Quantum Evolutionary Financial Economics

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THE MODEL

This is an integrated model of a coevolving real economy and financial market, proposed in the article:

"Financial Turbulence, Business Cycles and Intrinsic Time in an Artificial Economy", Gon_alves, C.P.; Algorithmic Finance (2012), 1:2, 141-156.

The model is among one of the first applications of quantum econophysics to combine quantum chaos and quantum optimization to the integrated financial modeling of real economy and financial market coevolving dynamics, this is done within an integrated modeling approach that shows the emergence of turbulence and scaling patterns in the financial returns' dynamics, resulting from the coevolution of competing companies in a market economy and a financial market comprised of value investors and arbitrageurs, thus establishing a bridge to evolutionary arbitrage theory.

Throughout the model review and tutorial, below, we will sometimes mention equation numbers, these correspond to equations in the article, available online at:

http://algorithmicfinance.org/1-2/pp141-156/

APPROACH TO QUANTUM EVOLUTIONARY FINANCIAL ECONOMICS

The current model is aimed at a modeling gap between artificial financial markets finance-driven modeling and economic nonlinear dynamics' modeling.

Indeed, a model of financial dynamics without the underlying business economic dynamics is incomplete, since the value-driver dynamics is either absent or tucked away to the form of a global noise term, on the other hand, a model of economic dynamics without the financial dynamics is also incomplete, since the financial system is not simply an evaluator it is also a source of business facts and dynamics.

Thus, the quoted article, and the present model move to the next direction in financial and economic modeling: to build a model of an artifical economy in order to address the coevolution between the real economy and the financial system.

Thus, we are dealing with the disciplinary context of evolutionary financial economics, defined as the scientific discipline that concerns itself with the evolutionary processes and dynamics that take place within a systemic framework of coevolution between real economies and financial systems.

A quantum econophysics-based approach to evolutionary financial economics is followed. Econophysics is concerned with applications of statistical and quantum mechanics to economics, the applications of quantum mechanics (including quantum statistical mechanics and quantum field theory) to economics is called quantum econophysics, it is this later framework that is adopted here.

There are three main advantages of the quantum approach to Evolutionary Financial Economics:

• The explanatory effectiveness is expanded by the fact that one does not need any prior probability assumption, instead, one models the system's inter-relations and dynamics and from that result dynamical probabilities.

• Probabilities can have evolutionary and game theoretical interpretations.

• The adaptation process of a Complex Adaptive System (CAS) can be fully integrated with the probability formation and quantum game equilibrium assumptions.

In the article and model, we follow the conceptual basis of Complex Quantum Systems Science and Quantum Chaos applied to Evolutionary Financial Economics, with methodological consequences regarding:

• Combination of quantum lattice field theory with quantum optimization;

• Applications of complex quantum systems models, in particular dissipative quantum chaos, to financial theory.

Besides these two contributions to the field of quantum econophysics, the model links financial modeling to business cycle modeling, making emerge financial turbulence from a coevolving dynamics with the real economy.

HOW IT WORKS

The model works with a business lattice fitness field described by a field operator X(i) (in the article, it is represented by "x-hat underscore i", i being the lattice site, in this model description we use the notation X(i) due to limitations of text format in placing the hat above the X).

The field operator is assumed to commute between each pair of different lattice sites, thus, [X(i),X(j)] = delta(i,j), and has a continuous spectrum that spans a Rigged Hilbert space at each site (equation (4)). At each lattice site (called "patch" in the 2x2 lattice of the Netlogo world), the field eigenvalues will work as the configuration space for a wave function (it is actually a wave functional over the fitness field).

The quantum game is divided in rounds, with a local harmonic oscillator Hamiltonian condition and optimization problem defined by the minimization of the potential energy.

The potential is fixed for each round, but changes from round to round through a nonlinear map update rule.

The resulting optimization leads to the Gaussian wave packet solution corresponding the harmonic oscillator ground state:

 ps_i,t(x) = [1 / (gamma * sqrt(2*pi))] ^ 0.5 * exp[-u(t)/(4 * gamma ^ 2)]


 u(t) = (x - F(x_i(t-1)) ^ 2

where gamma represents the square root of business cycle intrinsic time. The wave function leads, at the end of each round t, to a certain fitness field value at site i with a gaussian probability distribution.

F is a nonlinear map, whose structure is given by equations (16) to (18), on the other hand, equations (14) and (15) formalize the financial dynamics, which depends upon two types of agent:

• Value investors, that evaluate the local fitness field: aiming at reflecting the fitness value at each site (company);

• Arbitrageurs, that look for companies that are considered to be close in fitness to the company under valuation and, thus, act as a price-to-value convergence force.

Both arbitrageurs and value investor trading takes place with an intrinsic time rescaling, linked to financial valuation of financial volatility risk versus business fitness risk (equation (15)).

The model combines chaotic dynamics, linked to the nonlinear map F, with quantum fluctuations linked to the wave localization on a definite field value at each site.

This allows for previous quantum wave reductions coupled to nonlinear dynamics to affect the Hamiltonian for the next round of the game, at each site.

The combination of the quantum stochastic and chaotic dynamics affects the wave function at each round, through the quantum game optimization problem, such that the quantum state transition for the quantum game equilibrium solutions psii,t-1(x) -> psii,t(x) can be accounted for by a round-to-round connector in the form of a quantum Radon-Nikodym operator, that is, we have, for the quantum game equilibrium dynamics:

 |psi_i,t> = K(t,t-1) |psi_i,t-1>

where K(t,t-1) is an operator satisfying:

 K(t,t-1)|x> = K(x,t,t-1)|x>


 K(x,t,t-1) = exp{ (2*x*[F(x_i(t-1)) - F(x_i(t-2))] - [F(x_i(t-1))^2 - F(x_i(t-2))^2]) /(4 * gamma ^ 2) }

from where it follows that:

 psi_i,t(x) = K(x,t,t-1) * psi_i,t-1(x)

We are dealing, in this case, with a conditional wave functional, conditional upon the fitness value at the beginning of the round (xi(t-1)). The wave functional at the end of each round could either be obtained by a conditional Schrdinger wave equation with a rotated solution, so that at the end of each round the wave functional would coincide with the eigenfunction of the Hamiltonian problem without any chronological time-dependent phase of the form exp(iE(t-t') / h-bar_s), or, alternatively, we can work with the round-dependent Hamiltonian problem plus quantum optimization and connect the two round's quantum game equilibrium solutions by a quantum Radon-Nikodym operator.

It is called Radon-Nikodym operator because the effect of this operator, applying Born's rule to calculate the probability distribution, is similar to a Radon-Nikodym derivative applied to connect two probability distributions (in this case two Gaussian densities).

NOTE: in quantum computation one can define the transition between two superposition states in a diagonalized form on the "logical" basis in terms of quantum Radon-Nikodym operators, this can be easily shown for a single qubit, for instance, in the transition:

 |psi> = psi(0)|0> + psi(1)|1> -> |phi> = phi(0)|0> + phi(1)|1>

As long as psi(0) and psi(1) are both different from zero we can write:

 K(|phi>,|psi>)|s> = (phi(s)/psi(s))|s>, s=0,1

from where it follows that: K(phi,psi)|psi> = |phi>. The Radon-Nikodym operator cannot, however, produce the following entanglement operation:

psi(0)|00> + psi(1)|10> -> psi(0)|00> + psi(1)|11>

This is because the initial state is a pointer state. On the other hand, entanglement can also take place in a case of a unitary transition that has an "equivalent" Radon-Nikodym operator, for instance, consider the following 2-qubit state:

|psi,phi> = phi(0)(psi(0)|00> + psi(1)|10>) + phi(1)(psi(0)|01> + psi(1) |11>)

One can obtain entanglement through a sequence of unitary quantum gates with the output state:

|psi> = psi(0) |00> + psi(1)|11>

In this case, there is also a corresponding quantum Radon-Nikodym operator, that can be defined as:

K(|psi,phi>,|psi>)|ss> = (1 / phi(s))|ss> 
K(|psi,phi>,|psi>)|s1-s> = 0|s1-s>

Applying this operator to |psi,phi> would yield the entangled state psi(0) |00> + psi(1)|11>.

One can check the above equivalence with a straightforward example of the initial state |+,+> = |+>|+>, in this case, the unitary transition is given by an inverse Haddamard transform on the second qubit, which would yield |+,0> followed by a C-NOT gate that would yield the desired entangled state |psi> = (1/sqrt(2))(|00> + |11>), the corresponding Radon-Nikodym operator would be:

K(|+,+>,|psi>)|ss> = sqrt(2)|ss> 
K(|+,+>,|psi>)|s1-s> = 0|s1-s>

This example allows us to illustrate that only some unitary transitions can be addressed by an equivalent quantum Radon-Nikodym operator.

In the current work, the quantum Radon-Nikodym operator is assumed in the formalism as it allows for a useful link between standard financial mathematics (which works with Radon-Nikodym derivatives) and it is effective for conditional quantum optimization problems leading to wave function trajectories incorporating end-of-round wave packet reduction and chaotic dynamics in a path-dependent quantum adaptive computation framework.

HOW TO USE IT

The interface is divided in a number of sliders that allow the user to control both the economic and financial parameters, as is now explained.

A) ECONOMIC PARAMETERS:

miu, theta0 and theta1 are the three main parameters of equation (3), for the mean reversing dynamics of the logarithmic growth rate in the economic equilibrium price (see Eqs.(1) and (2) for the equilibrium and growth definition).

b corresponds to the cubic map's parameter, as per equation (18), while m is the coupling parameter between the business fitness dynamics and the financial returns (as per equation (17)), introducing an impact on business dynamics coming from the financial system (for further details see the "MASS DISCUSSION AND EXPERIMENTS" section below).

epsilon is a global coupling constant for a mean field coupling between companies in the fitness dynamics (as per equation (16)), accounting for the competition between firms.

gamma corresponds to the square root of a business cycle intrinsic time scale and defines the level of economic volatility (for further details see the "MASS DISCUSSION AND EXPERIMENTS" section below).

B) FINANCIAL PARAMETERS:

phi0 and phi1 correspond to the two main parameters for the financial intrinsic time dynamics, as per equation (15).

r0 is a fixed component in the financial returns.

epsilonA and deltaA are, respectively, the proportion of arbitrageurs in the financial system and the arbitrage threshold.

There are three influences upon the financial returns, in accordance with equation (14):

• The fixed average returns component (r0);

• The value investors component, which has an impact (or coupling) of (1 - epsilon_A)

• The arbitrageurs component, which has an impact (or coupling) of epsilon_A

Arbitrage leads to a dynamic networking of connections, in the sense that the arbitrage coupling for each company is not fixed, arbitrageurs try to identify which companies have value fitness in a close neighborhood of each other (of no more than delta_A, in terms of Euclidian distance) and, thus, trade in such a way that they take advantage of the mismatching of returns, leading to a returns' convergence that is modeled in terms of a mean taken over the arbitrage cluster of each company, this mean is evaluated for the companies that comprise the cluster and is the mean of the product of the square root of each company's financial intrinsic time by the respective fitness.

The main assumption is of a business fitness-seeking trading dynamics, so that returns with similar fitness should be closely matched, the arbitrage coupling is, thus, coevolving with the economic and financial dynamics.

The value investors, on the other hand, impact on financial returns by evaluating each company's fitness multiplied by the respective financial intrinsic time, keeping in line with the above assumption of business fitness-seeking trading dynamics.

Besides the parameters, the model's interface shows six outputs to the user:

• A market portfolio returns plot;

• The returns for a single company, chosen at random at the beginning of the simulation;

• The mean field of business fitness;

• The "Phase Space" plot, defined in terms of the companies' mean (economic) equilibrium price logarithm versus the companies' mean shares price logarithm at each round;

• The volatility synchronization plot, defined in terms of the mean absolute returns of the population of companies (this is a market-observable volatility, which can be calculated for actual financial data);

• The Netlogo world, itself, which makes correspond the absolute returns of each patch (company) to a greyscale color scheme, so that lighter colors correspond to stronger volatility, while darker colors correspond to lower volatility, as measured by the absolute returns.

MASS DISCUSSIONS AND EXPERIMENTS

As a first point, some of the parameters that we will be addressing in this section were not explicitly placed in the model's sliders, but they underlie some of the parameters that are present in the sliders. As we move along this section, we will address the sliders. Having made this first note, we now address the mass problem and experiments that the user can try out.

A) THE MASS ISSUE

In this market model, the mass (or the mass-like term) enters first in the Hamiltonian problem. Indeed, given the potential function for the i-th company at round t:

V(x) = V(x,i,t) = (a/2) * ( x - F(x_i(t-1)) ) ^ 2

The parameter "a" represents the business evolutionary pressure, such that the higher the parameter a's value is the more competitive is the business environment, as explained in the article.

Now, given the harmonic oscillator's mass "m" and "a" we obtain the oscillation angular frequency measured in radians:

omega = (a / m) ^ 0.5

from this last relation comes:

m = a * omega ^ -2

Since, in the economic setting, "a" is dimensionless, the business cycle mass-like term is expressed in units of inverse squared angular frequency and is assumed as characteristic of the industry itself.

Thus, the higher the mass is, the lower is the oscillation frequency, and vice-versa, so that faster-paced economic rhythms or slower-paced economic rhythms are being addressed by this relation between mass and angular frequency.

On the other hand, the parameter gamma is also related to the economic mass-like term, indeed, given that gamma = sqrt(tau) and tau is defined as the mean potential energy, calculated under the round's wave function, multiplied by 2 / a, then, it follows that:

tau = (h-bar_s * omega) / 2a = (h-bar_s / 2) * (a * m) ^ -0.5

Thus, the higher the mass is, the slower is the economic intrinsic time, such intrinsic time is not measured in clock time, rather it is interpreted in terms of economic rhythms that rescale volatility with the usual square root rule that holds for clock-based temporal intervals.

This intrinsic time scale, simultaneously, relates the three main constants of the economic game: h-bar_s, a and m.

In the model's sliders we omitted "a" and "h-bar_s" opting to work with "gamma" and "m".

Now, the user can perform different experiments regarding the massive coupling between the financial system and the economic system.

The first two experiments work with a fully decoupled setting. To begin the first experiment set the parameters as follows:

• miu = 3.7, theta0 = 0.01, theta1 = 0.9, b = 2.84, epsilon = 0, epsilon_A = 0, r = 1.0E-5, phi0 = 0.5, phi1 = 0.001, gamma = 0.003 and m = 0

In this case, we have set m = 0, just like in Fig.1 of the article, which means that we are considering a decoupling of the economic dynamics with respect to the financial dynamics. We must be careful, however, in interpreting this value for m, what is meant by m = 0, both in the article and in this experiment, is that we are setting the mass coupling in equation (17) to zero, of course, the mass "m" still enters the economic dynamics through gamma, it just does not enter into the equation for the coupling of the economic dynamics to the financial dynamics, so it is set to zero in that equation specifically.

Run the model for a while and watch closely the resulting dynamics, then, change the parameter gamma reducing it by -0.001 running again the model until reaching 0.

What you have done by reducing gamma is to zero in on the periodic window shown in figure 1 of the current article.

The basic meaning of the procedure that you just undertook is the following:

Allowing for finite and fixed "a * m" component, reducing h-bars, transforms the Gaussian wave packets that characterize each round to increasingly tight distributions, until, in the limit, the |psii,t(x)|^2 collapses to a Dirac distribution centered at F(x_i(t-1)), such that the system follows a classical trajectory, this means that, while the economic dynamics stabilizes in a periodic dynamics, in accordance with the classical periodic window (just like in Fig.1 bifurcation map), the financial dynamics suffers an oscillating explosive process, which busts the market in terms of volatility and produces an oscillating exponential growth in the portfolio returns.

The dynamics is, in this case, unsustainable.

The birth of the periodic window in the classical dynamics for h-bars -> 0, does not have a corresponding quantum dynamics for h-bars sufficiently large, where the quantum fluctuations lead to dynamical chaos "destroying" the classical periodic window.

Let us, then, allow for gamma = 0.001, running the model with this parameter produces the explosive pattern, but if we let m be equal to 0.01 and, then, run the dynamics again, the result is quite different. The massive interaction between financial system and the economic system's dynamics introduces sufficient feedback to break the oscillating explosive dynamics, however, the resulting dynamics does not show evidence of the usual financial turbulence pattern.

Raising b to 3 allows you to recover the usual turbulent pattern at the microscopic (i.e., company) level.

The portfolio, on the other hand, for this new parameter setting, smooths out the turbulence, this no longer holds, however, if the spatial coupling is increased. Even for epsilon = 0.05, implying a high level of differentiation between the companies, the increased coupling is enough to lead to turbulence at the portfolio returns level.

Increased coupling will change the economic and financial dynamics qualitatively, depending upon the coupling level.

Considering again a lower epsilon = 0.01, where the portfolio is still effective in reducing turtulence for the investor, and epsilon = 0.05, one may try to address the effects of arbitrage in each case.

As it turns out, for epsilon = 0.01 with the arbitrage coupling set to 0.1, and delta_A = 0.22, we see that the portfolio returns will start to show evidence of turbulence and higher volatility risk, including the occurrence of jumps. For epsilon = 0.05, we can see that arbitrage in no way removes the turbulence from the portfolio returns.

Thus, in this model, arbitrage does not necessarily lead to a lowering of financial turbulence, it can have the opposite effect.

ADDED MATERIAL

At http://youtu.be/IXBskXwr-c8 is shown a video with an added experiment set and discussion on arbitrage and the coevolving financial and economic dynamics.

REFERENCES

"Financial Turbulence, Business Cycles and Intrinsic Time in an Artificial Economy", Gon_alves, C.P.; Algorithmic Finance (2012), 1:2, 141-156.

There is only one version of this model, created over 12 years ago by Carlos Pedro S. Gonçalves.

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