Introducing financial market solitons

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when growing dislocations cause fast transition swings from one state of equilibrium to another. .... tically fit to the data in modified Geometrical Brownian Motion stochastic equations [17]. ... ence in building systems for stocks, forex and commodities. ..... Seasonal, Trend and Random components of Oil prices for a minute (a), ...
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Introducing financial market solitons © Authors, 2018 © Radiotekhnika, 2018

S.Yu. Eremenko − Dr.Sc. (Eng.), Professor, Director of Soliton Scientific Pty Ltd (Sydney, Australia) E-mail: [email protected] Based on a generalization of Frenkel-Kontorova soliton model, a novel concept of financial markets solitons can be introduced in a few ways – as topological dislocations, dynamic equilibrium and transitional effects between highly correlated markets. Market soliton dislocations exist in the form of orders distribution irregularities along discrete lattice of price level, and generally, misbalance of committed funds reflecting demand-supply deficiencies which like wave-particles can grow, shrink, group and cause consistently impulsive market swings. The market can be conceptualized as a constantly evolving self-organizing ‘ocean of dislocations’. Multiple computational models are proposed to explore solitonic dynamic equilibrium effects in spreads between highly correlated markets when growing dislocations cause fast transition swings from one state of equilibrium to another. Numerous statistical experiments in R allow identifying ‘big dislocation waves’ in many commodities and currencies decorrelating from underlying pricing currency. The theory can be the foundation of new trading indicators and systems exploiting soliton dislocation patterns. Soliton market dislocation theory is universally applicable for all liquid asset classes and may contribute to both theory of solitons (extending it to markets) and financial markets theory (introducing solitons), and can be considered as a symbiosis of two theories first discovered in this paper.

Keyw ords: soliton, soliton dislocation, financial markets, market soliton, soliton market model, forex, commodities, R, trading systems. На основе обобщения солитонной теории дислокаций Френкеля−Конторовой, новая концепция солитонов в финансовых рынках может представлена в виде дисбаллансов в распределении ордеров на дискретных уровнях, состояний динамического равновесия и переходных процессов между тесно коррелирующими рынками. Разработаны компьютерные модели для исследования солитонных дислокационных эффектов между коррелирующими рынками, позволяющие идентифицировать дислокационные волны в ценах на многие валюты и другие активы.

Клю чевы е слова: солитон, солитонная дислокация, модели финансовых рынков, трейдинговые системы.

DOI: 10.18127/j20700970-201804-07

1. Introduction The concept of ‘solitons in financial markets’ raises reasonable scepticism – how something ‘solitary’ and ‘stable’ may exist in volatile and unpredictable financial markets? This paper tries to address these controversies by introducing some new concepts of solitonic market dislocations, solitonic effects between correlated markets and solitonic transitional swings from one state of demand-supply equilibrium to another which do exist in modern nonlinear, impulsive and interconnected liquid markets. Traditionally a versatile term ‘soliton’ relevant to many branches of physics, hydrodynamics, quantum mechanics, string theory, fiber optics, solid state physics, plasticity has been used to describe the whole class of nonlinear wave-particle phenomena like solitary waves, vortices, and group waves. However, since the introduction of the concept of a ‘topological dislocation’ in 1938 Frenkel-Kontorova soliton model, the meaning of the ‘soliton’ has been significantly extended to include not only traditional waves but also all kind of dynamic and static dislocation defects (Toda-lattice, compactons, micro-dislocations) defined on regular lattices [1−5]. And in this particular way, the concepts of solitons may be introduced in some non-traditional areas like financial markets driven by orders defined on a ‘regular lattice’ of price levels in market order queues of liquid markets. So, market solitons are not mysterious stand-alone waves but rather ‘dislocations’ capable to grow, shrink, group and cause consistently observable impulsive price swings. Furthermore, the concept of dislocations can be generalized to abstract from ‘irregularities in orders’ to ‘misbalanced funds’ or ‘demand-supply deficiencies’ and be the foundation of new market dislocation theory described in this paper. Apart from topological dislocations, the solitonic effects may also exhibit themselves as a dynamic equilibrium and transitions swings in tightly correlated markets. The introduction of solitons in markets is a challenging task due to many reasons. Firstly, we have to invent the new meaning for a ‘soliton’ for a market system which seemingly does not have spatial dimension, and the concept of a stable travelling spatial wave does not make sense here. Secondly, we need to extend the physical meaning of ‘spatial solitons on lattice’ with something more generic taking into consideration that there are other lattices for systems like financial markets moving via orders defined on discrete price levels. Thirdly, 54

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driven by orders from both emotional humans and emotionless trading systems, the markets seem chaotic, unpredictable and non-deterministic. It means that standard mathematical methods of soliton mathematics [1-5] are hardly applicable here and only sophisticated statistical approaches may have a chance to uncover some statistical solitonic patterns. Fourthly, the solitons are typically associated with the states of dynamically stable equilibrium of repelling and consolidating forces, and it is necessary to understand the statistical meaning of these forces in chaotic and unpredictable markets. Fifthly, there are many ‘types’ of solitons introduced in natural sciences [1] – waves, group waves, vortices, dislocations, topological solitons - and it is unclear which soliton concepts may have some relevance for markets. These challenges explain lack of theories and books on solitons in financial markets. However, the idea has been elaborated in a few papers. For example, theoretical physicists Jin-long MA and Fei-te MA in 2007 paper [7] suggested that Yang-Mills theory of quantum mechanics may be applicable to financial markets and lead to financial solitons, and authors provide examples of a Chinese stock and an oil future implying the existence of '…kind of new substance and form of energy existing in financial trade markets' and ‘…traveling wave solution (soliton)’. Another interesting article [8] by G.Dhesi and M.Ausloos implies the existence of '…kink-like effect reminiscent of soliton behavior' in their model of markets where the soliton-like function is introduced and supposed to be statistically fit to the data in modified Geometrical Brownian Motion stochastic equations [17]. The authors have introduced the interesting term 'psychological soliton' related to irrational trader's behavior and market overreaction to news. The concept makes good sense in markets which are often 'overshoots' on news, and 'psychological soliton' is a useful way of thinking as a reflection of trader's emotions from greed to fear. However, apart from occasional news, the modern markets are also driven but emotionless trading systems where the concept of a ‘psychological soliton’ is hardly applicable. Both papers do not explain what is and what causes the ‘market solitons’ in a first place. Also, in both theories the authors try to explain the appearance of wave-like [7] or kink-like [8] functions either in prices of individual securities or logarithmic returns - which are actually ratios – with some questions arising why solitons, which should reflect a wave-particle behavior and dynamic equilibrium of repelling and consolidating forces, should appear in ‘ratios’ and why only for individual securities. The theory elaborated in this paper extends the ideas of predecessors about soliton-related ‘…form of energy existing in financial trade markets', and nonlinear ‘kink-like soliton behavior’ but uses significantly different soliton dislocation approach which allows not only explaining what kind of solitons may exist in financial markets but also propose multiple solitonic models applicable for all liquid markets. Firstly, we introduce market solitons as constantly evolving ‘dislocations’ (rather than travelling waves [7]) following a generalization of Frenkel−Kontorova (FK) soliton model [1−3]. This matches the theory to the universal concept of topological solitons widely used in many branches of science [1−5]. Secondly, we introduce an idea that nonlinear solitonic effects appear not in prices of individual securities but in ‘spreads’ between tightly correlated markets which naturally incorporates solitons as a dynamic balance between consolidating and repelling forces. Thirdly, the macro-model of correlated markets behavior becomes quite close to solitonic equations for a large pendulum swing when a ‘correlated markets pendulum’ oscillates around neutral levels until significant dislocations cause large transitional swing towards another level of dynamic equilibrium. Fourthly, the concept of the dislocation solitons is quite close to chaos and fractals theories previously explored for self-organizing markets in some publications overviewed in [6]. And lastly, we propose multiple computational solitonic models and real trading systems exploring nonlinear dislocation effects. The main focus of the paper is to offer some novel ideas on how and what kind of solitons and solitonic models can be introduced in financial markets utilizing statistical data language R [9], knowledge of waves in inhomogeneous media from author’s books [10, 11], decade-long observations of financial markets and experience in building systems for stocks, forex and commodities. 2. Frenkel-Kontorova soliton dislocation model. Introduced in 1938 by Y.Frenkel and T.Kontorova, FK-model describes a topological defect/dislocation in a periodic ‘harmonic chain’ or lattice consisting of two layers of atoms, or generically, any particles connected with neighbors by a periodic force (Fig1, a). In this schematic model finally leading to sine-Gordon differential equation [1, 2] “Нелинейный мир”, № 4, т. 16, 2018

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'' ytt'' − y xx = f ( x) = − f 0 sin(2π x / a ),

(1)

the top layer atoms interact with each other by a strong force while the bottom layer affecting the top one with a weaker periodic force depending on size a of harmonic chain structure. Removing an atom from the top layer creates a ‘hole’, or so-called soliton dislocation [1] (Fig. 1,a) which causes the displacement of the system from the position of equilibrium (Fig. 1,b) described by expression [1, 2] a  a  y ( x, t ) = π − 4arctan exp ( x − vt / lv )  . +  F ( t , x ) ,  F ( t , x ) = 2  2π 

(2)

It contains the function shown in Fig. 2,a which also describes a large swing of a pendulum F (t ) = π − 4arctan exp ( −ωt )  . 

(3)

This ‘soliton function’ describing one of the most economical moves in nature frequently appears in many soliton theories of dislocations, pendulum swings, vortices, strings, chaos, topological twists [1−5]. The concept of a dislocation seems quite ‘strange’ [1] − because it describes not a ‘matter’ but the absence of it (hole, void, nothingness), but the presence of it creates a local disorder, chaos, misbalance. This dislocation behaves like a soliton wave-particle [1] which can grow, shrink, collide with other dislocations and ultimately produce microand macro-defects causing material creeps and plasticity effects widespread in nature [1−5]. Also important is that a dislocation can be stationary [1], exist forever and described by one-dimensional equation (2) at t = 0 making it somewhat different from familiar multi-dimensional travelling solitons [1−5].

b)

а)

Fig. 1. a − Two layers of atoms in typical crystal structure; b − Removing of one atom creates dislocation soliton causing misbalance in a system. Sources: https://commons.wikimedia.org/wiki/File%3ABoron-nitride-(hexagonal)-side-3D-balls.png

а)

b)

Fig. 2. a − Soliton dislocation function and its first derivative; b − Combination of two-soliton swings with different parameters produces wide variety of shapes

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FK-model simply described in A.Filippov’s book [1] and systematic mathematical book [2] also introduces the concept of a ‘positive dislocation’ associated with ‘rareness of atoms’ and ‘negative’ or ‘anti-dislocation’ for ‘compact zones’ of atoms [1] prescribing that the dislocations can be static or moving creating all sorts of interaction patterns (Fig. 2,b), but a dislocation can only be eliminated by collision with an anti-dislocation [1]. Interestingly, this is conceptually similar to market orders producing compact and rarefied zones in order queues of financial markets, and this idea leads to micro- and macro-soliton dislocations described hereafter. Another remarkable property of an FK-dislocation is the expression of impulse and energy [1, 2] similar to 2 Einstein's famous formula E ( v ) = m ( v ) c from special theory of relativity: = E

m0v02 = ,  P 1 − v 2 / v02

m0v0 2 a = ,  m0 m. 2 2 π 2 l0 1 − v / v0

(4)

It highlights a solitonic wave-particle duality of a dislocation, easy association of its energy with size/mass and the conclusion that an energy supplied to a dislocation increases its size. In order for an atom to occupy a dislocation spot (Fig. 1), it must overcome Peierls-Nabarro energy potential [1, 2] E > E *.

(5) Conceptually this is also quite similar to micro-behavior of financial markets when buyers must commit more funds/energy for current prices to occupy certain levels measured in discrete ‘ticks’. 3. Generalized Statistical Soliton Dislocation Model (SDM). The great significance of FK-soliton model is that it has inspired many well-researched generalizations ultimately leading to the concept of a topological dislocation/defect [1−5] universally applicable to many branches of science and, as we propose, to markets. A topological defect/dislocation is a universal term for some kind of irregularity/deficiency/misbalance/chaos in a regular discrete structure like atomic lattices, biological cells, chains etc. Interestingly, it can be applied not only for ‘spatial’ but for all kind of lattices, for example, price grids of orders in financial markets or even regular periodic processes. A dislocation introduces a piece of disorder in orderly structure. If we consider a distribution of the countable matter/quantity/funds denoted as E(x) along the grid with discrete levels xk = ka and denote ρ ( x ) = dE / dx as a density of a distribution, a local defect – like a removal of an m -element from a chain − will create a density fluctuation dE dx

ρ ( x ) == ρ0 ( x ) + δρ ( mx / a ) ;  for δρ ( x ) = pulse ( x )   x ≤ 1, δρ ( x ) = 0  for   x > 1

(6)

where pulse ( x ) is some kind of finite pulse function like shown in Fig. 2. Being defined on a regular lattice ( a = const ) with distribution of countable quantity E ( x ) which obey the rule of addition E=   E1 + E2 + E3 +…,

(7)

the defects and density fluctuations δρ ( x ) can grow, shrink, group and produce ‘rarefied zones’ D+ ( x ) - ‘positive dislocations’ and ‘compact zones’ D− ( x ) – negative, or anti-dislocations [1] = D+ ( x )

= D ( x ) ∫ pulse ( x ) w ( x ) dx > 0  ∫ pulse ( x ) w ( x ) dx < 0,   +





(8)

where w+ ( x ) , w− ( x ) are some distribution functions. These dislocations D+ and D− behave like ‘oppositely signed’ particles [1] − can collide, group, be eliminated in collisions and produce all sorts of interaction patterns shown in Fig. 2,b, and this is actually the essence of a topological soliton idea [1−4]. It is always applicable where a matter/quantity/process is defined on a regular grid and able to produce compact and rarefied density fluctuations which can eliminate each other, and financial markets moving by orders defined on discrete price ‘ticks’ are not an exception. “Нелинейный мир”, № 4, т. 16, 2018

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FK-model introduces another important concept – two connected layers (Fig. 1) − which can be easily generalized into ‘two linked systems’, let’s say primary system S1 and support system S 2 . The nature, universe, life, markets and even social and political ‘structures’ based on ‘relations’ offer wide variety of examples of connected systems and processes − stars in galaxies, atomic structures, biological cells - and tightly correlated markets. “Linked systems” means the existence of some strong relationships/ratios either between two systems variables/vectors E1 and E2 or their variations: = E12 ( t ) E1 ( t ) / E2 ( t= ) ,  J12 ( t )

∂E1 ∂E2 = / E1' ( t ) / E2' ( t ) .  ∂t ∂t

(9)

For complex systems the relationship functions E12 ( t ) and J12 ( t ) are inherently unknown, but often can be obtained by statistical and correlation analysis of historical observations. This allows building regression models expressing relations (9) in linear E1 ( t ) E1' ( t ) A B ' =+ R12 ( t ) ,  ; =+ R12 ( t ) R12 ( t ) , R12' ( t ) ~ NID 0,σ 2   ' ' E2 ( t ) E2 ( t ) E2 ( t ) E2 ( t )

(

)

(10)

or nonlinear forms  1   1  E1 ( t ) E1' ( t ) =LA  + =LB  ' P t ,  ;  + P12' ( t ) P12 ( t ) , P12' ( t ) ~ NID 0,σ 2 , ( )  12 '  E (t )    E2 ( t ) E2 ( t )  2   E2 ( t ) 

(

)

(11)

where constants A, B and nonlinear functions LA ( x ) , LB ( x ) can be obtained in such a way that residual functions ' R12 ( t ) , R12 ( t ) , P12 ( t ) , P12' ( t ) becomes normally independently distributed ( NID ) [8, 14, 17]. Modern statistical tools like R [9] allows building these relationships quite effectively. Equations (10), (11) with statistically calculated parameters define ‘linkage forces’ between two systems generalizing the FK two-layers soliton model.

A soliton is traditionally defined as an effect of nonlinear equilibrium between repelling and consolidating forces, and to introduce it into generic statistical soliton dislocation model we need to define some kind of an equilibrium function ES ( t ) . It can be done in multiple ways, for example, using residual statistical function from (10) ES = ( t ) R= 12 ( t )

E1 ( t ) A −   E2 ( t ) E2 ( t )

(12)

which ‘by construction’ would oscillate around the long-term average value E1 ( t ) ~ A except some periods when growing dislocations would cause ‘violent swinging behavior’ in function E1 ( t ) . To model oscillating and swinging behavior, we can split research time interval on small periods ∆t and use solitonic equation for a pendulum swing on each interval: EStt'' = −ω 2 ( t0 ) sin( ES ), . t0 < t < t0 + ∆t

(13)

It will describe both small vibrations and large swings ES small ( t ) ≈ A sin(ω ( t0 ) t )   when ES < ES * ;

ESlarge ( t ) ≈ A(π − 4arctan(exp ( −ω ( t0 )  t ))) when  ES > ES *.

(14)

Depending on the value of potentially calculated ‘inflection point’ ES * [1] we would get either oscillating or large solitonic swing behavior (Fig. 2,a). Large swings can be modelled by a kink-like function frequently appearing in many soliton theories [1−5] 58

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Swing ( t , a, b, c, d ) =+ d c*(π − 4arctan(exp ( a ( t + b ) ))  

(15)

a are the height, width and d , b are the shifts of the curve. The summation of many swing functions where c,   would compose an evolution wave Wave ( t ) =

∑Swing (t, a , b , c , d ). i

i

i

i

(16)

i

Let’s note that for really complex chaotic systems like financial markets the exact equations (13) - (16) are not that important and need to be understood in some statistical sense. The more important is the definition of ‘inflection points’ which can be estimated by special backtesting statistical experiments defining, like PeierlsNabarro potential (5), the turning points between dynamic equilibrium and transitional swinging behavior between two connected systems. Generalized Soliton Dislocation Model (SDM) unites the concepts of ‘topological’ and ‘equilibrium balance’ solitons and explains the evolution of complex systems from the viewpoint that growing dislocations/misbalances/irregularities may break dynamic equilibrium between tightly correlated systems forcing the system to evolve into new equilibrium state via some transition swings. SDM may provide universal calculation framework to study complex dislocation effects relevant to many branches of sciences, including physics, medicine, life sciences, evolution, astronomy, biology and financial markets where impulsive swings are especially pronounced. 4. Impulsive micro-structure of liquid financial markets. The financial market is perceived to be a complex chaotic dynamical system consisting of thousands of trading instruments the prices of which are driven by emotional humans reacting on predictable and unpredictable news and emotionless trading systems implementing a diverse range of technical algorithms. Interaction of these factors creates complex patterns of prices evolution. However, the markets are not totally random - they are actually connected to each other and often move in sync. Also, there are a few typical features defining the market’s structure on all timeframes - namely impulsiveness, trendiness, and randomness. Statistical tools like R allow to visualize this effect like shown in Fig. 3 for a ~170,000 minute, hourly and daily observations for prices of Oil/USD. Both ‘seasonal’ and ‘random’ components of prices are distinctively impulsive, especially on 1minute level closest to market orders. And the intriguing question arises what causes these impulses and they passed from one timeframe to another.

a)

b)

c)

Fig. 3. Seasonal, Trend and Random components of Oil prices for a minute (a), hourly (b) and daily (c) charts show the impulsive character of financial markets on all timeframes

5. Micro-soliton dislocations in markets Multiple concepts from generalized Frenkel−Kontorova model can be applied to study the solitonic effects in liquid financial markets instruments which on the micro-level are driven by orders located on predefined levels in a market orders queue provided by major brokers (Fig. 4). These ‘discrete’ levels naturally define a regular grid/lattice structure with a grid size equal to minimal ‘tick’ for market prices (0.0001 for Forex, 1 for Indices, 0.01 for stocks). Introducing an elementary Frenkel-like dislocation in this discrete ‘market system’ is quite easy. Let’s consider an ideal case of the uniform distribution “Нелинейный мир”, № 4, т. 16, 2018

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of orders on the sell side and the process of a steady accumulation of an asset which will yield a straight line in observable chart prices. But removal of all orders from a certain price level would create a ‘hole/gap’ - similar to ‘missing atom’ in FK soliton model (Fig. 1). Now the accumulation price wave will quickly ‘jump through’ this mini-gap and create a price swing shown in Fig. 5. The larger the gap, the faster the price swing. And this gap/nothingness/void is exactly the FK-soliton dislocation analog in markets. Being defined by the distribution of countable quantities (7) on a periodic grid these gaps can widen, shrink, group with other gaps and produce all kind of ‘macro-defects’. Opposite to gaps (‘positive dislocations’ [1]) we can consider ‘orders compounding’ (compactons [2] or ‘negative dislocations’ [1]) which also can widen, shrink and group with others. As we can see, there is a striking similarity between the definition of dislocation solitons in regular atomic, or market, or in fact all regular lattices, and this underscores the universality of a topological dislocation concept.

Fig. 4. Market Depth table of liquid stocks shows uneven distribution of orders on different levels which creates micro-dislocations and buying and selling pressure. Source: http://www.solitontrading.com.au/SRT/Default.aspx

a) b) Fig. 5. a − Elementary market price swing with pulsing derivative; b − Market prices evolve via series of swings

In the real situation like shown for the real stock sample in Fig.4, the orders are not uniformly distributed. Moreover, there is a well-documented tendency of orders to stay close and form groups [16], and there are multiple explanations why this is happening. 1. Massive orders from big institutions may be spread over multiple neighboring price levels. 2. Sometimes, there are large ‘static’ orders (especially for stocks) from companies persistently supporting their stock prices at certain levels. 3. Multiple algorithmic systems may place limit or60

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ders at predefined calculated levels. 4. Retail traders use large institution’s orders as a ‘downside protection’ and place their orders ahead (so-called front-running technique). In a nutshell, the tendency of orders to gravitate to each other is a well-defined feature of liquid markets [16], and Fig. 4 shows the real examples of order consolidations for a few stocks. The orders form ‘rarefied zones’ corresponding to “positive dislocations” and ‘compact zones’ - ‘negative dislocations’– exactly like in physical soliton model in atomic structures [1]. These dislocations can grow up, shrink, disappear, produce all kind of interactions in between and grow into macro-defects – similar to the evolution of defects in nature. There are a few mathematical models capable to describe prices evolution around ‘dislocation gaps’. Let’s examine how the supply of accumulation funds F ( t ) changes the prices P ( t ) via orders unevenly distributed along the discrete price ticks. From equation dP dF = P (t )   P ( F ( t )= = * ) , dP dt dF dt

dF dF dt , E   = E dP

(17)

it is clear that the price changes dP ( t ) / dt   would depend not only on a speed of fund’s supply dF ( t ) / dt but also on the coefficient E = dF / dP   which for discrete price levels Pk = ak would be defined by an orders density distribution like shown in Fig. 4. The denser the zones the slower the price changes while the prices move faster over rarefied order zones. So, observable price swings not only reflect impulsive supply of funds but also uneven distribution of orders which typically creates ‘buying and selling pressures’ tending to move prices in certain direction. The model describing pressure σ = Eu x' and price changes ut'  waves can be borrowed from the dynamic theory of elasticity [10] which leads to the equation '' utt'' − c 2 ( x, t ) u xx f ( x, t ) , c 2 = E / ρ =

(18)

for price increment u ( t ) assuming the liquid market medium can be simulated by an elastic medium where E , ρ are the elasticity modulus and density of orders distribution along the direction x for price levels in market orders queue. From the expression for a speed of wave c we can see that the pressure wave would naturally accelerate at low order’s density areas and slow down around big orders. Let’s note that equation (18) looks quite similar to equation (1) from FK-soliton model, but in case of very dynamic markets the velocity-related coefficient c ( x, t ) is not a constant and the function f ( x, t ) typically meaning ‘an external force’ affecting the system is not necessarily periodic. From equations (17), (18) it is possible to see that the price swings typically observable on market price charts (See Fig. 5,b) can be explained either by uneven orders/funds distribution or by instant supply of large funds dF ( t ) / dt which accumulate all orders on many levels as it is happening during market news releases. These ‘shocks’ may instantly eliminate micro-market dislocations however in dynamic liquid markets they will reappear again on other price levels − because the market prices are only driven by orders. While the market shocks are occasional, the dislocations in order distributions are the permanent feature of the markets. In summary, the dislocations in markets are very dynamic, may frequently appear/disappear, but regardless of periodic news shocks they always exist in the form of misbalanced orders and deficiency in committed funds which in turn reflect unbalanced ‘opinions of humans’ (perhaps ‘psychological solitons’ [8]) and diversity of trading systems constantly reacting on price changes. Dislocations drive market prices on every level. They also can grow up, shrink, group with others like ‘particles’ – underscoring the wave-particle duality [1−5] typical for all solitons. Let’s also note the deep connection between the concepts of dislocations (‘missing atoms’, ‘holes’) and ‘chaos’ – presence of a dislocation creates a ‘local disorder/chaos’ which can grow up, disappear, group and evolve. But markets are not completely chaotic and, as Fig. 3 shows, the impulsiveness is passed from one level to another. Rather, the market can be considered as a self-organizing chaotic system [6] where dislocations and pressure waves are eventually driving random market moves in a certain direction, and the theory of dislocations described here just enforces this view. “Нелинейный мир”, № 4, т. 16, 2018

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6. Mini- and macro- soliton dislocations The dislocations on the level of orders we would call micro-dislocations, but it is quite easy to generalize the concept to mini- and macro-dislocations remembering the ability of topological dislocations/deficiencies/irregularities to grow and produce macro-defects like in nature. In fact, the orders in markets represent the countable quantities of committed funds following (unlike prices) the simple rules of additions (7). So, we can substitute the notion of orders dislocation with the more generic concept of funds dislocations which can happen on every market timeframe – from 1 minute to hours, days and years. Growing supply of funds on a buying side of the market creates ‘buying pressure’, with market orders producing consolidated order groups as shown in Fig.4 – anti- dislocations in soliton dislocation terminology [1]. The deficit of funds creates ‘rarefied zones’ in order’s distribution which correspond to ‘positive dislocations’. For conceptual example, let’s imagine that Gold is in a strong demand, currently has reached $1200 level, strongly going up and $200M of orders want to buy it today, but knowing strong demand the sellers want to sell only $100M of orders at 1220 level. So, around 1200-1220 level we may observe the significant ‘committed funds misbalance’ - like a 'hole' in FK-soliton terminology [1]. In absence of adverse condition this misbalance would result in a fast price swing. This $100M hole/gap/deficiency between 1200-1220 levels is an example of a the mini-soliton dislocation beyond the level of orders. Interestingly, we can include in the definition of fund deficiency not only real visible orders funds but also uncommitted funds from active traders who are ready to deploy the funds at certain levels. The deficiency generally means the demand-supply deficiency for an asset. Daily deficiencies may extend into weekly, monthly and even yearly dislocations which we would call macro-dislocations. Like in nature where the dislocations can grow up from tiny to cosmic defects, the macro-market dislocations related to structural market deficiencies/oversupply/misbalances can also reach enormous proportions and many years to evolve (like 2014 Oil Crisis or 2008-2009 Global Financial Crisis). The idea that micro-market dislocations can evolve into a mini- and macro-dislocation can be illustrated by using time series decomposition technique (using R function STL [9]) applied for the same instrument (like Oil/USD) on different time frames – 1-minute (Fig. 3,a), 1-hour (Fig. 3,b) and 1-day (Fig. 3,c). Comparison of charts shows that on all timeframes both seasonal and random components are distinctively impulsive implying that the nonlinear ‘energy of impulses’ (let’s recall the proportionality (4) of energy with dislocation size in FKtheory) can be passed from one market level to another. Because the impulses are ultimately caused by orders and funds dislocations, it is reasonable to conclude that due to the ability of dislocations to add to each other and grow, the energy of small impulses is passed from one market level to another, and large impulses are typically composed of smaller impulses. This is not much a surprise because the nature behaves in exactly the same way – small dislocations causing the micro-swings (Fig. 1,2) have a tendency to grow, group and produce macrodisplacements and even destructions. Mathematical description of an effect of passing the energy of microdislocations to macro-dislocations is still needed to be formally developed – which is a challenging task for future research. The hint how it can be done is given by the formula (4) for energy of FK-soliton similar to E = mc 2 allowing easy association of a dislocation with a particle and its energy with size. Because particles can combine, the energies can be added. In a context of a discussion about the ‘energy of market dislocations’, it is quite interesting to recall a citing of theoretical physicists MA Jin-long and MA Fei-te in 2007 paper [7] about '…kind of new substance and form of energy existing in financial trade markets'. The theory described here suggests that ‘new substance’ is actually ‘the dislocations’ and ‘energy’ is the energy of impulses produced by a presence of dislocations. And this nonlinear energy exists everywhere in financial markets ultimately driven by growing dislocations. 7. Primary market dislocations Let’s note that association of the market dislocations with the imbalances in orders or deficiency in committed funds for one tradable stock/currency/instrument is not generic enough, but can be generalized to the level of a ‘primary market’ which represents a raw commodity – (Oil, Gold, Copper…), Currency (AUD, EUR, RUB, CHF…), Stock (AAPL, ANZ, HPQ, MSFT) or Index (DAX, FTSE, …) irrespective of underlying currency in which the asset is traded in. For example, British Pound GBP can be traded against different currencies (GBPUSD, GBPCAD, GBPCHF…), but the ‘primary market’ defined by the physical supply of Pound bank62

“Нелинейный мир”, № 4, т. 16, 2018

notes is the one – GBP. So, we need to embrace that the primary market dislocations appear in the deficiencies/disorders of countable quantities of real funds (GBP), but not in prices GBPUSD, GBPCFH or other ratios. To generalize and match the concepts against the physical Frenkel-Kontorova soliton model which deals with ‘primary’ and ‘secondary’ layers, we need to introduce the concept of a measurable dislocation/misbalance/deficit on the level of the primary market/raw commodity/currency assuming that it will be ultimately reflected in misbalanced orders for every participating market. For example, Oil oversupply dislocation would affect all markets – Oil/USD, Oil/RUB, Oil/GBP. In this sense, the concept of market soliton dislocation becomes quite universal, measurable and easily understood. For example, strong demand for a commodity can be associated with a ‘positive dislocation’ while weak demand/deficit with ‘negative dislocations’; they will, in turn, cause the orders and funds misbalances which will eventually cause the price swings. Because these dislocations are related to countable quantities (measured in tons, dollars, ounces) they can be added together, grow, shrink, evolve and produce all kind of composition patterns in between – similar to other topological solitons in nature. In general, last paragraphs introduce and generalize the concept of market soliton dislocations as potentially measurable dislocations/deficiencies/misbalances occurring on every market level – from order irregularities to misbalances of funds to demand-supply deficiencies in primary markets. Conceptually they behave like other topological wave-particles solitons in nature and are capable to grow, shrink, group, evolve and cause market swings. Let’s note that the theory of market dislocations described here does not contradict with modern market theories like efficient markets or price elasticity [18, 19]. It just highlights how previously disconnected theories – solitons and markets – can be merged and supplement each other for better understanding of prices evolution and nonlinear solitonic effects in markets. 8. Computational market soliton dislocation models The concept of market soliton dislocations described previously provides some insights into the nature of market impulses as a reflection of misbalances in orders, funds distribution and demand-supply deficiencies in primary markets. Like soliton wave-particles, these dislocations can grow, shrink, group, evolve into macrodislocations and cause market waves. However, while the dislocations in nature and markets are countable-inprinciple (for, example, it is possible to calculate exact misbalance of buy-sell funds in a current market depth of a stock or number of ‘missing’ atoms in an atomic structure), it is quite technically complex and ‘expensive’ exercise to monitor constantly evolving misbalances. Knowing the dislocation nature of market swings may have the theoretical significance, but the ultimate goal of financial markets models is to research not the dislocations themselves but their effect on tradable prices and creating models to profit from this knowledge. Interestingly, the Frenkel-Kontorova “two-layers” soliton model allows not only defining the concept of a dislocation but also creating the useful models to research on soliton dislocation effects. The soliton dislocation model (SDM) described in paragraph 2 provides universal research framework implementing the core idea that growing dislocations cause a displacement of tightly linked two-layers system from the state of equilibrium. In financial markets, it is quite easy to find both ‘two-layers systems’ - in the form of highly correlated markets and the ‘state of equilibrium’ - in the form of calculatable spread functions. This allows building multiple universal model of the behavior of financial markets for all major asset classes incorporating solitonic concepts of dynamic equilibrium between repelling and consolidating forces and pendulum-like solitonic swings appearing when growing dislocations cause the evolution of tightly correlated markets. 9. Soliton market dislocation model 1 – Spread Dislocation Waves All major markets – forex, commodities, equities, and bonds - offer a strong correlation with major underlying currency – typically US Dollar - in which the assets are typically priced in and traded. This allows building a universal market dislocation model to research on waves appearing when a significant misbalance in a primary market breaks this strong linkage – often with massive swing resembling the pendulum-like FK-soliton dislocation swing (1)−(3) studied in paragraphs 1 and 2. Let’s denote [COM / USD ]( t ) as the price of a generic commodity COM  (or a currency, stock, index) expressed in US Dollar. So, symbol COM may represent a commodity (Oil, Gold, Gas, Sugar, Coffee, Copper, Zinc etc), raw currency (AUD, EUR, GBP, NZD, SEK), or units of shares, stock indices, bond or future contracts available for a financial exchange. For the value of USD Dollar “Нелинейный мир”, № 4, т. 16, 2018

63

participating in ratio [COM / USD ] we can use some kind of USDollar Index − USDollar ( t ) , for example, FXCM Dow Jones Index (https://en.wikipedia.org/wiki/Dow_Jones_FXCM_Dollar_Index) or Intercontinental Exchange US Dollar Index (https://en.wikipedia.org/wiki/U.S._Dollar_Index). Major brokers provide both quotes [COM / USD ] ( t ) and USDollar ( t ) on all timeframes, and because the Dollar appears as a denominator in the ratio [COM / USD ] ( t ) , the two functions [COM / USD ] ( t ) and USDollar ( t ) would be mostly negatively correlated on every market level, from 1-minute to daily charts like shown in Fig. 6. The correlation coefficient Corr ( COM / USD ( t ) , USDollar ( t ) ) between two functions can be calculated in R, Excel or provided by broker platforms like Bloomberg Terminal or other providers, notably www.myfxbook.com. For example, for daily functions for Oil, Copper, AUD, NZD the correlation coefficients for 2011−2016 period are −0.86, −0.93, −0.85, -0.8 respectively. This “correlation linkage” acts as a ‘force’ connecting two layers of a market model which becoming conceptually similar to FK-soliton dislocation model in physics and its generalization in paragraph 2. Now it is possible to construct an equilibrium functions (12) which would have a financial meaning of a spread function = ES ( t ) Spread ([COM / USD ] ( t )= ,    USDollar ( t ) )

( t ) − A / USDollar ( t ). [COM / USD ]  

(19)

It can be calculated in Excel, Python or R [9] as a residual function of the linear regression analysis between [COM / USD  ] ( t ) and 1. / USDollar ( t )  and even directly traded as a pair in some trading platforms like Bloomberg Terminal (www.bloomberg.com). The Fig. 6 shows the spread function for Oil constructed in R. The goodness of function (19) is that due to negative correlation between COM / USD ( t ) and USDollar ( t ) it mainly excludes the effect of US Dollar on a commodity behavior focusing only on commodity-specific dislocation effects. In absence of commodity-specific deficiencies ( Com  ~ Const ) the function, like a pendulum (14), would oscillate against zero levels, and this state can be considered as a state of dynamic equilibrium typical for solitons which is perceived to be dynamic equilibrium formations [1,2]. Rallies in USDollar ( t ) will be suppressed while strong demand-supply misbalances/dislocations in the raw commodity COM ( t ) will be reflected in the spread function (19), and this is exactly the pattern we target in soliton market dislocation model. In case of Oil in Fig.6, it is pretty easy to see mild oscillating behavior in 2011-2014 followed by massive irreversible swing of 2014-2015 related to well-known Oil market oversupply (=macro-dislocation) crisis affecting many Oil producing countries around the world. Due to special construction of a function (19) we know that this swing has been caused not by appreciation of US Dollar but structural dislocation in the primary Oil market. The similar spread dislocation waves analysis was done for other commodities (Natural Gas, Copper) and currencies (NZD, AUD, EURO, GBP, JPY), with some results shown in Fig.7, 8. They all show that the spread function is not uniformly distributed in time, but evolving via the series of long-lasting waves caused by growing dislocations in primary markets. These waves can be calculated by multiple means, for example, using smooth derivatives depicted on Fig.7,8 with zero-crossings defining the buy-and-sell trading signals for actual prices COM / USD ( t ) . Let’s note that for currency pairs like USD/JPY or USD/CHF where the Dollar appears as a numerator in currency ratios the spread function (19) should be rewritten in the form ES ( t ) Spread ([USD / COM ] ( t )= ,  USDollar ( t ) ) =

( t ) −USDollar ( t ) / A. [USD / COM ]  

(20)

For equities typically exhibiting a strong correlation between a stock price (like MSFT)   and correlated stock index  StockPrice ( t ) Index ( t ) (like DOW) the spread function can be constricted as = ES ( t ) Spread ( StockPrice = Index ( t ) ) StockPrice ( t ) / Index ( t ) −  / A Index ( t ) .  ( t ) , 

(21)

In the similar way the model can be applicable to analyze behavior of correlated markets for all major asset classes. Let’s note that the model (15) can be also used to identify commodity-specific waves on lower (mi64

“Нелинейный мир”, № 4, т. 16, 2018

nute/hourly) timeframes as shown in Fig. 9, but as numerical experiments show, the lower the timeframe the smaller the correlation linkage between a commodity price and pricing currency (~0.9 for daily, ~0.3 for 1minute charts). The main idea of the model is that for tightly correlated markets the equilibrium function (19) - (21) would normally oscillate around neutral levels until significant dislocations in primary market would cause massive transition swings from one state of dynamic equilibrium to another (Fig. 6−8), and special trading indicators discussed later can be used to identify big market dislocation waves.

a) b) Fig. 6. a − The residual function of linear regression model behaving like spread; b − Negative correlation between USDollar Index (top) and USOil (bottom) on daily charts

Fig. 7. Spreads show distinct dislocation swings – NGAS, AUDUSD, NZDUSD

Fig. 8. Spreads show distinct dislocation swings – EUR, GBP, JPY “Нелинейный мир”, № 4, т. 16, 2018

65

Fig. 9. Spread function swings for Hourly and Minute Oil charts

10. Soliton market dislocation model 2 – Pendulum Swing Model Solitonic equations (3), (13), (14) for a large pendulum swing offer simple and useful conceptual model of behaviour for strongly correlated markets. Like a pendulum pushed by small impacts, the spread functions (19)−(21) would oscillate around neutral levels until significant misbalances would force the ‘market pendulum’ to swing from one equilibrium state to another (Fig. 2). Oil chart 6 shows exactly this kind of macro-behaviour. In this schematic model (4) described by differential equation EStt'' = −  *sin a ( ES ) the equilibrium function at low levels ES ( t ) <    ES * when  sin ( ES ) ~ ES

would be descibed by a sine wave (14). But when a spread ex-

ceeds potentially back-testable ‘inflection point’ ES ( t )   > ES * , the pendulum would change its behavior and start

swinging in a ‘solitonic way’ [1] with function arctan ( exp ( at ) ) (2), (3), (15) frequently appearing in many soliton theories [15]. Of cause, the market is not as simple as a pendulum, but the pendulum-like evolution pattern can be very useful analogy to describe ‘breaking of relationships’ between traditionally connected systems in markets and generally in nature and life sciences. 11. Soliton market dislocation model 3 – Sharp Dislocation Swings. Model (19) assumes the spread function is calculated as a residual from a linear regression between commodity price [COM / USD ]( t ) and inverse pricing currency 1. / USDollar ( t ) . But according to scatterplot in Fig. 10, a for Oil, the linear model is only approximately representing more complex relationship. Nonlinear local regression technique using R function loess [9, 14] which applies the model = [COM / USD ]( t ) LA (1. / USDollar ( t ) ) + R ( t )

(22)

to a small moving window along the charts provides sharply distinct identification of dislocation swings as shown in Fig. 10,b. In the last expression, LA ( x ) is a local polynomial (typically first or second order [9]) fitted for a moving window data, and R ( t ) is an automatically calculated residual function with Gaussian distribution. Sharp 2014 Oil swing in loess fitting function with distinctively pulse-like smoothed derivative similar to FKsoliton function in Fig. 2,a confirms the tendency of misbalanced markets to evolve via violent swings. As will be discussed later, both swing curve and its smoothed derivative can be used as trading signals to capture major price dislocation swings. 12. Soliton market dislocation model 4 – Commodity Function Waves Availability of both commodity prices [COM / USD ] ( t ) and pricing currency USDollar ( t ) allows constructing a commodity function 66

“Нелинейный мир”, № 4, т. 16, 2018

COM ( t ) =  / USDollar ( t ) [COM USD ]( t ) * 

(23)

which also excludes the effect of a pricing currency and theoretically reflects the supply-demand misbalances/dislocations on a commodity behavior as shown in Fig. 11 for Oil example.

Fig. 10. a − Scatterplot of Oil vs Inversed USDollar with linear and local regression lines; b − Swings identification using loess regression fitting function

a)

b)

Fig. 11. a − Oil Commodity Function with Oil/USD and USDollar charts; b − Normalised Commodity Function with smoothed derivatives

Also important is that the measurement units for a commodity functions (tons, euros, bitcoins) – can inprinciple allow matching them against demand-supply misbalances. To introduce an equilibrium commodity function, it is possible to use a standard statistical normalization procedure with easily calculatable mean and standard deviation constants ES ( COM = (t ))

( COM ( t ) −

COM

) /σ

(24)

which will bring the commodity function close to zero levels as shown in Fig. 11,b. Big commodity-related waves can be identified using different smoothed derivatives shown in Fig. 11,b which can also be the base of trading signals along with many other technical trading indicators like Bollinger Bands, Envelopes and others [12, 13]. In summary, commodity functions can be useful to research on the dislocation misbalances in primary commodity, forex, and stock markets. 13. Other prospective models The idea of soliton-like dislocations causing the displacement of highly correlated systems from the state of dynamic equilibrium can be incorporated in other market models briefly listed below. “Нелинейный мир”, № 4, т. 16, 2018

67

- Geometrical Brownian Motion model – well-known stochastic procedure [17] already used by G.Dhesi and M.Ausloos in their ‘psychological soliton’ model [8] – can incorporate ‘solitonic function’ arctan ( exp ( x ) ) appearing in many soliton theories [1−5]; - Geometrical Brownian Motion model [17] can be rewritten in terms of spreads between highly linked markets rather than logarithmic returns for one market – to implement the idea that random market walks would typically produce normally distributed oscillations of spreads around zero levels until significant primary market dislocations cause violent transition waves towards another level of equilibrium like shown in Fig.6-11; - Differential Models would replace spread functions (19)−(24) with their derivatives or logarithmic returns typically used in financial market theories [8, 12, 13, 17−19]; however, we should remember that soliton dislocations appear not in ‘derivatives’ or ‘ratios’ but in misbalanced orders, funds, and demand-supply deficiencies. 14. Soliton trading systems Soliton market models described above can be the foundation of trading systems (only briefly described here) specifically targeting dislocation waves and patterns. The core idea is to analyze not the prices but spreads between traditionally correlated markets - to profit from strong dislocation swings periodically happening for all markets. To trade forex, commodities, and indices it is possible to use ‘big waves’ depicted in Fig. 6−11 (buy on a derivative zero line crossing up, sell on crossing down) or create more sophisticated soliton indicators like the one shown in Fig. 12, a for AUDUSD example. The idea of indicators − ideally ranging from -100 to 100 - should be to use different kind of smoothed equilibrium functions (19)−(24) and identify ‘inflection points’ (14) ES * (like 20) by using backtesting algorithms which would deliver maximal statistical profit with minimal drawdown (Fig. 12,b).

a)

b) Fig. 12. a − Trading Signals generated by a Soliton Wave Indicator ranging from -100 to +100 for AUDUSD-daily; b − Backtesting performance (AUDUSD); c − Global Macro trend-following strategy backtesting for 40+ instruments (note: backtesting does not guarantee future performace) 68

“Нелинейный мир”, № 4, т. 16, 2018

Buy signal would be generated when an indicator crosses up ES ( t ) > ES * and sell-and-reverse if ES ( t ) < ES * . It will produce series of continuous trades shown in Fig. 12 which will pick long lasting waves presumably caused by structural misbalances in raw commodity markets (AUD, Oil, Gold…). Let’s note that widely used standard moving averages trend following techniques [12, 13] applied directly to prices AUD/USD, Oil/USD, Gold/USD would also react on all unwanted USD rallies creating false, weak signals on the way. Backtesting and optimization of Soliton Wave Indicators for dozens of available instruments allow building long-term ‘global macro’ trend following strategies automatically monitoring dozens of assets around the world, with optimizing portfolio risk-to-reward Sharpe ratio [19] by assets diversification and small capital allocation for every position based on back-tested drawdown factor. The details of multiple totally automated systems comprising of 50+ trading robots are described in fx.solitonscientific.com, www.solitonscientific.com/page_product_trad.html, and trading.solitonscientific.com. 15. View on markets from the positions of soliton dislocation theory From the viewpoint of the of market soliton dislocations theory, the financial markets can be viewed as a constant evolution of dislocations (irregularities, misbalances, deficiencies) happening on every market level and causing markets prices to evolve via nonlinear swings. Like in nature, the dislocations can grow, shrink, group and ultimately produce macro-market swings observable on liquid market charts for all timeframes. This suggests that an energy of nonlinear dislocation swings is not generally ‘wasted’ but passed from one level to another contributing to sustainable market trends. This resonates with comments of other authors suggesting the existence of 'kind of new substance and form of energy existing in financial trade markets' [7] and 'kink-like effect reminiscent of soliton behavior' [8]. The theory described here associates the ‘substance’ with ‘dislocations’ and ‘kink-like effects’ with ‘market dislocation swings’. Because of soliton dislocations, the markets are not random, but rather self-organizing chaotic systems with repetitive impulsive behavior where an elementary dislocation/misbalance/deficiency can be considered as ‘a particle of disorders’ capable to grow, group and evolve. In this sense the markets conceptually are not that different from other self-organizing phenomena in nature – the message also conveyed by fractal theory founder Benoit Mandelbrot [6]. Markets are also not random because of existence of strong correlation forces which, like true solitons as a dynamic balance between repelling and consolidating forces, keep the linked markets moving in sync. But when these dynamic equilibriums are broken, the markets evolve from one state to another via series of pendulum-like swings described by solitonic equations. As we can see, the concept of solitons is very relevant to markets, like to many other branches of science. Let’s note that it seems incorrect to associate all market waves with soliton dislocation waves. Similar to oceans, there are also market shock waves caused by major news events and small linear up-down waves not transferring much energy. But because the soliton dislocation waves are caused by genuine misbalances in primary markets, we should expect them to prevail, especially if we consider market charts on longer timeframes where long-lasting nonlinear waves supported by supply of big funds are typically visible. To summarize in a few words – market is an ocean of dislocations. 16. Summary and future research directions Based on universal generalization of Frenkel-Kontorova soliton model, the novel concept of financial markets solitons can be introduced in a few ways – as topological dislocations and dynamic solitonic effects in highly correlated markets. Some definitions and summaries regarding topological soliton dislocations are listed below. 1) Market soliton dislocations are potentially measurable dislocations/deficiencies/misbalances. 2) Micro-soliton dislocations appearing on the discrete levels of market order queues are typical topological lattice dislocations defined as potentially observable misbalances in orders distributions; they create buying and selling pressures causing micro-market swings. 3) Growing micro-dislocations produce Mini-soliton dislocations which can be associated with demandsupply misbalances in allocated and provisional funds. 4) Growing mini-soliton dislocations produce Macro-soliton dislocations which can be associated with significant long-lasting demand-supply deficiencies in primary markets. “Нелинейный мир”, № 4, т. 16, 2018

69

5) Being directly linked to countable quantities, the market dislocations/deficiencies/misbalances can be added together and, like in nature, produce macro-defects. 6) Presence of market dislocations/misbalances causes market prices to swing. 7) Soliton market dislocations behave like wave-particles capable to grow, group and cause market swings – matching other solitons in nature. 8) Market Dislocation Waves are the combination of market soliton dislocation swings. 9) Market soliton dislocations appear not in prices (returns, ratios) but in uneven distribution of countable orders, funds and generally demand-supply deficiencies. The significant difference of micro-market dislocation in comparison with other topological solitons in nature is that they are very dynamic (constantly evolving ‘ocean of dislocations’) and defined not on a spatial physical lattice but on the lattice of discrete price levels in market order queues. But ability of them to grow, group and evolve matches them to other topological solitons in physics. Apart from topological solitons, there are also Solitonic effects as ‘dynamic equilibrium’ and ‘transitional waves’ effects observable in spreads between highly correlated markets. Soliton market models allow analyzing the displacements of correlated markets from the state of equilibrium when growing dislocations cause pendulum-like transition swings from one state to another. Multiple computational models applicable for all liquid markets allow identifying the ‘big waves’ in spreads caused by structural dislocations in primary markets which can be powerful trading signals. New indicators and trading systems are proposed and is being built to exploit soliton dislocation patterns. For future research, it would be interesting to incorporate nonlinear soliton dislocation models into existing financial market theories and Geometric Brownian Motion stochastic procedure, obtain statistical differential equations for soliton dislocations and solitonic dynamic equilibrium effects and market ‘inflection points’, mathematically describe the effects of passing energy of dislocations from one level to another. Soliton market dislocation theory may contribute to both theory of solitons (extending it to markets), and financial markets theory (introducing dislocation solitons), and can be considered as a symbiosis of two theories first discovered in this paper. This paper does not pretend to offer a complete solitonic theory of financial markets which would require long term efforts of financial and soliton theories specialists, but rather introduces new concepts which may lead to appearance and future development of these theories. Soliton market dislocation theory may contribute to both theory of solitons (extending it to markets), and financial markets theory (introducing dislocation solitons), and can be considered as a symbiosis of two theories first discovered in this paper. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

70

Filippov A.T. The Versatile Soliton. Birkhauser. 2010. Braun O.M., Kivshar Yu.S. The Frenkel-Kontorova Model: Concepts, Methods, and Applications. Springer; 2004. Malomed B.A. Soliton Management in Periodic Systems. Springer. 2006. Nakahara M. Geometry, Topology and Physics. Taylor & Francis. 2003. Mermin N.D. The topological theory of defects in ordered media. Reviews of Modern Physics 1979; 51 (3): 591. Mandelbrot B., Hudson R. The (Mis)behavior of Markets. A fractal view of Risk, Ruin and Reward. New York: Basic Books. 2004. MA, Jin-long; MA, Fei-te. Solitary wave solutions of nonlinear financial markets: data-modelling-concept-practicing. Front. Phys. China 2007; 2: 368374. Dhesi G., Ausloos M. Modelling and measuring the irrational behavior of agents in financial markets: Discovering the psychological soliton. Chaos, Solitons & Fractals 2016. 88, July: 119–125. R-documentation on https://www.r-project.org. Eremenko S.Yu. Natural Vibrations and Dynamics of Composite Materials and Constructions. Kiev: Naukova Dumka. 1992. Eremenko S.Yu. Finite Element Methods in Mechanics of Deformable Bodies. Kharkov: Osnova; 1991. https://www.researchgate.net/publication/321171685_Eremenko_S_Yu_Finite_Element_Methods_in_Mechanics_of_Deformable_B odies_in_Russian. Kaufman P. Trading Systems and Methods + Website. Wiley Trading. 2013. Murphy J.J. Technical Analysis of the Financial Markets: A Comprehensive Guide to Trading Methods and Applications. New York Institute of Finance. 1999. Bates D.M., Watts D.G. Nonlinear Regression Analysis and Its Applications. Wiley. 1988. Veinik A.I. Thermodynamics of real processes. Minsk: Nauka i Technika. 1991. Yue P., Xu H., Chen, W., Xiong X., Zhou W. Linear and nonlinear correlations in the order aggressiveness of Chinese stocks. Fractals 2017; https://doi.org/10.1142/S0218348X17500414. “Нелинейный мир”, № 4, т. 16, 2018

17. Ross S.M. Variations on Brownian Motion. Introduction to Probability Models (11th ed.). Amsterdam: Elsevier. pp. 612–14. 18. Story J. Price Elasticity of Demand and Marketing: Mastering elasticity to market strategically. 2016. ISBN 1523257237. 19. Elton E.J., Gruber M.J., Brown S.J., Goetzmann W.N. Modern Portfolio Theory and Investment Analysis, 9th Edition. Wiley. 2014.

Поступила 24 сентября 2017 г.

Введение в солитонную теорию финансовых рынков © Авторы, 2018 © ООО «Издательство «Радиотехника», 2018

С.Ю. Еременко − д.т.н., профессор, директор, научно-технологическая компания Soliton Scientific Pty Ltd (Сидней, Австралия) E-mail: www.solitonscientific.com На основе обобщения солитонной теории дислокаций Френкеля−Конторовой, новая концепция солитонов в финансовых рынках может быть представлена в виде топологических дислокаций, состояний динамического равновесия и переходных процессов между тесно корреллирующими рынками. Солитонные рыночные дислокации могут быть представлены в форме дисбаллансов в распределении ордеров на дискретных уровнях, инвестиционных фондов и несбалансированного спроса и предложения на актив. Как другие солитоны в природе, эти дислокации могут расти, уничтожаться, группироваться, развиваться и приводить к волнам в ценах. Концептуально, финансовый рынок может быть представлен как само-организующийся «океан дислокаций». Предложено несколько компьютерных моделей для исследования солитонных дислокационных эффектов между кореллирующими рынками, когда растущие дислокации приводят к быстрому разбалансированию рынков и переходу в новое состояние равновесия. Многочисленные эксперименты на статистическом языке R позволяют идентифицировать дислокационные волны в ценах на многие валюты и другие активы. Солитонная теория рынков существенно дополняет как теорию солитонов (обобщает на финансовые рынки), так и теорию рынков (представляет концепцию солитонов), и можеь быть оценена как новая комбинированная теория впервые представленная в данной работе.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Filippov A.T. The Versatile Soliton. Birkhauser. 2010. Braun O.M., Kivshar Yu.S. The Frenkel-Kontorova Model: Concepts, Methods, and Applications. Springer; 2004. Malomed B.A. Soliton Management in Periodic Systems. Springer. 2006. Nakahara M. Geometry, Topology and Physics. Taylor & Francis. 2003. Mermin N.D. The topological theory of defects in ordered media. Reviews of Modern Physics 1979; 51 (3): 591. Mandelbrot B., Hudson R. The (Mis)behavior of Markets. A fractal view of Risk, Ruin and Reward. New York: Basic Books. 2004. MA, Jin-long; MA, Fei-te. Solitary wave solutions of nonlinear financial markets: data-modelling-concept-practicing. Front. Phys. China 2007; 2: 368−374. Dhesi G., Ausloos M. Modelling and measuring the irrational behavior of agents in financial markets: Discovering the psychological soliton. Chaos, Solitons & Fractals 2016. 88, July: 119–125. R-documentation on https://www.r-project.org. Eremenko S.Yu. Natural Vibrations and Dynamics of Composite Materials and Constructions. Kiev: Naukova Dumka. 1992. Eremenko S.Yu. Finite Element Methods in Mechanics of Deformable Bodies. Kharkov: Osnova; 1991. https://www.researchgate.net/publication/321171685_Eremenko_S_Yu_Finite_Element_Methods_in_Mechanics_of_Deformable_Bodies _in_Russian. Kaufman P. Trading Systems and Methods + Website. Wiley Trading. 2013. Murphy J.J. Technical Analysis of the Financial Markets: A Comprehensive Guide to Trading Methods and Applications. New York Institute of Finance. 1999. Bates D.M., Watts D.G. Nonlinear Regression Analysis and Its Applications. Wiley. 1988. Veinik A.I. Thermodynamics of real processes. Minsk: Nauka i Technika. 1991. Yue P., Xu H., Chen, W., Xiong X., Zhou W. Linear and nonlinear correlations in the order aggressiveness of Chinese stocks. Fractals 2017; https://doi.org/10.1142/S0218348X17500414. Ross S.M. Variations on Brownian Motion. Introduction to Probability Models (11th ed.). Amsterdam: Elsevier. pp. 612–14. Story J. Price Elasticity of Demand and Marketing: Mastering elasticity to market strategically. 2016. ISBN 1523257237. Elton E.J., Gruber M.J., Brown S.J., Goetzmann W.N. Modern Portfolio Theory and Investment Analysis, 9th Edition. Wiley. 2014.

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