Swingtum A Computational Theory of Fractal Dynamic Swings and Physical Cycles of Stock Market in A Quantum Price-Time Space

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1 Swingtum A Computationa Theory of Fracta Dynamic Swings and Physica Cyces of Stock Market in A Quantum Price-Time Space Heping Pan Schoo of Information Technoogy and Mathematica Sciences, University of Baarat Mt Heen, Victoria 3350, Austraia, Emai: h.pan@baarat.edu.au (Paper for the 2003 Hawaii Internationa Conference on Statistics and Reated Fieds) Abstract This paper presents the basic framework of a comprehensive computationa theory of stock market behavior, which we ca Swingtum, taking mutivariate stock index time series data as input, and producing probabiistic predictions of stock index movement at mutipe time frames. The theory shoud aso be appicabe to other iquid markets. The Swingtum theory is based on the view that the movement of the stock market as a whoe as represented by its benchmark index is driven by three types of forces: business dynamics, mass psychoogica dynamics, and news impacts, and consequenty the market movement can be decomposed into four types of components: dynamica swings, physica cyces, abrupt momentums, and random waks. Dynamic swings incude business cyces, stock ife cyces, and Eiott waves of different eves, which typicay have a fracta nature characterized by og-periodic power aws. Physica cyces incudes anniversary days, seasonaity cyces, and weeky cyces, which have reativey constant periodicity. Abrupt momentums may be caused by endogenous critica points or driven by exogenous news shocks or impacts. Random waks correspond to remaining randomness not expainabe by any systematic force. The dynamic swings and physica cyces identified and modeed from the historica index time series wi most ikey define a quantum space of price and time in which the market wi most ikey trave from one quantum price eve to another or from one time zone to another. There is a fundamenta symmetry between price and time. The actua path is not ony determined by dynamic swing phase and physica cyce phase, but aso by the possibe news impact. The more genera theory of Swingtum extends the fracta and cycica modes of a univariate benchmark index to the mutivariate time series modes of intramarket and intermarket dynamica anaysis. Keywords: Computationa finance, stock market index, dynamic swing, physica cyce, abrupt momentum, random wak, fracta, og-periodic power aw, time series prediction, Eiott waves, Gann cyces, Gann anges, price-time symmetry, quantum space, news impact. Acknowedgement: An eary stage of this research was sponsored by China s Nationa Natura Science Foundation under the grant Learning Bayesian networks for knowedge discovery and data mining through the Schoo of Remote Sensing and Information Engineering, Wuhan University, Wuhan, China.

2 . Introduction We observe that professionas and academics from each of technica anaysis, quantitative anaysis and fundamenta anaysis of the financia markets are beginning to recognize the importance of each other s approach, and there is a tendency for the three to converge. The aim of this research is to combine technica anaysis and quantitative anaysis into a unified theory, which is the first major step in this convergence. Another motivation comes from our rea-data neura network experiments which indicate a significant performance of over 90% correctness in predicting the next-day index change direction. It turns out both necessary and feasibe to deveop a comprehensive computationa theory of stock market behavior, which coud predict the market direction at mutipe time frames using ony index or price time series data. The ideas presented in this paper constitute a basic framework of such a theory which we sha ca the Swingtum theory in the foowing and future discussions. Stock markets are compex dynamica systems whose eements are investors and traders with varying capita sizes using different investment and trading strategies. The behavior of the stock market of a given country is generay measured by a benchmark index such as ASX S&P 200 index for Austraian stock market. The index is recorded in a time series whose time unit is usuay one day or one minute. The daiy index time series is the primary data source for technica anaysis and quantitative anaysis for short-term traders and mid- to ong-term investors with time frames ranging from days through years to decades. The minutey index time series is the main data source for day traders with a time frame ranging from a few minutes through hours to a few days, normay ess than a week. Each data point contains usuay 5 numbers: the open, high, ow, cose price and the traded voume of that time unit. In genera, if we ony consider one number, it is the cose price. Sti the time series of an index is considered a univariate time series, even the singe variabe may be a vector of 5 eements. Looking inwards, a stock market has its interna structure consisting of a number of sectors, so the compete coection of a the sector indexes for a stock market may be caed the intramarket time series which must be mutivariate. Looking outwards, a stock market is positioned in the internationa stock market ecoogica systems. It may have its super markets that infuence itsef in an asymmetrica way, and there are aso a number of other interreated stock markets on the same eve, so the infuences are not uni-directiona. Taking the Austraian stock market as an exampe, the US stock market as represented by three indexes Dow Jones, S&P 500, and NASDAQ may be considered its super market, whie the Japanese, Hong Kong, Korean, German, French and British stock markets are obviousy its neighbors on the same eve. The super and neighbor stock markets for a given stock market as we as typica bond markets, god and oi markets and other interreated financia markets may form the intermarkets of this stock market. The univariate time series of the stock market and the mutivariate time series of its intramarkets and intermarkets provide the compete primary data source for prediction of the stock market movement. The secondary data source incudes mainy the news (incuding a kinds of textua information) avaiabe from the Internet. However, this secondary data source is extremey irreguar, and requires sophisticated techniques for automatic processing and interpretation. 2

3 This research aims at deveoping a comprehensive computationa theory about the dynamics of the stock market for predicting the stock market index movement on mutipe time frames from the primary data source of the index time series of this market, its intramarkets and its intermarkets, as we as from the automatic anaysis of the news from Internet. We sha ca the compete coection of our views to the stock market, our assumptions, our mathematica modes as we as computationa procedures the Swingtum theory. Athough this theory wi keep evoving, our basic view to the market is fundamenta. Our inspirations come from a century-od technica anaysis and the recent progresses in quantitative anaysis of the financia markets, incuding econometrics and econophysics. A companion paper (Pan 2003) provides a joint review on technica and quantitative anaysis and aso has pointed out the possibiity of a unified science of inteigent finance. More references on quantitative finance are provided by Sornette (2003), Farmer (998), Farmer and Joshi (2002), Lo and McKinay (999), Campbe et a (997), Mandebrot (982), Zhou and Sornette (2003). Comprehensive coverage on technica anaysis can be found in Murphy (999), Acheis (2000), Bukowski (2002), and Prechter (2002). The paper is organized as foows. Section 2 estabishes our fundamenta view to the stock market which is expressed in the Swing Market Hypothesis. Section 3 formuates a simpe but genera dynamic mode of the market returns. Sections 4 and 5 presents the parametric modes for characterizing fracta and cycica market movements. Section 6 describes the quantum price-time space in which the market is supposed to trave as driven by endogenous dynamics and exogenous news shocks or impacts. Section 7 outines a computationa approach of nonparametric nearest neighbor pattern recognition expoiting mutidimensiona chaos in price time series. Finay section 8 concudes the paper. 2. The Swing Market Hypothesis A ong-standing conventiona view of the mainstream economists to the financia markets is expressed in an Efficient Market Hypothesis (EMH), which views asset prices and their associated returns from the perspective of the specuator, and assumes the market is amost aways efficient, meaning that the prices aready refected a current information that coud hep anticipating future events. Under the EMH, the stochastic process of market returns is modeed as uncorreated random wak, so any profitabe prediction of the market returns woud be considered impossibe. Athough the strong form of EMH has been shown not true by many studies as reviewed by Pan (2003), EMH does provide a perfect reference with which more reaistic market views can be deveoped. We consider that the market is driven by various forces of different origins, wave engths and magnitudes, as we as different durations. Each force ony has one dimension: either positive for upwards or negative for downwards. At any given time, a the effective forces combine to form a joint force which has a joint market impact eading to changes in price (or index). For generaity, we wi just tak about the dynamics of the price of a market. If the market is a stock, the price of the market is just the price of that stock. If 3

4 the market is the whoe stock market for a nationa economy, the price of the market refers to a benchmark index such as ASX S&P 200 index for the Austraian stock market. Foowing Mandebrot s discovery of fractas in the financia markets, Peters (994) proposed a Fracta Market Hypothesis (FMH) which basicay says that the market is stabe when it consists of investor covering a arge number of investment horizons, and information is vaued according to the investment horizon of the investor. Because the different investment horizons vaue information differenty, the diffusion of information wi aso be uneven. At any one time, prices may not refect a avaiabe information, but ony the information important to that investment horizon (or time frame in our terms). The FHM owes much to the Coherent Market Hypothesis (CMH) of Vaga (99) and the K-Z mode of Larrain (99). Like the FMH and CMH, our view to the market, which we ca the Swing Market Hypothesis (SMH), is aso based on the premise that the market assumes different states and can shift between stabe and unstabe regimes. However, we take one step further in considering the dynamic structure of the market. The Swing Market Hypothesis (SMH) proposes the foowing: () The market aways consists of investors or traders with a possibe different capitas, different time frames, different information conditions, and different skis. The everasting and ever-evoving differences in investors or traders are the permanent drivers of market dynamics. (2) The market is sometimes efficient and other times inefficient, and the market tends to swing between these two modes intermittenty. Each mode, efficient or inefficient, may comprise mutipe regimes such as trending, cycing, spiking, consoidation, etc. (3) The market movement is driven mainy by three types of forces: business dynamics, mass psychoogica dynamics, and new impacts. (4) The market movement can be decomposed into four types of components: dynamica swings, physica cyces, abrupt momentums and random waks. Business dynamics incude goba and nationa business cyces, intramarket dynamics and intermarket dynamics, which usuay refer to the fundamenta economic and business conditions. Mass psychoogica dynamics incude the greed and fear dynamics of investors and traders defined by human nature and aso forged by existing popuar knowedge, methodoogies and technoogies of technica anaysis and fundamenta anaysis. For exampe, the appearance of a certain chart pattern may trigger simiar trading decisions made by many different technica traders because they have a acquired simiar technica anaysis education. Both business dynamics and mass psychoogica dynamics can produce simiar dynamica swings which can be characterized by mathematica fracta modes such as og-periodic power aws. Eiott waves of different eves are the visua and quaitative description of dynamica swings by technica anaysts. Athough not as strong as dynamica swings, physica cyces do exist in the market, which incudes anniversary days, seasonaity cyces and weeky cyces. For exampe, statistica studies show that Thursday of a week is often the reversa if the first three days have trended in the same direction. There are even intraday dynamica patterns 4

5 given the daiy market situation. Physica cyces have reativey constant periodicity. Abrupt momentums refer to drastic price movement which cannot be expressed in continuous anaytica forms. Momentums may be caused by exogenous forces such as news shocks or impacts, or as critica points or singuarities caused by endogenous dynamics. Random waks in the context of SMH correspond to remaining randomness not expicabe by any systematic force. The next section derives a simpe but fundamenta price impact mode of the stock market returns driven by market forces. 3. A Fundamenta Price Impact Mode of The Stock Market Assume the market operates in the continuous time t, i.e. a trading occurs in a continuous fow, so we sha not need to distinguish between each individua order. We wi ony consider the price (or index ) of the market p( as a continuous function of time. At any time t, the joint market force f ( acts upon the market and drives the price p( up or down in a continuous fux of movement. Note that the market force f ( may incude a kinds of suppy or demand, greed or fear, rationa decisions or emotiona reactions, etc. But it must reaize its impact to the market price through a kinds of orders and transactions. We sha not concern ourseves with a the order detais unike the agent-based modes such as one proposed by Farmer (998). However, we sha foow some part of Farmer s path in mode buiding but our deveopment is rather going into a direction quite different from his agent-based mode. Let p( be the price (or index) at time t, the reative return ( is defined as p( t + τ ) P( Rτ ( = () p( whereτ is the time scae. In genera, it is more common to use the og-return rτ ( defined as rτ ( = n p( t + τ ) n p( (2) For sma changes in p (, the og-return rτ ( and the reative return Rτ ( are approximatey equa. Suppose at time t the price p ( is changed to ~ p ( t ) immediatey after the force f ( acts upon the market, ~ and this process can be expressed as p ( = ~ p ( p(, f (, m ) (3) where m is the abstract mass of the market, which refects how much the price wi change reative to the magnitude of the force, and which may depend on the capitaization of the market and the past trading history. The price change shoud satisfy the foowing basic conditions ) The price is aways positive but finite 0 < ~ p( p, f, m ) < (4) R τ 5

6 2) p~ is an increasing function of the force f, meaning that the price impact is in the direction of the joint force and (maybe nonineary) proportiona to force magnitude ~ p ( p, f, m) f (5) 3) If there is no force at a there is no market impact, i.e. ~ p ( p,0, m) = p (6) 4) p~ is additive in the force ~ p ( ~ p ( p, f, ),, ) ~ m f 2 m = p ( p, f + f 2, m) (7) 5) ~ p is a decreasing function of the mass m, meaning that the price impact is inversey (maybe nonineary) proportiona to the mass ~ p( p, f, m) m (8) 6) The price return due to the impact of the force f is competey determined by f and m ~ p = δ ( f, m) (9) p δ is caed the market impact function, and it must be an increasing function of f and a decreasing function of m according to equations (5) and (8). Appying equation (9) into equation (7) gives a fundamenta equation of the market impact function δ ( f + f 2, m) = δ ( f, m) δ ( f 2, m) (0) This equation has a simpe but remarkabe soution satisfying equations (5) and (8) f / m δ ( f, m) = e () Therefore, the basic ~ price impact function is derived as f / m p = pe (2) The basic og-return dynamics function is thus ~ p r = = p f m Or we shoud remember the time, n (3) ~ p ( f ( r ( n = p( m( = (4) We see that this mode is simiar to Newton s Second Law in physics. However, The abstract mass m is not a constant, but a sowy varying quantity in time reative to the variation of force f (. It can be understood as a scae factor that normaizes the order size and be considered as the iquidity. However, we prefer to consider it as an abstract quantity whose variation may be reated to the custered voatiity. Considering the high autocorreation of the custered voatiity, m may be considered constant for a imited time period. However, we must be aware that m may vary sowy. A varying mass may be compared to the Einstein s reativity theory in physics. Athough professiona traders are aware of the reativity of price in mass psychoogy, a modeing of price reativity requires further research. 6

7 The og-price at the current time t can be obtained through the integra of the og-return from the starting time t = 0 to the current time t t n p( = n p(0) + n r( dt + 0 t f ( = n p(0) dt (5) 0 m We mainy consider two types of forces: dynamic swings and physica cyces. For generaity, assume there are L eves of dynamic swing forces f, =,2, L, L, and K eves of physica cyce forces g k, k =,2, L, K. The joint force f ( is the sum of mutieve dynamic swing forces and mutieve physica cyce forces L = K f ( = f ( + g ( (6) k = k Substituting equation (6) into equation (5) gives t L K f ( n p( = n p(0) + dt = n p(0) + n p ( + n qk ( t0 m = k = (7) where t n p ( = f ( dt m 0 for =,2, L, L (8) t n qk ( = g k ( dt m 0 for k =, 2, L, K (9) are respectivey the contribution to the current og-price n p( by the evoutionary impact of the dynamic swing force f ( and the physica cyce force (. Let A = n (0) (20) 0 p L Φ( = n p ( (2) = K Ψ( = n q k ( (22) k = equation (7) can be rewritten as n p( = A0 + Φ( + Ψ( (23) Apparenty, Φ (, Ψ(are the joint dynamic swing component and the joint physica cyce component in the og-price. g k 4. Mutieve Fracta Swings In Log-Periodic Power Laws Each -th eve swing force f ( must have a certain wave form. A number of studies have shown the existence of the og-periodicity in the stock market indexes and prices, thus we consider that f ( shoud incude a wave form in og-time f ~ sin( ω n( t τ ) + φ ) (24) ( 7

8 where ω, φ are the og-period and phase respectivey, and τ a time shift. Considering the existence of power aws and og-periodicity as demonstrated by Sornette s group and other researchers, we consider p ( shoud have a form which incudes a power component and a og-periodic wave component β n p ( A + B ( t τ ) [ + C cos( ω n( t τ ) + φ )] (25) where A, B, C, β, ω, φ are unknown parameters pertinent to eve. Sornette et a have aso shown the possibiity of using more compicated second-order and third-order Landau expansions, however, we consider our use of mutieve og-periodicities coud render higher-order Landau expansions unnecessary. Using equation (0), the joint infuence of dynamic swings can be expressed as L Φ( A + B ( t τ ) β [ + C cos( ω n( t τ ) + φ )] = There are two different ways for actua computation using this mode. The first is a genera regression of a the unknown parameters of this mode using the whoe time series data set. The second way is a recursive procedure which successivey fits a singe eve mode of (25) to one segment of the time series data corresponding to an Eiott wave in the context of the previous eve. However, we recognize that a singe ogperiodic power aw mode of (25) can ony fit to a singe wave, which is what was done by Sornette s group. Such a singe mode cannot predict the regime shift after an antibubbe a correction wave for a buish trend has finished. In a truy fracta procedure, on each eve, we shoud fit an eementary fracta made up of two waves: a trending wave foowed by a correcting wave. This eementary fracta for an up trending wave and its correcting wave is defined by a pair of og-periodic power aws β β n p A + B ( τ [ + + C ( τ cos( ω n( τ + φ )] for t < τ (27) ( ' ' ' ' β ' β ' ' n p ( A + B ( t τ ) [ + C ( t τ ) cos( ω n( t τ ) + φ )] for t > τ (28) In this case, τ shoud correspond to the critica point the time of reversa from the trending wave to the correcting wave. Equations (27) and (28) shoud be fitted to the up trending wave and the correcting wave respectivey. However, we consider there may exist certain geometrica reationships between the two sets of parameters: ' ' ' ' ' ' ( A, B, C, β, ω, φ ) versus ( A, B, C, β, ω, φ ). This geometry, if existent such as Fibonacci ratios, woud substantiay reduce the number of unknown parameters. Observations from technica anaysis show that this geometry is probabiistic. (26) 5. Mutieve Physica Cyces in Hibert Transform Unike the dynamic swings characterized by og-periodicity, physica cyces have inear periodicity and inear phase. Therefore, we can use the we-estabished signa processing techniques for detecting physica cyces incuding the periodicity and phase shift. Hibert transform f ˆ( t ) of a function f ( is defined for a t by f τ f ˆ t dτ π + ( ) ( ) = t τ (29) 8

9 Hibert transform f ˆ( t ) is orthogona to the origina function f (, so it can be used to create an anaytica signa from a rea signa iϕ ( z( = f ( + ifˆ( = A( e ( i = ) (30) where ˆ 2 2 A ( = f ( + f (, fˆ( ϕ ( = arctan (3) f ( are the instantaneous ampitude and the instantaneous phase. Therefore, Hibert transform provides a basic means for modeing physica cyces of inear periodicity and phase shift of the market price time series (taken as rea signas). Ehers (200) deveoped a set of practica agorithms for cacuating the dominant period and instantaneous phase from price time series data. However, his approach is imited to a singe eve. Our mode of physica cyces generaizes to mutieves. In essence, we first generaize a simpe Hibert transform to a Hibert waveet, and then the time series data are anayzed through a mutiscae waveet transform into a waveet pyramid. On each eve of this pyramid, we detect the dominant periodicity and cacuate the instantaneous phase from the Hibert transform on that eve. The outcome of this mutieve Hibert transform and detection procedure wi be a compete description of mutieve periodicities and instantaneous phases. Pan (996) conducted a comparative performance study for a number of compex waveets. 6. A Quantum Space of Price and Time Eiott wave theory suggests that the price, more often than not, traces back to certain Fibonacci ratios such as 38.2%, 50%, 6.8%, 00%, etc. The support or resistance eves have a quantum nature (or at east geometrica). Gann was probaby the first in discovering the quantum structure of the price-time space. Gann theory of price-time cyces (Pan 2003) suggests that there is a significant symmetry between the price and the time, and time is often more important than price. When certain time zones and price eves coincide, the reversa is imminent. On the other hands, the og-periodic power aw modes show cear geometrica properties that are consistent with the Gann anges deveoped in technica anaysis. The dynamic swings and physica cyces identified and modeed from the historica index time series wi most ikey define a quantum space of price and time. This space is divided by important price eves and time zones defined by the previous dynamic swings and physica cyces. The market wi most ikey trave from one significant price eve to another or from one significant time zone to another. The actua path is not ony determined by dynamic swing phase and physica cyce phase, but aso by the possibe news impact. 7. Mutidimensiona Embedding and Nearest Neighbour Agorithm for Prediction Based on the fracta mode of dynamic swings and the waveet anaysis of physica cyces in a quantum space of price and time, we wi investigate two different approaches for predicting the stock index movement The first is a direct appication of the fracta mode 9

10 and waveet anaysis whose parameters are estimated from the historica data, especiay the ast and current Eiott waves. The second approach is a pattern recognition approach based on chaos theory. Each time sampe from historica time series data is embedded in a mutidimensiona feature space, where the feature vector consists of a subset of the fracta and waveet parameters, especiay the phases at the mutipe time scaes. For any given current time, its feature vector is constructed from the fracta and waveet modes, and then a certain number of its nearest neighbors are searched out from the historica pattern space. Finay, a certain noninear regression mode can be estimated from the nearest neighbors. This mode then can be used for prediction. Note that this regression is oca to the nearest neighbors, and thus the predictive mode is adaptive. 8. Concuding Remarks The Swingtum theory outined in this paper provides a comprehensive dynamic mode of stock market integrating fracta dynamic swings and physica cyces as we as the quantum price-time space. The mode is computabe in terms of statistica parameter estimation and nonparametric mutidimensiona embedding and nearest neighbor pattern recognition. The theory is a step toward unifying professiona technica anaysis and academic quantitative anaysis into a science of inteigent finance. The more genera Swingtum theory shoud extend the fracta and cycica modes of a univariate benchmark index to the mutivariate time series modes of intramarket and intermarket dynamic anaysis. This is an ongoing effort, further theoretica deveopment, system impementation and rea-data experiment wi be reported in the future. References Acheis S.B. (2000): Technica Anaysis from A to Z. McGraw-Hi Trade. Bukowski T.N. (2002): Trading Cassic Chart Patterns. John Wiey & Sons. Campbe J.Y., Lo A.W., and Mackinay A.C. (997): The Econometrics of Financia Markets. Princeton University Press, Princeton, N.J. Farmer J.D. (998): Market Force, Ecoogy and Evoution. Santa Fe Institute Working Paper , Farmer J.D. and Joshi S. (2000): The price dynamics of common trading strategies. Journa of Economic Behavior and Organization, October 2002, preprint at Ehers J. (200): Rocket Science for Traders: Digita Signa Processing Appications. John Wiey & Sons. Iinskaia A. and Iinski K. (999): How to reconcie market efficiency and technica anaysis. Larrain M. (99): Empirica tests of chaotic behavior in a noninear interest rate mode. Financia Anaysts Journa, 47. 0

11 Lo A.W. and McKinay C. (999): A Non-Random Wak Down Wa Street. Princeton, NJ: Princeton University Press. Mandebrot B.B. (982): The Fracta Geometry of Nature. Freeman. Murphy J. (999): Technica Anaysis of the Financia Markets: A Comprehensive Guide to Trading Methods and Appications. Prentice Ha Press. Pan H.P. (2003): A joint review of technica and quantitative anaysis of financia markets towards a unified science of inteigent finance.. Proc Hawaii Internationa Conference on Statistics and Reated Fieds, June 5-9, Hawaii, USA. Pan H.P. (996): Genera stereo image matching using symmetric compex waveets. SPIE Proc. Vo.2825 on Waveet Appications in Signa and Image Processing IV, Denver, USA. Peters E.E. (994): Fracta Market Anaysis. John Wiey & Sons. Prechter R.R. (2002): Conquer the Crash. John Wiey & Sons.. Sornette D. (2003): Why Stock Markets Crash Critica Events in Compex Financia Systems. Princeton University Press, Princeton. Vaga T. (99): The coherent market hypothesis. Financia Anayst Journa, December/January. Zhou W.X and Sornette D. (2003): Renormaization group anaysis of the anti-bubbe in the US S&P 500 index: Expanation of the hierarchy of 5 crashes and prediction.

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