Chaos Theory and Application in Foreign Exchange Rates vs. IRR (Iranian Rial)

Transcription

1 World Academy of Sciece, Egieerig ad Techology Chaos Theory ad Applicatio i Foreig Exchage Rates vs. IRR (Iraia Rial) M. A. Torkama S. Mahmoodzadeh, S. Pourroostae ad C. Lucas Abstract Daily productio of iformatio ad importace of the sequece of produced data i forecastig future performace of market causes aalysis of data behavior to become a problem of aalyzig time series. But time series that are very complicated, usually are radom ad as a result their chages cosidered beig upredictable. While these series might be products of a determiistic dyamical ad oliear process (chaotic) ad as a result be predictable. Poit of Chaotic theory view, complicated systems have oly chaotically face ad as a result they seem to be uregulated ad radom, but it is possible that they abide by a specified math formula. I this article, with regard to test of strage attractor ad biggest Lyapuov expoet probability of chaos o several foreig exchage rates vs. IRR (Iraia Rial) has bee ivestigated. Results show that data i this market have complex chaotic behavior with big degree of freedom. Keywords Chaos, Exchage Rate, Noliear Models. example heart pulses, clock s oscillatig movemet like a pedulum ad ecoomic fluctuatios all show a dyamic o liear behavior. Based o this theory, happeig i world are such dyamic ad complicated that they soud to be chaotic, but i effect, chaotic systems have fudametal ad fixed order. Although idetifyig this covert order is ot impossible but it is difficult. Because may parameters ad factors i dyamic ad upredictable dealig, form pheomea s behavior ad make typical future behavior [2,3]. I this article, we have tried to do a quick review of basis cocept ad mathematical, furthermore differet methods of chaotic test to be itroduced ad assessed. I secod part we will itroduce. Parts III, has bee allocated to heo chaotic model. Parts IV, discusses chaotic tests. I part V foreig exchage rate has bee discussed. I sectio VI we have doe a case study of Ira exchage rate ad fially i part VII we have culmiated above metioed discussio. C I. INTRODUCTION HAOS theory was itroduced ad applied by Edward Lorez i 965 for the first time i meteorology []. Nowadays this sciece is the milestoe for fudametal chages i scieces especially meteorology, astrology, mechaics, physics, mathematics, biology, ecoomics, statistic ad maagemet. Eve though chaos theory i recet decades has bee part of survey i miscellaeous scietific fields, but its basic cocept has its root i primitive humas uderstadig of the uiverse. The Greek word of chaos that is traslated to disorder ad lawlessess shows aciet Greek uderstads of the uiverse. Accordig to this view of poit, although world s etities seem to be chaotic ad radom ad as a result upredictable, but at the same time they are i order ad determiistic. If we cut oe chaotic system i a specific time, we will be faced with chaos ad upredictability. While we have a process view poit ad look the developmet of chaotic system i a adequate time period, the after we ca discer a fixed degree of order i chaos. Chaotic process is a product of a dyamic o liear system. Such systems have bee oticed i ature ad also i huma behaviors. For M. A. Torkamai is with Azad uiversity, at Varami-Pishva, Tehra, Ira (correspodig author to provide phoe: ; torkamai@gmail.com). S. Mahmoodzadeh is with Ecoomic Departmet, Allameh Tabatabaee Uiversity, Tehra, Ira ( soheil.mz@gmail.com). S. Pourroostaei is with Computer Egieerig Departmet, Ira Uiversity of Sciece ad Techology, Tehra, Ira ( Saeed_Pourroostaei@yahoo.com). Prof. C. Lucas is with Electrical ad Computer Egieerig Departmet, Uiversity of Tehra, Tehra, Ira ( lucas@ipm.ac.ir). II. ECONOMIC DYNAMICS Historically, ecoomists have, wheever possible, used liear equatios to model ecoomic pheomea, because they are easy to maipulate ad usually yield uique solutios. However, as the mathematical ad statistical tools available to ecoomists have become more sophisticated, it has become impossible to igore the fact that may importat ad iterestig pheomea are ot ameable to such treatmet. There is a strog support i ecoomics for both the sigificace of liear models, ad the advatages of oliear models. But oliear models clearly outperform liear models. Clearly the ecoomic world is oliear, so it would appear that focusig o liear dyamics is of limited iterest. However ecoomists have typically foud oliear models to be so difficult ad itractable that they have adopted the techique of liearizatio to deal with them. Importat pheomea for which liear models are ot appropriate iclude depressios ad recessioary periods, stock market price bubbles ad correspodig crashes, persistet exchage rate movemets ad the occurrece of regular ad irregular busiess cycles. Therefore, ecoomic theorists are turig to the study of o-liear dyamics ad chaos theory as possible tools to model these ad other pheomea [4]. The most excitig feature of oliear systems is their ability to display chaotic dyamics. Much ecoomic data has this radom-like behavior, but it comes from agets ad markets that are presumably ratioal ad determiistic. Radom-like data that ecoomists ofte ecouter might ot be comig from a radom system. The geeratig system 328

2 could be determiistic ad perhaps the ecoomy ca be explaied by a relatively simple oliear system. Chaos is widely foud i the fields of physics ad other atural scieces. However, the existece of chaos i ecoomic data is still a ope questio. Sice the mid eighties several ecoomists have tried to test for oliearity ad i particular for chaos i ecoomic ad fiacial time series [5]. Oe route toward fidig a oliear uderlyig system i the ecoomy would be to show that the data itself demostrates oliear or chaotic properties Researchers developed tests for chaos ad oliearity i data. There are two major classes of tests for chaos withi data. The first is ways to look at the paths or trajectories of the data whe the system s iitial coditios are adjusted slightly. This ca be doe by estimatig a Lyapouv expoet. The Lyapouv expoets are a measure of the average divergece (or covergece) betwee experimetal data trajectories geerated by systems with ifiitesimally small chages i their iitial coditios. If the data poits deviate expoetially whe there is a very small tweak o a determiistic model, it will have a positive Lyapouv expoet. If the paths coverge back to a steady state, the the Lyapouv expoet will be egative. A positive expoet sigals that the system must have sesitive depedece to iitial coditios ad therefore it is chaotic [6]. The secod type of test for chaos examies the dimesioality of the system. It may seem easy to explai that a square has two dimesios, ad a lie has oe, but it is sigificatly more complicated for chaotic systems sice they have o-iteger dimesioality. The fractioal dimesioality is what coied the term fractal for shapes geerated by chaotic data. Dimesioality aalysis becomes extremely complicated with realistic chaotic systems. Chaos exists i may differet fractal dimesios. Ufortuately, the aalysis gets more ad more complex the larger the fractal dimesio beig searched ad the tests for chaos become weaker. Uless data displays low-dimesioal chaos, it may be udetectable to curret tests. This is a sigificat obstacle to chaotic-ecoomic theory, ad oe of the mai reasos the literature has ot reached a cosesus o the existece of chaotic dyamics i data [7]. III. HENON CHAOTIC MODEL Chaotic series ca be cosidered as a complex of oliear processes that make irregular ad very complicated results. Oe of chaotic map is Heo model. This model is 2 variable, o liear ad quadratic. Relatios of this model are as follows: Y + = + a = b 2 + Y That i chaotic regio a=.4 ad b=0.3. Primary coditios of is x 0 =0.8, x =0.63. I Figs. ad 2, relatio of variable i oe dimesio ad two dimesio spaces has bee show [2]. World Academy of Sciece, Egieerig ad Techology () 329 Fig. Oe dimesioal Heo map Fig. 2 Two dimesioal Heo map IV. CHOATIC TESTS Geerally, 2 view poits have bee itroduced for evaluatio of situatio of complicated time series. I first stadpoit, this issue is examied whether the time series i questio have bee made by a defiite process or radom. I 2 d viewpoit it is tried to recogize whether the time series is a result of a chaotic behavior or a o chaotic behavior. Methods that are utilized i st view poit are based o aalysis of system's correlatio dimesio. Methods pertiet to 2 d viewpoit are mostly icludig biggest lyapuov expoet. A. Attractor Dimesio Test This test is based o oe of the special specificatios of radom process i compariso with chaotic process. Radom processes iclude ulimited dimesios. But a chaotic process has more limited dimesios. It meas that it icludes a complex of poits that time series would result i them. Therefore by calculatig dimesios of a series, it is possible to uderstad its makig process. Accordig to this method if series domai was high, it would show a radom process, otherwise, it would be a chaotic process. Attractor dimesio by usig a variable called itegratio correlatio that was itroduced by procaccia ad grassberger i 983 is calculated as follows [7]. Dimesio is omiated as low boud of ecessary idepedet variable's quatity for describig the model. Attractor is the developed cocept of all equilibrium paths i phase space like equilibrium poits ad limit circles i stable systems that have got correct dimesio. I cotrast, chaotic systems attractor has fractal dimesio ad are called strage attractor. I most primitive method of desigatig of fractal dimesio, M(L) are cosidered as

3 quatity of ultra cubes with the dimesio of M ad legth of the lie "L" that covers attractor, based o this: D M () l ~ l (2) Ad "D" that is fractal dimesio is obtaied: log[ M () l ] D = lim (3) l 0 log l Accordig to defiitio, poit, lie ad plate i 2 dimesio space have dimesios of 0, ad 2 respectively. This defiitio of fractal has practical limits ad solely attractor's geometrical structure has bee cosidered i it. Correlatio dimesio is the most commo estimatio of attractor dimesio that is simply calculated by procaccia ad grassberger method. Based o this method, vectors "m" are part of i coditio of (t) time series that are made with the legth of "N". i = [ x( ti ), x( ti + ) x( ti + m) ] (4) Correlatio itegral for "N" vector with the distace less tha "r" from each other is calculated like this: N N C( r, m) = lim I( r i j ) (5) N 2 N i= j= I which C(r,m) is a estimatio of a probability that 2 vectors of time series with legth of N, have a distace less tha "r" from each other. "I" also is a fuctio of heavy side ad is defied as below: 0 x y > r I( x, y) = (6) x y < r I case quatities of N poits are big eough, distributio is Expoetial fuctio υ ( r; m) r C ~ That υ is correlatio dimesio υ = logc lim r 0 logr ( r, m) This test is based o this fact that chaotic maps do ot fill the space i big dimesios, but radom data are ot like this. Whe the chaotic process is more complicated, it is ecessary that data should be cosidered i bigger ad higher dimesios. A chaotic process ca fill a space with "" dimesios but it leaves big holes i "+" dimesio. It is clear that this method is ot practical i big dimesio graphically. B. Test of Lyapauv's Biggest Expoet lyapauv's expoet has bee recogized as oe the most suitable methods for recogitio of dyamic processes owadays. Lyopauv's expoet is a average expoet of dyamic processes ad shows rate of divergece ad covergece of coditio routes i phase space. Divergece of coditio route shows that a system with slight differeces i primary coditios, with elapsig of time has very differet coditio routes from each other ad predictability capability i these kids of processes disappears quickly. Accordig to defiitio, each system by havig at least a positive lyapauv's expoet is a chaotic system. Coversely the size of related expoet is commo fitted with a time that after it, dyamical process would become upredictable [8,9]. World Academy of Sciece, Egieerig ad Techology (7) (8) Level of beig chaotic of time series is measured based o lyapauv's expoet. This measure states that with alteratios i primary coditios or model parameters, produced series up to which level differs from origial series. Lyapauv's expoet shows certaity of short term time series ad therefore predictability of series. I other words this expoet shows level of beig chaotic of a series. High quatities of this expoet show high sesitivity of series to primary quatities. If differece of primary quatities is a kow quatity, differece of series quatity after certai quatity of stage, is equal to expoetial fuctio these quatities. The less the quatity of this expoet, the less is the growth of this expoetial. Lyapauv's biggest expoet test is oe of the most importat methods for recogizig chaos i time series. I order to calculate lyapauv's expoet, the vectors cotai m compoets are used. i = [ x( ti ), x( ti + ) x( ti + m) ] (9) From vectors with distace less tha "r" it is calculated as per follows: m m r0 j) = i j r (0) The below term is calculated: m m i+ j+ d ( m i j) () ;, = r0 j) Tha after, lyapauv's biggest expoet is calculated: log[ d j) ] Le ( m, ) = (2) i j N( N ) "Le" sig shows etity of time series i questio. Positive amout of "Le" shows chaotic ess of process ad difficulty of predictability ad whe it is egative, it shows that process i the log ru is o chaotic ad predicable. If "Le" moves towards positive quatity ear zero, chaotic system is weak ad middle term predictability is possible [0]. C. Test Result o Heo Map As it was metioed, Heo map is oe of the chaotic maps. This model complies with followig certai behavior patter. With applicatio of attractor's dimesio ad lyopauv's biggest expoet tests o Heo map, followig results were obtaied. With the applicatio of attractor's dimesio test o Heo map this quatity proved to be equal to.2. I Table I, lyopauv's test results have bee calculated i restructured dyamics of to 0 dimesioal developed vectors. TABLE I LYOPANUV S BIGGEST EPONENT TEST RESULTS ON HENON MAP Max D= D= D= D= D= D= D= D= D= D=

4 World Academy of Sciece, Egieerig ad Techology V. FOREIGN ECHANGE RATE A exchage rate represets the value of oe currecy i aother. A exchage rate betwee two currecies fluctuates over time. The foreig exchage market is the largest ad most liquid of the fiacial markets. Foreig exchage rates are amogst the most importat ecoomic idices i the iteratioal moetary markets. The forecastig of them poses may theoretical ad experimetal challeges. Foreig exchage rates are affected by may highly correlated ecoomic, political ad eve psychological factors. The iteractio of these factors is i a very complex fashio. Therefore, to forecast the chages of foreig exchage rates is geerally very difficult. Researchers ad practitioers have bee strivig for a explaatio of the movemet of exchage rates. Thus, various kids of forecastig methods have bee developed by may researchers ad experts. Techical ad fudametal aalyses are the basic ad major forecastig methodologies which are i popular use i fiacial forecastig. Like may other ecoomic time series, forex has its ow tred, cycle, seaso, ad irregularity. Thus to idetify, model, extrapolate ad recombie these patters ad to give forex forecastig is the major challege. Foreig exchage rates were oly determied by the balace of paymets at the very begiig. The balace of paymets was merely a way of listig receipts ad paymets i iteratioal trasactios for a coutry. Paymets ivolve a supply of the domestic currecy ad a demad for foreig currecies. Receipts ivolve a demad for the domestic currecy ad a supply of foreig currecies. The balace was determied maily by the import ad export of goods. Thus, the predictio of the exchage rates was ot very difficult at that time. Ufortuately, iterest rates ad other demad} supply factors had become more relevat to each currecy later o. O top of this the fixed foreig exchage rates was abadoed ad a floatig exchage rate system was implemeted by idustrialized coutries i 973. Recetly, proposals towards further liberalizatio of trades are discussed i Geeral Agreemet o Trade ad Tariffs. Icreased Forex tradig, ad hece speculatio due to liquidity ad bods, had also cotributed to the difficulty of forecastig Forex [0]. Geerally, there are three schools of thought i terms of the ability to profit from the fiacial market. The first school believes that o ivestor ca achieve above average tradig advatages based o the historical ad preset iformatio. The major theory icludes the Radom Walk Hypothesis ad Efficiet Market Hypothesis. The secod school's view is that of fudametal aalysis. It looks i depth at the fiacial coditio of each coutry ad studies the effects of supply ad demad o each currecy. Techical aalysis belogs to the third school of thought who assumes that the exchage rates move i treds ad these treds ca be captured ad used for forecastig. It uses such tools as chartig patters, techical idicators ad specialized techiques like Ga lies, Elliot waves ad Fiboacci series []. VI. CASE STUDY This method is applied o some foreig exchage rates data of day time slice sampled vs. 0RIAL, sice March 25, 2002 to May 23, Results have show i Table II. TABLE II STRANGE ATTRACTOR AND LYOPANUV S BIGGEST EPONENT TEST RESULTS USD vs. 0RIAL Max Max GBP vs. 0RIAL Max Max EUR vs. 0RIAL Max Max CAD vs. 0RIAL Max Max CHF vs. 0RIAL Max Max Strage Attractor USD GBP EUR CAD CHF VII. CONCLUSION Progresses i calculatio tools i recet decades have provided us with the possibility of utilizig theories based o existece of certai or chaotic o-liear patters. Chaotic theory with more through study of specificatios of complicated behavior ad data that seem to be radom, try to recogize order ad patter goverig them ad use them for predictability future tred i short term. Nowadays this kowledge with the help of data behavior aalysis has provided the base of structural chages i future performace predictio. I this article with reviewig the cocepts of this theory ad testig for kowig chaos existece, we have examied a case study. Results obtaied from this study shows existece of complex chaotic behavior i foreig exchage 33

5 rate market i Ira. We ca say data have big degree freedom i their behavior ad shows radom like behavior. World Academy of Sciece, Egieerig ad Techology REFERENCES [] B. Davies; "Explorig Chaos: Theory ad Experimet (Studies i Noliearity S.)", Westview Press, [2] J. Baks, D. Valetia, J. Arthur; "Chaos: A Mathematical Itroductio", Cambridge Uiversity Press, [3] W. Brock, P. Lima, "Noliear Time Series, Complexity Theory, ad Fiace", Hadbook of Statistics, Vol 4, Elsevier Sciece Publisher, B.V [4] Y. Wu, D. Z. Zhag, "Demad fluctuatio ad chaotic behaviour by iteractio betwee customers ad suppliers", It. J. Productio Ecoomics 07, , [5] Julie Clito Sprott; Chaos ad Time-Series Aalysis, Oxford Uiversity Press [6] W. Barett, P. Che, "The Aggregatio-Theoretic Moetary Aggregates are Chaotic ad Have Strage Attractors", Proceedigs of the Third Iteratioal Symposium i Ecoomic Theory ad Ecoometrics, Cambridge, Cambridge, Uiversity Press, 987. [7] P. Grassberger, I. Procaccia, "Measurig the Strageess of Strage Attractors", Physica. 9D, 30-3, 983. [8] S. Boccalett C. Grebog C. La H. Maci D. Maza, "The cotrol of chaos: theory ad applicatios", Physics Reports 329, 03-97, [9] M. Shita O. Lita "Noparametric Neural Network Estimatio of Lyapuov Expoets ad a Direct Test for Chaos", Discussio Paper, No. EM/02/434, [0] H. Dewachter, M. Lyrio "The cost of techical tradig rules i the Forex market: A utility-based evaluatio" Iteratioal Moey ad Fiace, Vol.25, pp , [] B. Davies; "Explorig Chaos: Theory ad Experimet (Studies i Noliearity S.)", Westview Press, [2] W. Barett, A. Serletis; "Martigales, oliearity, ad chaos", Joural of Ecoomic Dyamics & Cotrol ,