 6.

2. Economic Space notion We introduce Economic Space notion with goal to study economic variables of economic agents and economic variables of entire macroeconomics as functions of time and Economic Space coordinates. Economic variables can describe Supply and Demand, Loans and Debts, Labor and Taxis, Production Function and Capital and so on. Our definition of Economic Space notion has nothing common with spatial econometrics (Anselin, 2009) and our approach completely differs from agent-based economics (Judd and Tesfatsion, 2005). We assume that different economic states allow define different “intrinsic” Economic Spaces those describe different approximations of economic and financial processes. Existence of such Economic Space and opportunity describe economic variables as functions of time and Economic Space coordinates allows use functional analysis, stochastic functions and modern methods of mathematical physics for economic and financial modeling. Introduction of Economic Space for economic modeling arises two problems. First: how to define Economic Space? Second: how to measure coordinates of economic variables and how to define the partition of economic variables over certain Economic Space? To solve these problems we suggest use methods that are well known in economics for many years. We refer to Lee (1999), BIS (2011) and BIS (2013) as a small amount of economic and financial risk management studies. Risk management deals mostly with risk ratings of banks and corporations. Risk ratings are provided by international rating agencies like Fitch (2006), Moody’s (2007), S&P (2012) and DBRS (2013). We state that Economic Space notion and economic and financial modeling on Economic Space can be developed on base of risk ratings practice. To show that let outline that risk ratings procedures distribute economic agents as companies, corporations, and banks over a finite number of risk grades, like AAA, BB, CCC and so on. Finite number of risk grades can be treated as finite number of points of discreet space. Risk grades of economic agents like banks or corporations can be treated as their

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coordinates on discreet space. Risk ratings estimations of economic agents are similar to measurements of coordinates of economic agents on discreet space. Ratings of single risk can be treated as coordinates on one-dimensional discreet space. Simultaneous estimations of ratings of n risks of economic agents are similar to measuring coordinates on n-dimensional discreet space. Thus existing risk ratings practices can be treated as procedures that distribute economic agents on discreet space. We suggest mention such space as Economic Space. Let associate risk ratings or coordinates of particular economic agent with coordinates of its economic variables. That defines economic variables of economic agent as functions on Economic Space. Let propose that current risk ratings methodologies can be extended to estimate risk ratings of any economic agent as banks and corporations, householders and personal investors, etc. Let assume that risk methodologies may allow measure risk ratings of economic agents on discreet or continuous spaces. Risk ratings may define probability that economic agent has certain coordinates on discreet or continuous space. Let study risk ratings of economic agents that are under simultaneous action of n risks on Euclidian space Rn. Location of economic agent on Economic Space Rn, or the risk grade of economic agent can be defined by probability distribution. Let define Economic Space as any mathematical space that is used to map economic agents by their risk ratings as space coordinates. Further let study economic modeling on Economic Space Rn. To describe economics on Economic Space Rn one should determine n risks that disturb economics now. It seems impossible to take into account all existing risks that can affect current economics. Risk ratings procedures contain internal uncertainty and that uncertainty will grow up with growth of number of simultaneously measured risks. If one takes into account too many different risks then simultaneous measurements of all these risk ratings will have too high variability and hence model description can be too uncertain. To determine reasonable Economic Space one should estimate current risks and select two, three, four most important risks as main factors affecting contemporary economics. That allows define Economic Space that has dimension two or three and derive appropriate initial distributions of economic agents and their economic variables. To select most valuable risks one should establish procedures that allow compare influence of different risks on economic processes. That determines initial state of Economic Space Rn. To describe economic evolution in time term T it is necessary to foresee m main risks that will play major influence on economics in particular time term and define Economic 4

Space Rm. The set of m risks can be the same as for initial state, or different one. This set of m risks defines target state of Economic Space Rm. Then it is necessary define transition dynamics, transition model that describes move from initial set of n main risk and define initial representation of Economic Space Rn to target set of m main risk and target representation of Economic Space Rm. Such transition model describes how new risks grow up and how initial set of risks decline its action on economic processes. Transition dynamics from initial set of n main risk to target set of m risks describes evolution of initial representation Rn to target one Rm. And to complete the list of problems we outline that one should develop model of evolution of economic distributions under action of initial set of n risks, develop a model of evolution under target set of m risks and describe transition dynamics from initial risks to target risks. That short description arises a lot of difficult problems. The selection of main risks simplifies economic description and allows neglect “small risks”. On the other hand the selection process becomes a part of validation procedure. As one can select and measure main risks, than it is possible to validate initial and target set of risks and prove or disprove initial model assumptions. The procedures that measure and compare influence of different risks on economics should be determined and that is a separate tough problem. Transition process from initial set of risks to target risks becomes measurable. Hence it might be possible compare transition observations with model assumptions and estimate accuracy of model predictions. Financial modeling and forecasting on Economic Space are splitting into set of verification procedures. It gives a chance to make financial modeling more measurable and it’s forecasting more faithful. To prove that one should establish reasonable modeling. The success or failure will be best argument. With this in mind we move forward in developing modeling on Economic Space. To demonstrate possible advantages of Economic Space notion usage for economic and financial modeling we present some cases. Further we assume that methodologies and procedures that allow measure risk ratings of economic agents, as their coordinates on ndimensional Economic Space Rn exist.

3. Option pricing on Economic Space As first example of financial modeling on Economic Space we present option pricing. Option pricing theory is based on the Black-Scholes-Merton equation (BSM) (Black and

5

Scholes, 1973; Merton, 1973; Merton, 1998) and that is one of the most recognized equations in financial theory. We present an extension of the BSM equation on n-dimensional Economic Space. Further we shall mention Economic Space as e-space. The BSM equation for price V of option on underlying asset with price a has form: 𝜕

𝜕𝑡

𝜕

+

+ 𝜎

𝜕

𝜕

𝜕

=

(1)

Here, r is risk-free interest rate. A simple way to derive the BSM equation (Merton, 1998; Hull, 2009) is based on assumption that asset price a obeys Brownian motion dW(t)
= ;
=

c – is instantaneous rate of return on security, and σ2 – is instantaneous variance rate. Option price V = V(t,a) is function of time t and security price a. Operator denotes averaging procedure. Our contribution to option pricing theory concerns modeling on e-space. For simplicity let threat here options on stocks of economic agents like corporations or banks. Stock price is determined by Value of selected economic agent. Let mention any economic agent on espace as e-particle. Extensive economic variables of e-particle as Value and Capital, Demand and Supply etc., are functions on e-space. Extensive economic variables are additive and admit direct averaging under probability distributions. Contrary to them, intensive economic variables as Price or Interest Rate cannot be averaged directly. Random walks of e-particles on e-space mean random change of risk ratings of economic agents. Economic variables of eparticles on e-space Rn are under the action of n risks. Hence changes of e-space coordinates of economic agents (e-particles) induce changes of their economic variables. Value of economic agent a=a(t,x) on n-dimensional e-space Rn determines it’s stock price a and it is possible to replace option dependence on stock price by dependence on Value of economic agent. Hence option price V also becomes a function of coordinates x on n-dimensional espace Rn and option price takes form V=V(t,x,a). To derive the BSM equation on e-space Rn let suggest obvious extension of relations (2). Let assume that Value and stocks price of eparticle have linear dependence on dx: =

+ 𝜎

+

∙ 𝒙

(3)

Vector k describes the input of e-space coordinates variation dx on Value of e-particle and thus on stocks price; kdx denotes scalar product. Let assume that coordinates x of e-particle also obey Brownian walk dZ(t) on n-dimensional e-space 𝒙= 𝒗

+ 𝒁

(4)

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Vector υ defines regular speed of e-particle on e-space. Brownian motion dZi(t) along each axis of n-dimensional e-space follows
= ;
=

(5)

,

Vector η=(η1,…ηn) determines instantaneous variance rate along each axis on e-space Rn. For simplicity let assume that there are no correlations between dW(t) and dZi(t)
=

Assumptions (3-5) allow derive equation on option price V=V(t,x,a) as an extension of the BSM equation (3) on n-dimensional e-space Rn. 𝜕V 𝜕𝑡

+

𝜕V 𝜕

+ 𝑥

= (𝜎 +

𝜕V

∙

𝜕𝑥

+

);

𝜕 V 𝜕

= ,..

+

𝜕 V

𝜕 𝜕𝑥

+

𝜕 V 𝜕𝑥

= V

(6)

Derivation of (6) is similar to Merton (1998), Hull (2009) (Appendix A). Equation (6)

has many additional parameters ki, ηi ; i=1,…n. All these factors have influence on behavior of option price V(t,x,a). Brownian walks on n-dimensional e-space may depend on stock price of e-particle. For example, ηi may depend on “risk appetite” of management of eparticle. “Risk appetite” can decline if stock price of e-particle increases and can grow up if stock price of the e-particle falls. Equations (1,6) are valid if initial set of n risks is constant. Due to Section 3, initial selection of n risks that determines initial e-space Rnini can be different from final set of m risks and final e-space Rmfin. If during time to expiration forecasts predict that new risks can affect e-particle Value, then option pricing should be corrected. For such a case option pricing should depend on description of transition from initial e-space Rnini to final e-space Rmfin. Thus equation (6) is correct if initial risks remain constant. If some new risks grow up during the term to expiration than assumptions (3-5) and equation (6) should be modified. Thus, original assumption (2) and the BSM equations (1) seems fail to describe option pricing for the case with different sets of initial and final risks that affect Value a of the underling assets. Modification of (1,2) and (3-6) should describe transition from initial espace Rnini to final Rmfin and that requires additional studies and considerations.

4. Markov processes and Fokker-Plank equations on Economic Space Markov processes are used for finance and economic modeling (Shiryaev 1999, Steele 2001, McCauley 2013) for many years. Introduction of Economic Space allows describe positions of e-particles and economic variables of e-particles within Markov

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processes approximation. The walks of e-particles on e-space change risk ratings of economic agents (e-particles) and hence induce changes of economic variables of e-particles. Thus one should jointly describe variations of e-space coordinates and economic variables within Markov process properties. Let describe economic state of e-particles on n-dimensional espace Rn by distribution Ψ(t,x,u) that defines probability that e-particles have coordinates x=(x1,..xn) and their l economic variables equal u=(u1,..ul). At different moments e-particle has different coordinates x on e-space and different values u of economic variables. Let denote 1,N(t1,x1,u1; t2,x2,u2;… tN,xN,uN) as probability that at moments t1>t2>..>tN, e-particle has coordinates x1,…xN and economic variables equal u1,…uN. Let denote e-space coordinates and economic variables as state p=(x,u). 𝛹

,𝒑 ; ,𝒑 ;….

,𝑁

𝑁 , 𝒑𝑁

=𝜑

,𝑁

,𝒑 | ,𝒑 ;….

𝑁 , 𝒑𝑁

𝛹

,𝑁

,𝒑 ;….

𝑁 , 𝒑𝑁

Here 1,N(t1,p1|t2,p2;…tN,pN) is the conditional probability that e-particle at time t1 will be in state p1 if at previous moments t2 >…>tN it was in the states p2, … pN. Markov property 𝜑

,𝑁

,𝒑 | ,𝒑 ;….

𝑁 , 𝒑𝑁

=𝜑

,𝒑 ; ,𝒑 ;….

𝑁 , 𝒑𝑁

=

,𝒑 | ,𝒑

means that state of e-particle at time t1 depends on state at previous moment t2 only. For such

a case 𝛹

,𝑛

=𝜑

,𝒑 | ,𝒑 𝜑

,𝒑 | ,𝒑

…𝜑

𝑁−

, 𝒑𝑁− | 𝑁 , 𝒑𝑁 𝛹

𝑁 , 𝒑𝑁

For transition probability φ(t1,p1|t2,p2) Chapman-Kolmogorov equation (Klyatskin, 2011; Gardiner, 2009) has form: 𝜑

, 𝒑| , 𝒑

Ψ

=∫ 𝒑 𝜑

,𝒑 = ∫ 𝒑 𝜑

, 𝒑| , 𝒑 𝜑

, 𝒑| , 𝒑 Ψ

,𝒑

,𝒑 | ,𝒑

For diffusion process equation on φ1(t,p|t0,p0), takes form (Klyatskin, 2011) 𝜑

, 𝒑| , 𝒑

=−

𝒑

𝐵

Fokker-Plank equation

,𝒑 𝜑

, 𝒑| , 𝒑

+

𝒑

𝐵

,𝒑 𝜑

, 𝒑| , 𝒑

Derivations of Fokker-Plank equation for Ito processes and the Black-Scholes-Merton equation (1,2) were presented in numerous papers (Harrison, 2005; Conze, Lantos and Pironneau, 2007; Kohn, 2011; McCauley, 2013). Let derive Fokker-Plank equation for distribution Ψ(t,x,a) that describe probability that e-particle has e-space coordinates x and Value a and they follow (2-5) on n-dimensional e-space Rn. Further let show how derive

8

Fokker-Plank equation for distribution Ψ(t,x,u) in the assumption that all extensive economic variables u of e-particle follow relations similar to (2). Distribution Ψ(t,x,u) describes probability that e-particle has l extensive economic variables u=(u1,..ul) and coordinates x=(x1,..xn) on e-space Rn. Assumptions (2-5) allow derive the BSM equation. Same assumptions lead to FokkerPlank equation for probability distribution Ψ(t,x,a) that e-particle has coordinates x and its Value equals a. Assumptions (2-5) deal with e-particle’s share price a. It is obvious that one can treat (2-5) as relations on e-particle’s Value a. Let reformulate (2-5) as follows:

a= aμ = c+

+a

∙𝒗 ;

= 𝜎

< Y t >= ; < Y


= ;< +

Y +

>=

+

+

∙ 𝒁

+ 𝒁

>= 𝜎 +

(7)

>=

,

If coordinates x of e-particle on e-space Rn and Value a of e-particle obey (7), then Fokker-

Plank equation for probability distribution Ψ(t,x,a) has form (Gardiner, 2009; Kohn, 2011;

Moll, 2012; McCauley, 2013): 𝛹

𝑛

c+ ∙𝒗 𝛹 + ∑

+

𝑛

−∑

𝑥

=

[

=

𝑥

𝛹] −

𝛹 − 𝑛

∑ =

[ 𝑥

𝜎 +

𝛹]

𝛹 =

Let assume now that all extensive economic variables u=(u1,..ul) follow Brownian motion relations similar to (7). If l extensive economic variables u=(u1,..ul) and coordinates x=(x1,..xn) of e-particle follow Brownian walk, then Fokker-Plank equation can be derived for distribution Ψ(t,x,u) that describes probability that e-particle has coordinates x and economic variables u. Vector u=(u1,..ul) can define different extensive variables as Value and Capital, Supply and Demand, Value Added and Production Function, Debt and Loans and more. If dYj and dZi -Brownian processes and (x,u) follow: =

𝑥 =


=

>=

,

9

then Fokker-Plank equation on distribution function Ψ(t,x,u) has form: 𝛹

+∑ =

(

𝑛

𝛹) + ∑

;𝑛

− ∑

= ;=

=

𝑥

𝜎 𝛹

𝛹 − ∑

𝑥

=

𝜎

𝛹 −

𝑛

∑ =

𝑥

𝛹 =

These results demonstrate extension of financial modelling formalism and present supplementary approach for treatment of economic processes. That is only a first step for research of financial modeling on e-space.

5. Wave equations on Economic Space In Section 3 we suggested treat economic agents as e-particles on e-space. Each eparticle is described by set of l extensive economic variables u=(u1,…ul). As we mentioned in Section 4, economic modeling of e-particles can be described as evolution of probability distributions of economic variables on e-space Rn. Probability is used in economics for more than hundred years (Bachelier, 1900). Applications of probability and statistical methods for description of economic problems were presented in many studies (Diebold, 1998; Schenk-Hoppé, 2000; Stanley, 2003; Special Issue, 2008). Definition of economic variables as functions on e-space allows use random functions methods instead of random values and modern technique of stochastic dynamic systems (Klyatskin, 2011) for economic and financial modeling on e-space. Introduction of e-space and e-particles allow establish economic models that have parallels to Physical Kinetics and Hydrodynamics. Let denote Economic Kinetics as approximations that model economics as system with interacting economic variables of numerous e-particles on e-space. Let denote Economic Hydrodynamics as approximations that neglect existence of separate e-particles (separate economic agents) and study relations between interacting macroeconomic variables.

5.1. Economic Kinetics Description of economic variables of numerous e-particles as companies and banks, householders and personal investors, etc., is a very complex problem. Certain methods of physical kinetics may help simplify that difficult problem. Let describe each e-particle by l extensive economic variables that form vector u=(u1,…ul). Contrary to physical kinetics,

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dynamics of e-particles does not determined by physical collisions of e-particles. Dynamics of e-particles, the behavior of economic agents on e-space may reflect risk “appetite” of managers and shareholders as well as the influence of commodities and financial markets, government regulation risks, credit risks, and other risks that determine behavior of economic agents in economics. Let describe each e-particle (each economic agent) on n-dimensional e-space Rn at moment t by coordinates x=(x1,…xn), velocity υ=(υ1,… υn), and l extensive economic variables u=(u1,…ul). Let regard extensive (additive) variables because it is possible average them within probability distribution. Intensive variables like Prices or Interest Rates cannot be averaged directly. Enormous number of extensive variables like Value and Capital, Demand and Supply, Profits and Production Function etc., describe each e-particle and that increase model complexity in comparison with physical kinetics. Contrary to physics, economic variables do not obey conservation laws and change their values due to economic processes and motion of e-particles on e-space. Let assume that there are N(x) e-particles at point x. Let state that velocities of eparticles at point x equal υ=(υ1,… υN(x)). Each e-particle has l economic variables u=(u1,…ul). Let assume that economic variables equal (u1i,…uli), i=1,..N(x). Each extensive economic variable uj at point x generate macroeconomic variable Uj as sum of economic variables uj of all N(x) e-particles at point x =∑

;

= ,.. ;

= ,…𝑁 𝒙

For each macroeconomic variable Uj let define analogy of impulse Pj as 𝑷 =∑

𝝊𝒊 ;

= ,.. ;

= ,…𝑁 𝒙

Impulses Pj are extensive variables – they are additive. That allow define velocity υj that correspond to macroeconomic variable Uj as Uj υj = Pj . Let follow Landau and Lifshitz (1981)

and

introduce

economic

analogy

of

Boltzmann’s

distribution

function

f=f(t,x;U1,..Ul,P1,..Pl) on n-dimensional e-space that determine probability to observe macroeconomic variables Uj and impulses Pj at point x at time t. Let define macroeconomic density function Uj(t,x) as ,𝒙 = ∫

𝑓 , 𝒙,

,…

,𝑷 ,..𝑷

𝑷 .. 𝑷 ;

= ,…

(8.1)

and impulse density function Pj(t,x) as

..

𝑷

, 𝒙 = ∫ 𝑷 𝑓 , 𝒙,

,…

,𝑷 ,..𝑷

..

𝑷 .. 𝑷 ;

= ,…

(8.2)

That allows define e-space velocity υj(t,x) of density Uj(t,x) as

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,𝒙 𝒗

,𝒙 = 𝑷

,𝒙

(8.3)

Densities Uj(t,x) and impulses Pj(t,x) are determined as mean aggregates of

corresponding economic variables of separate e-particles. Functions Uj(t,x) can describe espace macroeconomic density of Demand and Supply, Assets and Debts, Production Function and Value Added and so on. Integral by macroeconomic density Demand over e-space defines Demand of entire macroeconomics. Thus (8.1-8.3) determine e-space densities of common macroeconomic variables. That allows replace relations (like regression or matrix models, etc.) between values of entire macroeconomics as Demand and Supply by relations between macroeconomic densities functions on e-space and use mathematical physics equations. E-space densities Uj(t,x) as Value and Capital, Supply and Demand play role similar to mass density function ρ(t,x) in physical kinetics (Landau & Lifshitz, 1981). Thus espace densities Uj(t,x) can be treated as origin of corresponding economic fluids (e-fluids). Let use (8.1-8.3) as tool to establish transition from approximation of economics as description of economic variables u=(u1,…ul) of separate economic agents (Economic Kinetics) to description of interaction between macroeconomic densities Uj=Uj(t,x) on espace (Economic Hydrodynamics) and develop economic models alike Hydrodynamics.

5.2. Economic Hydrodynamics Similar to microscopic and macroscopic description in physics, economics should be modeled by different approximations. Kinetic approximation studies dynamics and distributions of separate e-particles and their economic variables. Such description is similar to microscopic kinetic level in physics. Averaging of economic variables of separate eparticles at point x on e-space and description of densities functions of macroeconomic variables on e-space is similar to Hydrodynamics in physics. Transition from microscopic kinetic level to macroscopic description in physics requires certain roughness or averaging procedures. Such averaging procedures permit reduce number of degrees of freedom of initial problem (Kinetic approximation) and develop a model that operates with averaged variables (Hydrodynamic approximation). Similar procedures in macroeconomics allow move from description of Economic Kinetics as modeling of economic variables of many e-particles system on e-space to Economic Hydrodynamics as description of averaged economic densities on e-space. Such transition permits neglect behavior of separate e-particles, separate economic agents and their economic variables and describe macroeconomics as dynamics of economic fluids generated by economic densities on e-space Rn. Further let mention macroeconomic densities Uj(t,x) like 12

Value and Production Function, Demand and Supply as corresponding economic fluids (efluids). So, phenomenological

considerations lead to conclusion that

Economic

Hydrodynamics can describe large number of e-fluids generated by extensive economic variables of e-particles on e-space. Numerous of interacting e-fluids terribly increase complexity of Economic Hydrodynamics in comparison with physical Hydrodynamics. Below we present simple models of Economic Hydrodynamics. Parallels between physical and economic densities permit obtain equations on e-fluids that are alike to Continuity Equation and Equation Of Motion for physical fluids (Landau and Lifshitz, 1987). We present phenomenological derivation of e-fluid equations and underline differences between physics and economics that make e-fluid dynamics much more complex. Continuity Equation on economic density Ui(t,x), i=1,..l takes form 𝜕

𝜕𝑡

+

𝒗

=𝑄

(9.1)

υ(t,x) - velocity of economic density Ui(t,x) (8.3). Left side describes flux of economic density Ui(t,x) through unit volume surface and right hand side Q1 describe factors that induce changes of density Ui(t,x). Contrary to physics, Continuity Equation on economic density does not conform values or even integral of economic density over e-space. Economic density can increase or decrease due to movements of the selected volume on espace or due to various economic processes. Equation of Motion takes form 𝜕𝒗

[ 𝜕𝑡 + 𝒗 ∙ ∇ 𝒗] = 𝑄

(9.2)

Left side (9.2) describes flux of Ui(t,x)υ(t,x) through unit volume surface, taking into account equation (9.1). Right hand side describes factor Q2 that induce changes of Ui(t,x)υ(t,x) flux. To derive closed form equations (9.1-9.2) let make additional assumptions. As a first approximation let assume that extensive economic variables of different e-particles do not interact with each other. For example, let assume that Production Function of e-particle does not depend on Production Function of other e-particles, but can be determined by Labor, Capital, Demand, Market Share, Investments and so on. Let denote economic variables that induce changes of particular economic variable as Conjugate Variables. For example, Production Function can have Conjugate Variables like Capital, Labor, etc. Our assumptions mean that there are no interactions between same economic variables: Demand of e-particle does not depend on Demand of different e-particle. Physics analogy of absence of any interaction between same economic variables can be treated as lack of any interactions and collisions between gas particles and collisions with “walls”. That allows neglect economic

13

analogies of physical variables like pressure and viscosity and avoid here treatment of economic parallels to thermodynamic relations. To derive closed form for equations (9.1-9.2) let assume that Q1 and Q2 describe action of Conjugate economic fluids only. That makes Economic

Hydrodynamics

equations

(9.1-9.2)

similar

to

complex

multi-fluid

Hydrodynamics.

5.3. Economic wave equations Wave generation, propagation and interaction determines wide diversity of observed physical phenomena’s. Waves play significant role in physical phenomena’s from micro to macro scales. In economics and finance terms “waves” are used for Kondratieff waves, Inflation waves, Demographic waves and so on. All these issues describe time oscillations of economical variables, not wave propagation. Waves and wave propagation require space. Introduction of Economic Space notion permit derive wave equations on macroeconomic variables and study economic wave propagation processes. Existence of economic and financial wave processes on e-space permit develop Wave Economics that describes economic wave generation, propagation, interactions, amplification and dissipation that can be extremely important for macroeconomics and crises modeling, forecasting and managing. To demonstrate derivation of economic wave equations on e-space let present simple two conjugate e-fluids interaction model: Demand-Price model. Demand-Price relations are most familiar in economics. As usual it is assumed that Price grows up with Demand growth and Demand falls down as Price increases. Let derive wave equations for that system.

Two conjugate e-fluids: Demand-Price model. Demand is extensive variable and due to above consideration it is possible to derive equations on Demand e-spacial density. Price is intensive variable and does not generate espace density directly. Meanwhile Price is a coefficient that determines relation between Supply measured in currency units and Supply measured in physical units as “pieces” like tons, cars, shares, houses etc. As we study Demand-Price relations it is reasonable assume that Supply 2 in physical units is constant 2 = const., and Supply 2 in currency units depends on price p only 2=p2. In such simplified assumptions Demand and Supply can be treated as Conjugate extensive economic variables. Let repeat that Supply density function is proportional to Price due to relations 2=p2 , 2 = const. To describe Demand-Price model by equations (9.1-9.2) one should determine

14

interaction model between Conjugate e-fluids of Demand and Supply and define Q1 and Q2 factors. Let state that Q1 factor on the right hand side (9.1) for ρ1 Demand density Continuity Equation is proportional to ~ α2 2/t with negative α2 and Supply density 2. Right hand side (9.1) for 2 Supply density Continuity Equation is proportional to α1 1/t with positive α1. These assumptions model growth of Supply Price with growth of Demand and decrease of Demand with growth of Supply Price. To determine Q2 factor for Equation of Motion (9.2) let state that right hand side for Demand velocity υ1 equation is proportional to gradient of Supply density 2 with negative

20. Let state that Supply Price velocity grows in the direction of higher Demand and Demand velocity decreases in the direction with higher Supply Prices. For these assumptions two e-fluids system equations for Demand-Supply model with Supply e-fluid density being determined by Supply Price take form 𝜌

+

>

𝜌

𝒗𝜌 =

;




;

𝒗




Wave equations (10.2), or more correct bi-wave equations describe wave propagation with more complex behavior than common wave equations of second order. For example, Green function of equation (10.2) is determined as convolution of Green functions of two separate wave equations that establish (10.2) and thus response on δ-function source is more complex, that for second order wave equation. To make our models easier to be accepted let present one simple solution of above equations. Exponential growth of wave amplitudes of small disturbances Let outline that due to our assumptions Supply density is proportional to Price due to relations 2=p2, 2 = const. Thus Supply disturbances are proportional to Price disturbances and we can regard 2 as Price disturbances. Let take Price disturbances 2 as: 𝜎 = cos

∙𝒙−

exp

+𝒑∙𝒙

(11.1)

Here kx denote scalar product of vectors k and x. Let denote a=1- α1α2>1 ; b=α2

1

+α1

2

0 wave amplitude (11.1) grows up as exp( t). Thus wave propagation of small Price

disturbances 2 can propagate along e-space with exponential growth in time of its wave amplitude and that means exponential growth of Price fluctuations on e-space. Case 2. = 0. Then system (11.2) takes form: −

∙𝒑 +4

−

− [

−

−

∙𝒑 =

Hence from the second equation: −

=

>

𝑑

;

>

∙𝒑

=

The first equation takes form:

For (11.3)

+

=−

+4

𝑑 6𝑑

∙𝒑

𝑑+

−4

;

+

∙𝒑

𝑑

]=

= −4

∙𝒑

>

>

Let define  as angle between vectors k and p ∙𝒑=

;

Absolute values of vector

=

=

| | |cos 𝜃 |

√−

√

𝑑+

+ −

Wave amplitude (11.1) can exponentially grow up as exp(px). Vector p belongs to cone that has angle θ with wave vector k. Exponential growth of wave amplitude of Price perturbations in time and along vector p on espace describes growth of fluctuations of Price perturbations and that may reflect growth of crises trends in macroeconomics. Two conjugate e-fluids: Investments-Production model and vortex Q2 factor. Introduction of e-space and development of Economic Hydrodynamics permit describe economic variables and relations with help of vector fields on e-space. On e-space with dimension more than one, velocities of e-fluids can be potential or vortex vector fields. In previous paragraph we treated Q2 factor as potential field determined by gradient of the Conjugate density ~. Now we present Conjugate e-fluids model with vortex Q2 factor. Let regard model relations between Investments and Production.

17

Let assume that interaction between these Conjugate e-fluids has form of vortex vector field that is determined by rotor of e-fluids velocities. Let assume the following relations between e-fluids: Production density grows up with rise of Investments density. As Production growth saturates market Demand, than growth of Investments density goes down. Let propose that frequency of Investments turnover reflect the speed of growth of new production capacities and thus defines growth of Production. Let state that Q2 factor for Equation of Motion (9.2) for velocity υ1 of Production density is proportional to ~2rotυ2 of speed υ2 of Investments density with positive factor 2>0 and Equation of Motion takes form 𝒗

~

∙ 𝒗

On the other hand let assume that Q2 factor for Equation of Motion (9.2) for velocity of Investments density υ2 is proportional to ~1rot υ1 of Production density velocity υ1 on espace with negative factor

< . Indeed, rot υ1 can reflect the rate of saturation and thus

decrease the speed of Investments growth. 𝒗

~

𝒗

In other words: the higher is the frequency of circulating of produced commodities, the higher will be saturation of the market and that reduce speed of Investments. Our hypothesis on interaction of two Conjugate e-fluids in linear approximation on velocities leads to Equations of Motion: 𝜕𝒗

𝜕𝑡

=

𝒗 ;

𝜕𝒗

𝜕𝑡

=

𝒗 ;




;

;

= , ;

>

(12.1)

(12.2)

Equations (12.2) are common second order wave equations and describe wave propagation with speed c. We outline that wave behavior of vortex vector fields in linear approximation do not induce any perturbations of potential vector component and have no influence on efluids densities perturbations. It is obvious that for Investments – Production system vortex interaction model equations (12.1) are similar to Maxwell equations, but any further analogies are absent.

6. Conclusion

18

Economic and financial modeling problems are discussed for years. Each new economic crisis gives start for new debates on modeling methods and concepts. Econophysics studies during last decades present many contributions of theoretical and statistical physics methods for economic and financial modeling. These works enlarge formalism and schemes for financial modeling, enrich forecasting methods and develop advanced management for financial crises. Modern methods of mathematical physics help develop description and forecasting of financial processes, but their usage should be based on pure economic phenomenology and ground. With this in mind we introduce Economic Space notion as a definite analogy of physical space-time to enlarge economic and financial theory methods. We assume that risk ratings procedures that are widely used in economics and finance can become good background for Economic Space notion definition as an intrinsic economic notion. To demonstrate certain advantages for Economic Space usage we presented examples for financial and economic modeling on e-space. Our treatments of option pricing theory on n-dimensional e-space Rn allow derive extension for the BSM equation and uncover additional difficulties of option pricing modeling. These problems concern requirement to take into account possible changes of main risks that determines current e-space representation. Random dynamics of main risks during time to expiration means that options pricing equations (6) should be transformed into other ones on e-space with different axis. That effect might explain variances between predicted and observed option price dynamics. Correct description of these effects might rise up accuracy of option pricing. Introduction of e-space notions allows describe random walks of separate e-particles (economic agents) on e-space. Random change of e-space coordinates means random movements of risk grades and that has effect on economic variables of e-particle under consideration. Thus random Markov processes should jointly describe changes of e-space coordinates and economic variables of selected e-particle. The same happens with derivation of Fokker-Planck equation for distribution Ψ(t,x,u) that defines probability that particular eparticle has coordinates x=(x1,..xn) and that its l extensive economic variables equals u=(u1,..ul). Introduction of e-space permits apply random functions theories for financial and economic description and that definitely expand macroeconomic modeling capacities. The most attractive option that becomes available with introduction of e-space notion concern ability to study macroeconomic approximations that have certain analogies with physical Kinetics and Hydrodynamics. Such approach allows derive wave equations on 19

economic variables on e-space. Derivation of wave equations for simple models permits suppose existence of wide range of linear and nonlinear wave interactions processes that describe generation and amplification of economic and financial waves on e-space and may be cause of crises behavior. Thus, observation and modeling of macroeconomic wave processes on e-space might help for economic and financial crises studies, for taking correct investment decisions and managing sustainable monetary policy. E-space notion uncovers extreme internal complexity of financial and economic systems. Main difficulties concern econometric problems and observation, choice of most valuable risks and measurements of e-particles distributions on e-space. At present, there are no risk ratings methodologies that allow distribute economic agents on Rn. Development of econometric and economic statistics that may establish empirical ground for e-space modeling is the most crucial problem. Cooperative efforts of Rating Agencies and Businesses, Government Statistical Bureaus and Research Communities, Banks and Regulators, etc., are required to establish distributions of economic agents on e-space with one or two dimensions. It is necessary to unify risk ratings methodologies, develop methods to measure and compare different risks, methods to chose most valuable risks and so on. We propose that even simplest model of e-space distributions of macroeconomic variables can allow observe dynamics of economic variables, visualize macroeconomic distributions like Supply and Demand, describe e-space states of Inflation, Financial Markets, Derivatives Markets and more. A lot of problems should be solved to establish the econometric models on e-space and we hope that our research may give start for further studies.

Acknowledgement I am very thankful to my spouse Irina for her patience and support of my studies.

20

Appendix A. Derivation of Black-Scholes-Merton Equation on Economic Space Rn Option price V(t,x,a) depends on time t to expiration, e-space Rn coordinates x and Price a=a(t,x) of underlying stocks of certain e-particle. Let assume that stocks price a of eparticle follows regular move and Brownian walks dW(t)
= ;
=

c – is instantaneous rate of return on security, and 𝜎 – is instantaneous variance rate.

Operator denotes averaging procedure. Vector k describes action of e-space coordinates variation dx on change of e-particle stocks price. k·dx is a scalar product. Let assume that coordinates x of e-particle that defines underlying stocks price also follows regular move and Brownian walk dZ(t) on the n-dimensional e-space Rn: 𝒙= 𝒗

+ 𝒁

(A.2)

Vector υ defines regular velocity of e-particle on e-space. Each component of Brownian process dZ(t) obeys relations:
= ;
=

For simplicity let assume no correlation between
=

(A.4)

These assumptions lead to equation on option price V(t,x,a): =

+

+

+ 𝑥

𝑥

𝑥 +

𝑥 +

+

𝑥 +

𝑥

+

𝑥 𝑥 ;

𝑥 𝑥

𝑥

𝑥 +

, = ,…

The sum is taken over repeated indexes. Taking account of (A.1-A.4) receive: =

+

+

+

+ ∙

a

𝑥 +

+ 𝜎a

+a

a

+

+

+ 𝑥

+

∙

+ 𝜎a

+

a

+

+

+

+a

+ 𝑥 𝑥

+ ∙

+ 𝜎a 𝑥

𝑥

a

+

+

+

+a

+ 𝜎a

∙

+a

+

21

With accuracy dt obtain: =

+

Hence: =[

+

+a

+

+a

+a

∙

+ 𝜎a

∙

+

+

∙ 𝑥

]

+

𝑥

+ 𝜎a

+a

𝑥 +

(𝜎 + +

According to common “portfolio” function form 𝐵=

𝜕

t, x, a −

𝜕

−𝑥

𝑥

∙

+ 𝑥 )

+

𝑥

+

𝑥

𝜎

𝑥

𝜕

𝜕𝑥

equation on portfolio B does not depends on Brownian walks 𝐵=[

+

𝜎 +

∙

+

𝑥

+a

𝑥

]

Similar to standard derivation of the BSM equation for simplest case it is assumed that portfolio B is growing with riskless rate r: 𝐵= 𝐵

As a result extension of the BSM equation on n-dimensional e-space Rn takes form: 𝜕

𝜕𝑡

+

𝜕

𝜕

+ 𝑥

= (𝜎 +

𝜕

𝜕𝑥

∙

+

);

𝜕

𝜕

= ,…

+

𝜕

𝜕 𝜕𝑥

+

𝜕

𝜕𝑥

=

22

Appendix B. Derivation of Wave Equations On Economic Space We present derivation of wave equations for the first order linear approximation of disturbances of density and speed. It is assumed that velocities υ1,2 and density disturbances

1,2 are small. E-space densities 1,2 take form: 𝜌

=𝜌

,

,

+𝜎 , ; 𝜌

=

,

Continuity Equations on density disturbances 1,2 take form: 𝜎

+𝜌

𝜎

𝒗 =

;

= , = ; = , =

Equations of Motion for velocities υ1,2 take form: 𝜕𝒗

𝜌

𝜕𝑡

𝜌 =𝜌

=

> ;

𝜌

∇𝜎 ;

+𝜎 ;

< ;

𝒗𝒊 = −

= , ;

𝜕𝜎

𝜕𝑡

> ;

= , 𝜕𝜎

+

𝜕𝑡

Thus obtain equations 𝜎 −

=

𝜎

−

𝜎

+

< ;

; 𝜌 𝜎

𝛥𝜎 ;

Δ𝜎 = −

𝜕𝑑 𝑣 𝒗

=

𝜕𝑡

𝜎

=

Δ𝜎

Δ𝜎

−

𝛥𝜎 ;

Taking second derivative by t obtain: 𝜎

−

+

Δ

𝜎

=−

Equations on density disturbance: [

𝜕4

−

𝜕𝑡 4

+

𝜎

Δ[

𝜕

+

Δ 𝜕𝑡 −

−

Δ ] 𝜎

Δ𝜎 ] ,

=

(B.1)

To derive wave equations perform Fourier transform by time and coordinates or study the wave type solution for 1 = 1 (x-ct). Than equations take form −

+

+

−

Wave equations exist for positive roots c2 ,

=

−

−

>

+

+/−√ −

;


0

If factor under square root is positive, than two positive roots and two different wave propagation speeds

,

exist. Density disturbance equations (B.1) take form

23

𝜕

𝜕𝑡

−

Δ

𝜕

𝜕𝑡

−

Δ 𝜎

,

=

(B.2)

Equations (B.2) describe propagation of waves with two different speeds c1 and c2 and are called bi-wave equations. In case α1 and α2 equals zero, there are no wave equations. Density disturbance equations take form [

−

∆ ]𝜎

,

= ;