Multiple commodities in statistical microeconomics: Model and market

Physica A 462 (2016) 912–929

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Multiple commodities in statistical microeconomics: Model and market Belal E. Baaquie a,b , Miao Yu a,∗ , Xin Du a a

Department of Physics, National University of Singapore, 2 Science Drive 3, 117542, Singapore

b

INCEIF, The Global University of Islamic Finance, Lorong Universiti A, 59100 Kuala Lumpur, Malaysia

highlights • • • • •

The Empirical study of statistical pricing model is investigated. The Pricing of multiple commodities is made by the empirical model. The Pricing of single commodities is described by the empirical model. The Cross-Correlation of different commodities is determined by the model. The Auto-correlation of single commodities is studied by the model.

article

info

Article history: Received 21 September 2015 Received in revised form 23 May 2016 Available online 22 June 2016 Keywords: Path integral Statistical microeconomics Action functional Commodities Market

abstract A statistical generalization of microeconomics has been made in Baaquie (2013). In Baaquie et al. (2015), the market behavior of single commodities was analyzed and it was shown that market data provides strong support for the statistical microeconomic description of commodity prices. The case of multiple commodities is studied and a parsimonious generalization of the single commodity model is made for the multiple commodities case. Market data shows that the generalization can accurately model the simultaneous correlation functions of up to four commodities. To accurately model five or more commodities, further terms have to be included in the model. This study shows that the statistical microeconomics approach is a comprehensive and complete formulation of microeconomics, and which is independent to the mainstream formulation of microeconomics. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The theory of prices proposed in Ref. [1] is based on the concept of the action functional; the subsequent publication [2] provides strong empirical evidence in support of this formulation for the case of single commodities. The present paper extends the analysis to multi-commodities by modifying the single commodity model in a parsimonious manner. The theory of commodity prices [3] is one of the bedrocks of microeconomics and usually starts with the concept of the utility function of a typical consumer [4,5]. A maximization of the utility function with a budget constraint yields the demand for the commodities as a function of price. The supply function is obtained by maximizing the profit for the producers and the market prices of commodities in conventional microeconomics are fixed by equating supply with the demand [4,5].

∗

Corresponding author. E-mail addresses: [email protected] (B.E. Baaquie), [email protected] (M. Yu).

http://dx.doi.org/10.1016/j.physa.2016.06.102 0378-4371/© 2016 Elsevier B.V. All rights reserved.

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In contrast to conventional microeconomics, in statistical microeconomics [1] the prices of all commodities are taken to be intrinsically random—and the probability distribution function of prices is fixed by the exponential of the so called action functional. The action functional in turn is the sum of two parts, a ‘kinetic’ term that determines the dynamical evolution of commodity prices and a microeconomic potential that is the sum of the supply and demand functions. The action functional contains all the information of the market and determines the distribution of market prices as well as the change in market prices as the prices evolve in time [6–8]. The primary focus in the statistical microeconomic formulation is to describe the unequal time correlation functions of market prices. The auto- and cross-correlation [9,10] functions for multiple commodities is modeled using the action functional and the Feynman path integral. The action functional is calibrated by matching the prediction of the model’s correlation functions with the observed market and provides a stringent test of the accuracy of the model. The microeconomic potential for commodity with price p is given by V [p] and has been introduced in Ref. [1]; the potential has its minimum value at its extrema pˆ , given by

∂ V [ˆp]/∂ p = 0. The price pˆ is taken to be the average commodity price. What happens when the price p is not equal to the average price pˆ , that is, p ̸= pˆ ? The microeconomic potential V [p] in this case causes the prices to ‘move’, that is, to change and tend towards pˆ . Clearly, the more abrupt the change, the more unlikely it is; the change of price should, for normal market conditions, be gradual and relatively ‘smooth’. To achieve this smooth movement of the prices in general, a ‘kinetic term’ T [p(t )] is introduced. Although the concept of the kinetic term is taken from physics, it finds a natural expression in the evolution of the prices of commodities: the specific form of the kinetic term is determined by the study of market data. The kinetic term in the action functional is seen to be strongly supported by market data, and as of now has no clear theoretical explanation. One can only speculate that demand and supply are determined by consumers and producers, respectively and that the kinetic term reflects the process of circulation, distribution and exchange – as well as the degree of market liquidity – that is necessary for the products to make a transition from the producer to the final consumer in the market. One rather unexpected result is that the kinetic term in the action functional has a dominant role in the evolution of commodity prices; due to the high time derivative of prices in the kinetic term, the short term evolution of commodity prices is completely dominated by the kinetic term, with the microeconomic potential, containing the supply and demand functions, that come into play for the long term evolution. 2. The microeconomic action functional Consider N commodities, with market prices given by pI ; I = 1, . . . , N. Prices are always positive and can be represented by exponential variables as pI = p0 exI ; the normalized logarithm of prices, denoted by xI , is defined as follows xI (t ) = ln(pI (t )/p0I );

pI = p0I exI ;

I = 1, . . . , N .

The demand function and the supply function are modeled to be [1]

D [p] =

N 

d˜ i p0i e−˜ai xi ;

S [p] =

i=1

N 

˜

s˜i p0i ebi xi ;

d˜ i , s˜i > 0; a˜ , b˜ > 0.

(1)

i=1

The coefficients d˜ i , s˜i , according to Ref. [1], are determined by macroeconomic factors such as interest rates, unemployment, inflation and so on. For the purpose of modeling, prices in statistical microeconomics are expressed in terms of variables that are measured from the average value and normalized by the volatility of the stock. yi (t ) =

xi (t ) − x¯ i

σi

;

i = 1, . . . , N

(2)

x¯ I and σI are the average value of yI . The volatility of xI (t ) for the time period being considered and are given by

 2 σi2 = E [ xi − x¯ i ].

x¯ i = E [xi ];

The normalized variables yi are all of O(1) and hence one can model and compare commodities with vastly different volatilities and prices. In the statistical microeconomic approach, the microeconomic potential is the fundamental quantity that combines supply and demand by considering their sum [1]. The supply and demand yield the microeconomic potential given by

V =

N  

d˜ i p0i ea˜ i x¯ i e−˜ai σi yi + s˜i p0i e−bi x¯ i ebi σi yi ˜

˜



i=1

≡

N  

di e−ai yi + si ebi yi

i=1



(3)

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where ˜

ai = a˜ i σi ;

si = s˜i p0i e−bi x¯ i ;

di = d˜ i p0i ea˜ i x¯ i ;

bi = b˜ i σi .

For the case of multiple commodities, the microeconomic potential for the N-commodities is further generalized by including a term that depends on the product of the prices of commodities—and which cannot be placed either in the demand or in the supply component of the microeconomic potential. The multiple commodity microeconomic potential is given by

V [p] = D [p] + S [p] + correlation term

=

N  

di e−ai yi + si ebi yi +



i =1

N 1

2 ij;i̸=j

∆ij yi yj .

(4)

The ∆ij term is introduced to model the cross-correlation of the different commodities. The motivation for the ∆ij term is the following. The fit for the single commodity using the microeconomic potential is very accurate [2]. Hence, one would expect that the effect of multiple commodities should be a perturbation on the single commodities potential. This is the reason that the simplest modification of the single commodity microeconomic potential is used for modeling multiple commodities, and for consistency we expect ∆ij to be small. The dynamics of the prices for N-commodities is determined by the kinetic term T [p(t )] that, in general, is given by

T [p(t )] =

N 1

2 i,j=1

 Lij

 ∂ 2 yi ∂ 2 yj ∂ yi ∂ yj + β . ij ∂t2 ∂t2 ∂t ∂t

Similar to the reason that led to modeling the cross-correlations by the ∆ij term in the microeconomic potential V , we continue to model the kinetic term to be solely determined by the single commodity, with all the correlation coming from the ∆ij term. Hence, the kinetic term is chosen to be diagonal, with no cross-terms amongst the different commodities and is given by

T [p(t )] =

N 1

2

  Li

i

2

∂ 2 yi ∂t2

+ L˜ i



∂ yi ∂t

2 

.

(5)

The Lagrangian is given by the sum of the kinetic and potential factors and yields [1]

L(t ) = T [p(t )] + V [p(t )]. The Lagrangian, from Eqs. (4) and (5), is the following N 1

L(t ) =

2

  Li

i

∂ 2 yi ∂t2

2

+ L˜ i



∂ yi ∂t

2  +

N  

di e−ai yi + si ebi yi −



i=1

N 1

2 ij;i̸=j

∆ij yi yj .

(6)

The Lagrangian given in Eq. (6) is nonlinear. The action functional determines the dynamics (time evolution) of market prices and is given by



+∞

A[p] =

dt L(t ) =

−∞



+∞





dt T [p(t )] + V [p(t )] .

−∞

All prices of commodities are considered to be stochastic variables and the action functional is assumed to determine the probability distribution, which is given by Probability distribution for a specific time evolution ∝ e−A[y] . All correlation functions of the prices are given by the Feynman path integral [1,11] D123...n (t1 , t2 , . . . , tn ) = E [y1 (t1 )y2 (t2 ) · · · yn (tn )] = with

 Z =

Dye−A[y] .

1 Z



Dye−A[y] y1 (t1 )y2 (t2 ) · · · yn (tn )

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3. Correlation function We study the leading terms in the Lagrangian by doing a Taylor expansion of the potential term V about its minima, which will turn out to coincide with an expansion of V in a power series in yi . The minima xˆ i is defined by

∂ V (ˆx) = 0. ∂ xi Hence from Eq. (4)

 ∂ V (ˆx) ˜ ∆ij = −a˜i d˜i p0i ea˜i xˆ i + b˜i s˜i p0i ebi xˆ i − ∂ xi j,i̸=j



xˆ j − x¯ j

σj



= 0.

(7)

In our model, we assume that the equilibrium price of the commodities xˆ is given by its average value x¯ and yields xˆ i = x¯ i .

(8)

Hence, from Eq. (7) ˜ − a˜i d˜ i ea˜i x¯ i + b˜i s˜i ebi x¯ i = 0.

(9)

Note that Eq. (9) is independent of p0i and hence p0i does not enter the calibration of the model’s parameters. Eq. (9) yields

 x¯ i

e =

a˜ i d˜ i

(1/(˜ai +b˜ i )) .

b˜ i s˜i

(10)

Eqs. (3), (8) and (10) yield ai di = bi si .

(11)

3.1. Expansion of potential From the definition of yi given in Eq. (2), the minima of the action is about yi = 0. Hence, expanding the Lagrangian about yi = 0 yields

L=

 1 γi 2 αi 3 βi 4 1 2 2  ¨ ˙ ∆ij yi yj . Li yi + Li yi + yi + yi + yi + · · · − 2 2 2 3! 4! 2 ij,i̸=j

 1 i

Define the Lagrangian in terms of the quadratic and nonlinear terms as follows

L = L2 + L3 + L4 + O(y5 ) L2 (x) are the quadratic terms in the expansion of the Lagrangian given above and L3 (x), L4 (x) are the cubic and quartic terms. The quadratic Lagrangian is given by L2 = L0 + Lc L0 =

1  2



Li y¨i 2 +  Li y˙i 2 + γi y2i ;

i

and the nonlinear terms are

L3 =

αi 3 yi ; 3!

L4 =

βi 4 yi . 4!

The action is given by the following

 A = A0 + Ac + AI =

 A0 = AI =

 dt L0 ;



dt L

Ac =

dt (L3 + L4 ).

dt Lc

Lc = −

1 2 ij;i̸=j

∆ij yi yj

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From above we have

γi =

1 2

(di a2i + si b2i )

(12)

αi = (−a3i di + b3i si ) = (bi − ai )γi

(13)

βi = (a4i di + b4i si ) = (a2i − ai bi + b2i )γi .

(14)

The linear term in yi is zero due to Eq. (11). We will determine the values of α, β, γ , y¯ from market data; the potential parameter of ai , bi , si , di is then given by the following ai =

 ± 4βi γi − 3αi2 − αi

2γi γi si = ; b i ( ai + b i )

;

bi = ai +

γi . ai (ai + bi ) The positive branch for ai is used since ai > 0.

αi γi

di =

3.2. Auto-correlation The correlation function for the A0 is given by the Gaussian propagator (0)

D

(t − t ) = ′

1



Z

Dye−A0 [y] yI (t )yJ (t ′ )

and the auto-correlation function is given by (0)

(0)

DII (t − t ′ ) ≡ DI (t − t ′ ) =

1



Z

Dye−A0 [y] yI (t )yI (t ′ ) + O(∆2 ).

Using a Fourier transform to evaluate the propagator for the prices, and dropping the subscript I, yields D(0) (t − t ′ ) ≡



∞

dk

e

ik(t −t ′ )

4 Lk2 + γ −∞ 2π Lk + 

=

√ a− |t −t ′ | e− √ a−

−

√ a+ |t −t ′ | e− √ a+

2L(a+ − a− )

;

    4Lγ L . a = ±  1−   2L  2L L2 ±

 L

Case I: Real branch. 4Lγ <  L2 and a± is real; let

ω=

 γ  14 L

,

a± =



γ L

 e±2ϑ ,

e± 2 ϑ =

 L2 4Lγ

 +

 L2 4Lγ

− 1.

Hence D(0) (t − t ′ ) is given by

ωe−ω|t −t | cosh(ϑ) sinh[ϑ + ω|t − t ′ | sinh(ϑ)]. 2γ sinh(2ϑ) Case II: Complex branch. 4Lγ >  L2 and a± are complex; let     γ  14 2   γ L L2 ω= , a± = e±i2φ , cos(2φ) = , sin(2φ) = 1 − . L L 4Lγ 4γ L ′

D(0) (t − t ′ ) =

(15)

We hence obtain the complex branch propagator

ωe−ω|t −t | cos(φ) sin[φ + ω|t − t ′ | sin(φ)]. 2γ sin(2φ) ′

D(0) (t − t ′ ) =

Define the normalization constant

N =

ω

2γ sin 2φ

and yields the complex branch propagator ′ D(0) (t − t ′ ) = N e−ω|t −t | cos(φ) sin{φ + ω|t − t ′ | sin(φ)}.

(16)

The auto-correlation function of commodities will be seen to follow the behavior given by the complex branch. The real branch cannot describe the data from market.

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Fig. 1. α = 0.1, β = 0.15, φ = 30, θ = 20.

3.3. Cross-correlation The cross-correlation function is given by I ̸= J. The model yields E [yI (0)yJ (τ )] = DIJ (t ) = 1

=



Z



1 Z

Dye−(A0 +Ac ) yI (0)yJ (τ )



Dye−A0 [y] yI (0)yJ (τ ) 1 +

1 2 ij;i̸=j

∆ij





dtyi (t )yj (t ) + O(∆2 ) .

The first term is zero and hence (0)

DIJ (τ ) ≃ DIJ (t )

(17)

where (0)

DIJ (t ) ≡



∞

(0)

(0)

DI (t )DJ (t − τ )dt . −∞

From Appendix Eq. (28): (0)

DIJ (t ) =



C LI LJ

1 −|t |α cos φ e



α

1 + e−|t |β cos θ

β



1 P

1 R

˜ − {(h1 /R) cos φ˜ − (h2 /R) sin φ}

˜ − {(h5 /P ) cos θ˜ − (h6 /P ) sin θ}

1 Q

1 T

˜ {(h3 /T ) cos φ˜ − (h4 /T ) sin φ}

˜ {(h7 /Q ) cos θ˜ − (h8 /Q ) sin θ}



 (18)

with C =

−1

1

4 α 2 β 2 sin 2φ sin 2θ

φ˜ = φ + |t |α sin φ;

;

θ˜ = θ + |t |β sin θ .

Coefficients h1 − h8, P , Q , R, T are given in Eq. (31). Figs. 1 and 2 are plots of the cross-correlator for some typical values of the model’s parameter of the complex branch. The shape of the cross-correlator given by the model will be seen to be consistent with the result obtained by fitting the model to market prices. 3.4. Nonlinear terms As discussed in detail in Ref. [2], the correlation function to leading order for the nonlinear coupling yields (0)

E [y2I (t )]c = DI (0) − E [y3I (t )]c = −2αI

∞



βI 2

(0)

DI (0)



(0)

dz (DI (z ))2 + O(∆2 )

(0)

dz (DI (z ))2 + O(∆2 )

(19) (20)

0

(0)

E [y4I (t )]c = 3(DI (0))2 − 2βI

∞



(0)

dz (DI (z ))4 + O(∆2 ). 0

(21)

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Fig. 2. α = 0.1, β = 0.15, φ = 20, θ = 20.

Some integrations that are useful to solve the potential parameters a, b, s, d are the following [11] ∞



D(0) (τ )dτ = N

sin 2φ

0

∞



(D(0) (τ ))3 dτ = N

ω

,

∞



∞

(D(0) (τ ))4 dτ = N

sec φ − cos 3φ 4ω

0

3 3 2 sin φ(11 cos φ + 2 cos 3φ)

4ω

0



(D(0) (τ ))2 dτ = N 2

3 4 sin φ (50 cos 2φ + 6 cos 4φ + 47) tan φ

0

16ω(3 cos 2φ + 5)

.

Using four Eqs. (11), (19)–(21), potential parameters ai , bi , si , di can be obtained. 4. Market data and model The empirical correlator is denoted by the notation of GIJ (t ) and is defined by the expectation value of the market prices. For a time series data set with time interval of ϵ , the prices are given by yI (t ) = yI (n), where t = nϵ and we have τ = kϵ ; for N data points, the correlator is given by the moving average

  GIJ (τ ) = GIJ (k) = E [yI (0)yI (k)]c 

= market

N −k 1 

N n =0

yI (n)yJ (n + k).

The numerical evaluation of the correlators is obtained by taking the moving average over the data set. The model always yields a correlator that is a symmetric function of IJ, which is not necessarily the case for the market correlator [1]. To equate the market correlator with the model, it is made symmetrically, namely GIJ (τ ) = GJI (τ ) and in terms of the underlying data we have GIJ (τ ) =

1 2



N −k 1 

N n =0

yI (n)yJ (n + k) +

N −k 1 

N n =0

 yJ (n)yI (n + k) .

The parameter of time for the market and model are not the same. The reason being that time for traders is determined by the liquidity of the market and rate of transactions [12]. To reflect this feature of the market, define market time z (τ ) by

τ → z (τ ). The empirical correlator G(τ ) is given by the exact model correlator D(τ ) by the relation GIJ (t ) = DIJ (z (t )). For single commodity fit (0)

DII (z (t )) = N e−ωI z (t ) cos(φI ) sin{φI + ωI z (t ) sin(φI )}. For cross-correlation DIJ , we use Eq. (18) to fit the market cross-correlator.

(22)

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Table 1 Number and type of commodity. Number

Commodity

Type

Number

Commodity

Type

1 2 3 4 5 6 7 8 9

Crudeoil Heatingoil Brentoil Natural gas Copper Gold Silver Platinum Palladium

Energy

10 11 12 13 14 15 16 17 18

Cocoa Soybeansoil Orangejuice Livecattle Wheat Corn Soybean Roughrice Cotton

Food

Metal

Grain

Misc

According to Ref. [2], for single commodity fit, using GII (t ) ≡ GI (t ), we have (0)

GI (t ) = DI (z (t )) − Let



βI 2

(0)

DI (0)



(0)

(0)

dτ DI (z (t ) − τ )(DI (τ )) + O(∆2 ).

(23)

(0)

dτ (DI (τ ))2 = CI . When τ equals to zero, (0)

GI (0) ≃ DI (0) −

βI 2

(0)

(0)



DI (0)CI = DI (0) 1 −

βI 2

 CI

.

(24)

For the auto-correlation GII , note that the empirical definition of x¯ and σ implies that GI (0) = 1. Hence (0)

1 = GI (0) = DI (0) −

βI 2

(0)

(0)

DI (0)CI ⇒ DI (0) =

1 1−

βI 2

CI

.

(25)

(0)

Eq. (25) is consistent with Eq. (24). Substituting DI (0) into GI (t ), for t > 0 – using Eq. (16) for the numerator and denominator – we obtain1 GI (t ) = (0)

(0)

DI (z (t )) (0)

DI (0)

=

1 sin φ

e−ωz (t ) cos(φ) sin{φ + ωz (t ) sin(φ)}.

(26)

(0)

We use DI (z (t ))/DI (0) to fit GI (t ) in the single commodity case. Hence, once we have obtained φI , ωI all the parameters of the complex branch of the model can be determined. 5. Fitting with market data As a rule, the correlators evaluated from the data are denoted by GIJ and the result obtained by fitting the model are denoted by DIJ . The numbering for the indices I , J for the various commodities is the one given in Table 1. We analyze 18 commodities daily data drawn from four major groups of energy, metal, food, grain, from 2014/01/01 to 2015/02/03 download from Investing.com/commodities/real-time-futures.2 Since the correlators are symmetric, there are 153 correlators in total, with 18 auto-correlation functions and 153 cross-correlation functions. All the auto-correlation functions can be fit to a high degree of accuracy and confirms the results found in Ref. [2] for single commodities. The model can fit the majority of the cross-correlators Gij (i ̸= j) which are generically similar to the shape that the model generates from Eq. (18) and shown in Figs. 1 and 2. Of the 153 cross-correlators, 110 is of the shape that the model can fit quite well. The rest of the cross-correlators Gij have features that the model cannot fit; in particular, if the crosscorrelator has a maximum value at a time lag that is non-zero, then there are no choice of parameters for Gij that can fit the cross-correlator. The fitting is based on Eqs. (17) and (22) GIJ (τ ) = DIJ (z (τ ));

I ̸= J .

1 The approximation makes the result consistent with the value of G (0) = 1. I 2 Real-time streaming quotes for the top commodities futures CFDs. The quotes are available for a variety of futures such as Gold, Crude Oil, Silver, Copper and many more Metals, Energies and Softs futures. The latest price as well as the daily high, low and the change for each future. The ‘‘Base’’ price is the last close of each future contract (as of 16:30 ET). The change is calculated from the ‘‘Base’’ price.

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Fig. 3. Matrix of ∆ij for 18 commodities. Note that for but one case |∆IJ | < 0.08.

(0)

To leading order we approximate DIJ (t ) by the Gaussian approximation DIJ (t ) given in the Appendix and obtain (0)

GIJ (τ ) ≃ DIJ (z (τ )) = ∆IJ



∞

(0)

(0)

dtDI (t )DJ (t − z (τ ));

I ̸= J .

−∞

Hence, the cross-correlation coupling ∆IJ is given by GIJ (0) = ∆IJ

∞



(0)

(0)

dtDI (t )DJ (t );

I ̸= J .

(27)

−∞

The empirical cross-correlation of 18 commodities has been studied. Anticipating results derived later on in the paper, the parameters ∆IJ are given in Fig. 3 for all the cross-correlators. The fitting of market correlator to the model is done in the following.

• Simultaneously fit all the auto-correlation functions DII (z (t )) = DI (z (t )) with the G(t ), as in Eq. (26), and evaluate the parameters ω, φ, η, λ. The value of η, λ does not enter into the calibration of the other parameters. • Using the values of ω, φ , as in Eq. (26), and the nonlinear terms given Eqs. (19)–(21), we evaluate L, L˜ , γ , α and β . • Use the parameters L, L˜ , γ , α and β as in Eqs. (11)–(14) and find [a, b, s, d]. • Evaluate the cross-correlation function DIJ ; I ̸= J and determine ∆IJ . • The correlators are fitted for a maximum time of lag τ = 200 days. Note the remarkable fact, that ignoring one cross-correlator, the value of all the ∆IJ ’s is such that |∆IJ | < 0.08. The fact |∆IJ | < 0.08 provides strong evidence of the correctness of our approach of considering the multi-commodities model as a perturbation of the single commodities, which requires |∆IJ | ≪ 1. Note in extending the statistical model from single to N commodities, N (N − 1)/2 new parameters ∆IJ are introduced, which in turn are fixed by only a single value of the cross-correlators GIJ (0), as given in Eq. (27). For our model, the entire dependence of GIJ (τ ) for time lag τ > 0 is determined by the auto-correlation functions (0) DI (z (τ )) and its convolution with itself—as given in Eq. (17). The fact that the model can accurately describe all the autoand cross-correlators, up to N = 4 commodities, provides evidence for the correctness of the model. We will discuss later how the model can be extended to accurately describe a collection of commodities with arbitrary N. 5.0.1. Calibration for cocoa 1. Use cross-correlation Eq. (18) and auto-correlation Eq. (16) to fit each data to find ω φ , λ, and η. 2. Substitute ω, φ , λ, and η into Eqs. (19)–(21) and (15) to get γ , L, L˜ , αI , βI . 3. Using the value of all parameters obtained above as in Eqs. (11), (12), (13), (14), we can determine the indices [a, b, s, d] [2] of supply and demand function. The result obtained in Table 6 for cocoa from the fitting of the Crude oil, Platinum and Cocoa group is used below to illustrate the fitting procedure.

ω = 0.1535, ⇒ γ = 0.113,

φ = 1.2229, L = 203.37,

λ = 0.700, L˜ = −7.35,

η = 0.54 α = 0.0199,

β = 0.0747

B.E. Baaquie et al. / Physica A 462 (2016) 912–929

921

Fig. 4. Silver and Gold with η = 0.7; λ = 0.1004. Table 2 Gold–Silver. η = 0.7; λ = 0.1004. Two commodities fit

R2

γ

L

 L

α

β

Gold (G11) Silver (G22) G12

0.837 0.863

0.136 0.126

20.81 37.13

−1.52 −2.32

0.0102 −0.0320

0.257 0.230

∆12 = 0.034

 ∞  ∞ ⇒ dτ G(τ ) = 4.4413; dτ G2 (τ ) = 6.9973; 0 0  ∞  ∞ dτ G3 (τ ) = 4.5490; dτ G4 (τ ) = 4.2909 0

0

⇒ a = 0.72,

b = 0.90,

s = 0.094,

d = 0.075.

6. Fits for GII , GIJ All the fits of the model with data are carried out using the equations for the correlation functions given in Eqs. (16) and (18). All auto-correlators are normalized such that GII (0) = DII (0) = 1. However, for displaying the results clearly, the entire auto-correlator is rescaled – such that sometimes we scale to GII (0) > 1 or GII (0) < 1 – so that the results for the different commodities do not overlap and can be viewed clearly. Similarly, the cross-correlators are also scaled and for four or more commodities are also shifted to avoid an overlap of the graphs. 6.1. Two commodities Any two commodities, from different types as given in Table 1, can be fitted to a high degree of accuracy. The fit is even better if the two commodities belong to the same type. Fig. 4 shows the fit for Gold and Silver. The parameters from the auto-correlator are the following (see Table 2) The three correlators for gold and silver are fitted well, with R2 given by the following R2 =



R2G11

∗

R2G12 R2G22



 =

0.837

∗

0.923 . 0.863



6.2. Three commodities fit We study commodities from the same group and from different groups as well. 6.2.1. Three commodities in same group Figs. 5–8 for three commodities in one group are given below.

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Fig. 5. Crude oil–Heating oil–Brent oil (a) Autocorrelation and (b) Crosscorrelation with η = 0.7; λ = 0.775.

Fig. 6. Orange juice–Cattle–Soybean (a) Autocorrelation and (b) Crosscorrelation with η = 0.7; λ = 1.132.

Fig. 7. Gold–Silver–Platinum (a) Autocorrelation and (b) Crosscorrelation with η = 0.7; λ = 0.344.

B.E. Baaquie et al. / Physica A 462 (2016) 912–929

923

Fig. 8. Crude oil–Platinum–Cocoa (a) Autocorrelation and (b) Crosscorrelation with η = 0.70; λ = 0.54. Table 3 Crude oil–Heating oil–Brent oil. η = 0.7; λ = 0.775. Three commodities fit

R2

γ

L

 L

α

β

Crude oil (G11) Heating oil (G22) Brent oil (G33) GIJ

0.804 0.797 0.798

0.0539 0.0537 0.0536

171.4 158.6 167.4

−1.434 −1.182 −1.329

0.0613 0.0677 0.0638

0.0813 0.0688 0.0738

∆12 = 0.032

∆13 = 0.031

∆23 = 0.032

Table 4 Orangejuice–Cattle–Soybean. η = 0.7; λ = 1.132. Three commodities fit

R2

γ

L

 L

α

β

Orange juice (G11) Cattle (G22) Soybean (G33) GIJ

0.609 0.733 0.685

0.0512 0.0477 0.0425

116.06 236.95 225.14

0.0032

−0.0049

−1.48 −0.308

0.0290 −0.0576

0.0057 0.0776 0.0595

∆12 = −0.030

∆13 = 0.021

∆23 = −0.019

Table 5 Gold–Silver–Platinum. η = 0.7; λ = 0.344. Three commodities fit

R2

γ

L

 L

α

β

Gold (G11) Silver (G22) Platinum (G33) GIJ

0.827 0.803 0.799

0.0908 0.0752 0.0726

58.6 88.1 112.1

−1.863 −4.17 −2.26

0.0071 −0.0217 −0.0320

0.179 0.159 0.180

∆12 = 0.033

Crude oil–Heating oil–Brent oil (see Table 3):

 2

0.804

∗ ∗

R =

0.918 0.797

∗

0.921 0.923 . 0.798



Orange juice–Cattle–Soybean (see Table 4):

 2

0.609

∗ ∗

R =

0.875 0.733

∗

0.727 0.781 . 0.685



Gold–Silver–Platinum (see Table 5):

 R2 =

0.827

∗ ∗

0.931 0.803

∗

0.910 0.895 . 0.799



∆13 = 0.031

∆23 = 0.025

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B.E. Baaquie et al. / Physica A 462 (2016) 912–929

Fig. 9. Gold–Silver–Crude oil–Natural gas (a) Autocorrelation and (b) Crosscorrelation with η = 0.70; λ = 0.260. Table 6 Crude oil–Platinum–Cocoa. η = 0.70; λ = 0.54. Three commodities fit

R2

γ

L

 L

α

β

Crude oil (G11) Platinum (G22) Cocoa (G33) GIJ

0.871 0.835 0.920

0.0560 0.0585 0.113

286.3 305.2 203.4

−3.54 −4.17 −7.35

0.0566

−0.0241

0.0742 0.1346 0.0747

∆12 = 0.021

∆13 = 0.045

∆23 = 0.023

0.0199

Table 7 Gold–Silver–Crude oil–Natural gas. η = 0.70; λ = 0.260. Four commodities fit

R2

γ

L

 L

α

β

Gold (G11) Silver (G22) Crude oil (G33) Natural gas (G44)

0.834 0.827 0.749 0.377

0.0973 0.0835 0.0769 0.0667

54.9 105.2 35.3 81.8

−2.051 −2.940 −0.041 −0.921

0.0074 −0.0220 0.0962 0.0850

0.186 0.159 0.128 0.015

6.2.2. Three commodities from different groups Crude oil–Platinum–Cocoa:

 2

0.871

∗ ∗

R =

0.612 0.835

∗

0.920 0.943 . 0.920



The R2 of three commodities fit are normally between 0.8 and 1. Although they are not quite high, the values are high enough to be convincing.

6.3. Four commodities When we consider taking more commodities into the fit, such as 4 and 6 commodities, the fit is not so good (see Figs. 9 and 10).



0.834

 ∗ ∗ ∗

R2 = 

0.885 0.827

∗ ∗

0.507 0.725 0.749

∗

0.074 0.435 . 0.809 0.377



As can be seen from Table 7, the R2 of four commodities fit goes down to 0.7–0.9 and for some is in the range 0.4–0.7. So the fit seems to fail to give a good result when we consider four commodities.

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925

Fig. 10. Gold–Silver–Crude oil–Natural gas–Soybean oil–Cattle (a) Autocorrelation and (b) Crosscorrelation with η = 0.70; λ = 0.699. Table 8 Gold–Silver–Crude oil–Natural gas–Soybean oil–Cattle. η = 0.70; λ = 0.699. Six commodities fit

R2

γ

L

 L

α

β

Gold (G11) Silver (G22) Crude oil (G44) Natural gas (G44) Soybean oil (G55) Cattle (G66)

0.825 0.566 0.512 0.443 0.447 0.447

0.0794 0.0637 0.0863 0.0520 0.116 0.198

92.02 15.36 6.09 185.1 2.51 2.270

−2.259

0.0062 −0.0269 0.1351 0.0654 −0.1718 0.0987

0.155 0.203 0.184 0.0118 0.268 0.2717

1.948 1.448 −1.394 1.078 1.050

6.4. Six commodities

0.825  ∗  ∗ R =  ∗  ∗ ∗ 2

0.935 0.566

∗ ∗ ∗ ∗

0.281 0.573 0.512

∗ ∗ ∗

0.011 0.483 0.789 0.443

∗ ∗

0.265 0.291 0 0 0.447

∗

0.727 0.701 0.033 . 0.064  0 0.447



The R2 of six commodities fit cannot be considered to be a good fit (see Table 8). 7. Comparison of single and multiple commodities fit There are two different procedures for doing the fit for a given commodity: by using the single commodity data and the other by doing a fit for multiple commodities. The purpose of this exercise is to ascertain how much do the prices of other commodities affect a given commodity’s prices. We also need to verify that the multi-commodity prices are in fact a perturbation to the single commodity fit of the prices of a given commodity.

• A given commodity is calibrated using only the data for the given commodity. This is denoted by S-commodity in Table 9. • The same commodity is calibrated by doing a fit for the commodities for all the groups and maximizing the goodness of fit for this collection of commodities, including choosing the best value for the cross-correlation parameters ∆ij ’s. This fit is denoted by M-commodity in Table 9.

• In Table 9, Group A is a simultaneous fit for three commodities, with two commodities from metal and the other from energy. In Group B of Table 9, three commodities from different groups, namely one from energy, one from metal and one from food are calibrated using our model. • The single commodity fit has an R2 above 0.94 whereas when the multiple commodity fit is done, R2 is still good but now about 0.80. Most of the parameters between single commodity fit and multiple commodities fit are similar. The potential constants a, b, s, d given in Eq. (1), as well as the market time parameters η, λ change by about 10% in going from the single to the three commodities fit. This verifies our intuition that the price of a single commodity is mostly, up to 90%, determined by its own dynamics, with the other commodities being a perturbation on its prices. Hence, while it is true that single commodity

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Table 9 Comparison of Single-Commodity fit(S-) with Multiple-Commodities fit(M-). Group 1 is Gold–Silver–Crude oil (GSC) and Group 2 is Crude Oil–Platinum–Cocoa (CPC). Group1 GSC

R2

γ

L

 L

α

β

η

λ

a

b

s

d

M-Gold S-Gold M-Silver S-Silver M-Crudeoil S-Crudeoil

0.78 0.95 0.83 0.94 0.80 0.99

0.076 0.056 0.063 0.056 0.060 0.058

111.7 22.6 59.7 168.1 350.1 144.6

−2.16 2.24 −2.30 −1.68 −4.97 −1.45

0.056 0.041 0.006 −0.021 −0.018 0.020

0.073 0.059 0.146 0.127 0.131 0.107

0.7 0.7 0.7 0.7 0.7 0.72

0.41 0.83 0.41 0.1 0.41 0.1

0.38 0.44 1.5 1.7 1.5 1.2

1.1 1.2 1.6 1.3 1.5 1.5

0.046 0.030 0.013 0.015 0.014 0.015

0.14 0.08 0.014 0.012 0.013 0.019

CPC

R2

γ

L

 L

α

β

η

λ

a

b

s

d

M-Crudeoil M-Platinum S-Platinum M-Cocoa S-Cocoa

0.75 0.90 0.94 0.83 0.94

0.056 0.059 0.056 0.11 0.061

286.3 305.2 167.7 203.4 193.4

−3.54 −4.17 −1.663 −7.35 −2.801

0.057 −0.024 −0.043 0.020 0.016

0.074 0.135 0.098 0.075 0.062

0.7 0.7 0.7 0.7 0.76

0.54 0.54 0.1 0.54 0.51

1.1 1.5 1.7 0.72 0.85

1.2 1.5 1.3 0.90 1.11

0.020 0.013 0.016 0.094 0.028

0.022 0.013 0.012 0.075 0.037

Group2

fit is more accurate than the multiple fit, the multiple commodity fit contains the influence from other commodities and hence reflects the market more accurately. In conclusion, the result encoded in Table 9 supports our basic premise that the multiple-commodity behavior of the market should be considered to be a perturbation on the prices of single commodities.

8. Conclusion The theory of commodity prices needs to explain the behavior of all commodities, including their cross-correlations, and the action functional based statistical microeconomic modeling must provide such a description. The study of multiple commodities provides empirical evidence supporting the approach of statistical microeconomics. The fits have R2 ≈ 0.8 for up to three commodities, which is reasonable but not excellent. One of the main empirical result of this paper is that the market of single commodities can be viewed as being partially complete, with other commodities affecting the price of any given commodity only perturbatively, with correlation terms contributing less than 10% to the price of a single commodity. This result provides an explanation for the excellent results obtained for the single commodities considered in isolation, as was obtained in Ref. [2]. Note a minimal extension of the single commodity action was made, motivated by the need to preserve the accurate results for the single commodities. One can improve the accuracy of the model by including cubic, quartic and higher order terms of prices and involving different commodities. This would make the calibration more difficult, but would have the advantage of being able to simultaneously fit a large number of commodities. The results of this paper place the statistical microeconomic theory of commodity prices on a firm footing. The significance of the various terms in the action functional in terms of the functioning of the underlying economy need further study. Future research can study after aspects of market prices. One is for some extreme cases in the history of the market. For example, the prices of all precious metals increased together in 1976–1980. This created a high cross-correlation between them. To see such a peak value of cross-correlation, one must use a long time interval, which the model in this paper cannot achieve. The other interesting aspect is to investigate the time delay between the behavior of two different commodities. It may happen that commodity A triggers a delayed change in the price of commodity B.

Acknowledgments We thank the referees for their helpful suggestions.

Appendix The cross-correlation function is evaluated analytically and a few consistency checks are made by reducing it to special cases obtained earlier for the single commodity auto-correlator.

B.E. Baaquie et al. / Physica A 462 (2016) 912–929

927

(0)

A.1. Derivation of DIJ

The Gaussian propagate is given by (0)

DIJ (t ) =

∞



(0)

(0)

dτ DI (τ )DJ (t − τ ) −∞ ∞



dk′

=

2π

−∞

∞



−∞

2π

dτ −∞

eik (t −τ )

eikτ

∞



dk

′

LI k4 + L˜I k2 + γI LJ k′4 + L˜J k′4 + γJ

.

(28)

Performing two integration yields



1

(0)

DIJ (t ) =

LI LJ

∞ −∞

eikt

dk 2π (

k2

+ λ )( 2

+ λ∗ )(k2 + ω2 )(k2 + ω∗2 )

k2

2

with



L˜I

λ = 2

1+

2LI

ω =

L˜I



1+

2LJ



4γI LI

1−



L˜J

2



1−

2

4γJ LJ L˜J

,

λ∗ =

,

ω∗ =

2

2LI



2



L˜I

L˜J

2

2LJ

 1−

1−



 1−

1−

4γI LI L˜I

 ,

2

4γJ LJ L˜J



2

.

Define 1

(0)

DIJ (t ) =

LI LJ

ζ (t ).

Then ∞



ζ (λ, ω, t ) =

(29)

2π (k2 + λ2 )(k2 + λ2∗ )(k2 + ω2 )(k2 + ω∗2 )

−∞

λ2 = α 2 e2iφ ;

eikt

dk

ω2 = β 2 e2iθ ;

λ2∗ = α 2 e−2iφ ;

ω∗2 = β 2 e−2iθ .

Hence

ζ (λ, ω, t ) =

∞



1

(λ2 − λ∗ )(ω2 − ω∗2 )

−∞

dk 2π

e

ikt



1 k2 + λ2∗

−



1

1

k2 + λ2

k2 + ω∗2

1

1

−

1 k2 + ω 2



.

Define the normalization constant C =

1

(λ2 − λ2∗ )(ω2 − ω∗2 )

=

−1

1

4 α 2 β 2 sin 2φ sin 2θ

.

Note the identity I (z ) =



∞

eikt

dk

2 2 −∞ 2π k + z

=

1 −|t |z e .

2z

Thus C −1 ζ (λ, ω, t ) = I (λ)



1



1

−

+ I (λ∗ )



−



λ2 − ω∗2 λ2 − ω 2 λ2∗ − ω2 λ2∗ − ω∗2     1 1 1 1 + I (ω) − 2 − 2 . + I (ω∗ ) ω2 − λ2∗ ω − λ2 ω∗2 − λ2 ω∗ − λ2∗

We make the following definition C −1 ζ (λ, ω, t ) = ζ1 (λ, ω, t ) + ζ1 (ω, λ, t ) where

ζ1 (λ, ω, t ) = I (λ)



1

1



+ C .C . λ2 − ω∗2 λ2 − ω 2   1 1 and ζ1 (ω, λ, t ) = I (ω) − + C .C . ω2 − λ2∗ ω2 − λ2 −

(30)

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B.E. Baaquie et al. / Physica A 462 (2016) 912–929

Define h1 = α 2 cos 2φ − β 2 cos 2θ;

h2 = α 2 sin 2φ + β 2 sin 2θ;

h3 = α 2 cos 2φ − β 2 cos 2θ;

h4 = α 2 sin 2φ − β 2 sin 2θ;

(31)

h5 = −α cos 2φ + β cos 2θ;

h6 = α 2 sin 2φ + β 2 sin 2θ ;

h7 = −α 2 cos 2φ + β 2 cos 2θ;

h8 = −α 2 sin 2φ + β 2 sin 2θ ;

2

2

and R = h21 + h22 ;

T = h23 + h24 ;

P = h25 + h26 ;

Q = h27 + h28 .

Let

φ˜ = φ + |t |α sin φ;

θ˜ = θ + |t |β sin θ; .

We obtain

ζ1 (λ, ω, t ) =

1 −|t |α cos φ e



1 −|t |β cos θ e



α

ζ1 (ω, λ, t ) =

β

1 R 1 P

˜ − [(h1 /R) cos φ˜ − (h2 /R) sin φ] [(h5 /P ) cos θ˜ − (h6 /P ) sin θ˜ ] −

1 T 1 Q

˜ [(h3 /T ) cos φ˜ − (h4 /T ) sin φ]



 [(h7 /Q ) cos θ˜ − (h8 /Q ) sin θ˜ ] .

We obtain the final result that is used for the cross-correlator

ζ (λ(α, φ), ω(β, θ ), t ) = C (ζ1 (λ, ω, t ) + ζ1 (ω, λ, t )) C

(0)

DIJ (t ) =

LI LJ

(ζ1 (λ, ω, t ) + ζ1 (ω, λ, t )).

(0)

A.2. Consistency check for DIJ

We take the limit of t = 0 and λ → ω as well as the limit of β → ∞. A.2.1. λ → ω; t = 0 Recall from Eq. (30)

ζ (λ, ω, t ) =



∞

eikt

dk

+ λ∗ )(k2 + ω2 )(k2 + ω∗2 )   1 1 1 I (ω) + C .C . C −1 ζ (λ, ω) = 2 ( I (ω) − I (λ)) + C . C . + I (λ) − λ − ω2 λ2 − ω∗2 λ2∗ − ω2 −∞

2π (

k2

+ λ )( 2

k2

2

We take the limit of t = 0; λ → ω taking care to cancel the divergent terms that appear in the expansion. This yields



  1 1 1 1 1 + C . C . + − + C .C . 2 λ2 − ω 2 ω λ 2 λ2 − ω∗2 λ λ2∗ − ω2 ω   1 1 1 1 1 = + C .C . + − + C .C . 2 (λ + ω)λω 2λ λ2 − ω∗2 λ2∗ − ω2

C −1 ζ (λ, ω, 0) =

=

1

1

1 e3iφ 4 α3 1

=−

2α 2

1

−

+ C .C . +

1



1 1

2 α 3 e2iφ − e−2iφ

(sec φ − cos 3φ).

We hence obtain

ζ (λ, λ, 0) =

1 8α sin 2φ 2 7

e− i φ

(sec φ − cos 3φ)

and we have recovered the result given in Ref. [2].

+ C .C .

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929

A.2.2. β → ∞ From Eq. (30) we obtain the following

ζ (λ, ω, t ) =

∞



−∞

eikt

dk

2π (k2 + λ2 )(k2 + λ2∗ )(k2 + ω2 )(k2 + ω∗2 )

.

Taking the limit

β → ∞ : ω2 = β 2 e2iθ → ∞ yields the single commodity auto-correlator

ζ (λ, ω, t ) =

1

β

4

∞



−∞

eikt

dk 2π (

k2

+ λ )(k2 + λ2∗ ) 2

.

(32)

In this limit, the coefficients are given by h1 = h3 = −h5 = −h7 = −β 2 cos 2θ ; h2 = −h4 = h6 = h8 = β 2 sin 2θ ; and R = T = P = Q = β 2. Hence, after some simplifications

ζ (λ, ω, t ) = =

−1

 1 −α|t | cos φ 1  e cos (| t |α sin φ + φ − 2 θ ) − cos (| t |α sin φ + φ + 2 θ ) 2

4α 2 β 2 sin 2φ sin 2θ α 1 e

−|t |α cos φ

β 4 2α 3 sin 2φ

β

sin(|t |α sin φ + φ).

The final result agrees with the result obtained in Ref. [2]. References [1] Belal E. Baaquie, Statistical microeconomics, Physica A 392 (19) (2013) 4400–4416. [2] Belal E. Baaquie, Xin Du, Winson Tanputraman, Empirical microeconomics action functionals, Physica A 428 (0) (2015) 19–37. [3] Jean-Philippe Bouchaud, Marc Potters, Theory of Financial Risk and Derivative Pricing: from Statistical Physics to Risk Management, Cambridge University Press, 2003. [4] William Baumol, Alan Blinder, Microeconomics: Principles and Policy, Cengage Learning, 2015. [5] Hal Varian, Microeconomic Theory, W. W. Norton & Company, New York, 1992. [6] Harold L. Cole, Maurice Obstfeld, Commodity trade and international risk sharing: How much do financial markets matter ? J. Monetary Econ. 28 (1) (1991) 3–24. [7] Emmanuel Haven, Andrei Khrennikov, Quantum Social Science, Cambridge University Press, 2013. [8] Andrei Khrennikov, Quantum-like microeconomics: Statistical model of distribution of investments and production, Physica A 387 (23) (2008) 5826–5843. [9] Rosario N. Mantegna, H. Eugene Stanley, Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, 1999. [10] Vasiliki Plerou, Parameswaran Gopikrishnan, Bernd Rosenow, Luís A. Nunes Amaral, H. Eugene Stanley, Universal and nonuniversal properties of cross correlations in financial time series, Phys. Rev. Lett. 83 (7) (1999) 1471. [11] Belal E. Baaquie, Path Integrals and Hamiltonians: Principles and Methods, Cambridge University Press, 2014. [12] Belal E. Baaquie, Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates, Cambridge University Press, 2004.