Iteratioal Joural of Maagemet (IJM), ISSN 0976 6502(Prit), ISSN 0976 6510(Olie) Volume 1, Number 2, July - Aug (2010), pp. 09-13 IAEME, http://www.iaeme.com/ijm.html IJM I A E M E AN ANALYSIS OF STABILITY OF TRENDS IN GOLD PRICES USING FRACTAL DIMENSION INDEX (FDI) COMPUTED FROM HURST EXPONENTS E. Priyadarshii Research Scholar Sathyabama Uiversity Cheai-119 E-Mail: priyaeb@gmail.com Dr. A. Chadra Babu Dept of Mathematics MNM Jai Egieerig College Cheai-97 E-Mail: chadrababu_a@yahoo.co.i ABSTRACT Chaos is a oliear, dyamic system that appears to be radom but is actually a higher form of order. All chaotic systems have a quatifyig measuremet kow as a fractal dimesio. The fractal dimesio idex (FDI) is a specialized tool that applies the priciples of chaos theory ad fractals. With FDI oe ca determie the persistece or ati-persistece of ay equity or commodity. I this paper we study the data from gold rates by computig the fractal dimesio idex. The fractal dimesio idex is computed from the Hurst expoet ad the Hurst expoet is computed from Rescaled Rage R/S. Keywords: chaos, fractals, rescaled rage, persistece. 1. INTRODUCTION I Chaos theory, we defie a apparetly radom evet i the marketplace that has some degree of predictability. I order to do this we use fractal as a tool that is a represetatio of order from chaos. The fractal is commoly defied as a object with self-similar idividual parts. I the markets, a fractal ca be thought of as a object or time series that appears similar across a rage of scales. Each frame may zigzag a little differetly, but whe viewed from afar they have similar attributes o each scale. 9
All chaotic systems have a quatifyig measuremet kow as a fractal dimesio. The fractal dimesio is a o-iteger dimesio that describes how a object takes up space. Fractals will maitai their dimesio regardless of the scale used. This is evidet i atural pheomea such as moutais, coastlies, clouds, hurricaes ad lightig. Similarity across scales is essetial i tradig because each time frame of a market will have a similar fractal patter. Such markets ca oly be forecasted reliably with priciples applicable to oliear, atural systems usig Fractal geometry as tool. 2. FRACTAL DIMENSION INDEX (FDI) The fractal dimesio idex (FDI) is a tool that applies the priciples of chaos theory ad fractals. This specialized idicator idetifies the fractal dimesio of the market by usig re-scaled rage aalysis ad a estimated Hurst expoet. It does so by usig all available data o the time/price chart to determie the volatility or treds of a give market. FDI is the same type of tool used by emiet fractal scholars Beoit Madelbrot, H.E. Hurst, ad Edgar Peters i their examiatios of time series aalysis. With FDI we ca determie the persistece or ati-persistece of ay equity or commodity that we display i our graphig program. A persistet time series will result i a chart that is less jagged, subject to fewer reversals, ad resembles a straight lie. A ati-persistet time series will result i a chart that is more jagged ad proe to more reversals. Essetially, FDI will tell us whether a market is a radom, idepedet system or oe with bias. The FDI is based o the work of Madelbrot ad Hurst. 3. TRADING WITH THE FDI The FDI is useful to determie the amout of market volatility. The easiest way to use this idicator is to uderstad that a value of 1.5 suggests that the market is actig i a completely radom fashio. As the market deviates from 1.5, the opportuity for earig profits icreases i proportio to the amout of deviatio. The etire scale is based o a rage of 1-2, suggestig extreme liearity to extreme volatility. If the FDI is closer 2, the probability is higher that the ext move will be i the opposite directio of the curret move. A FDI closer to 1 sigals a tredig market i oe directio. This kowledge aloe gives the trader a icredible advatage, because it ca idicate which markets have the most opportuity. 10
4. COMPUTING THE FDI For computig the FDI oe must first compute the Hurst expoet. For a time series x, x2,,, 1 x ( R / S ) = c * Hurst computed the Rescaled Rage (R/S) ad foud that log (( R / S) ) = log ( c) + H *log ( ) Where c ad H are costats ad H is the Hurst coefficiet. Thus H is obtaied as the slope of the Log (R/S) versus Log () plot. H = 0.50 implies that the time series is a radom time series ad implies the absece of log term statistical depedece. The case 0.50 < H < = 1.00 implies a persistet time series, a time series characterized by log memory effects. This meas that if the tred has bee positive i the last observed period, the chaces are that it will cotiue to be positive i the ext period. Coversely if it has bee egative i the last period it is more likely that it will cotiue to be egative i the ext period. The level of persistece is judged by how far H is above 0.5. The case 0 < = H < 0.50 implies ati-persistet time series. A atipersistet system reverses itself more frequetly tha a radom oe. After the computatio of the Hurst expoet, we derive the fractal dimesio of the time series which easily accomplished usig the formula D = 2 - H. The value of D is the fractio of a dimesio betwee 1 ad 2 that the price data represets. Logarithmic returs are used to compute FDI. Because logarithmic returs sum to cumulative returs, most aalyst agree that this is most appropriate for fiacial aalysis. Whe experimetig with the FDI, a large data set is required. The results may appear distorted if there is ot eough data. H 5. CORRELATION BETWEEN PERIODS The correlatio C N betwee periods is calculated as follows: C N = 2 (2H 1) 1 (1) (2) A radom time series (H = 0.5) has zero correlatio. A persistet time series (with H greater tha 0.5) results i positive correlatio where as a ati-persistet time series 11
results i egative correlatio. C N is a measure of the log-term memory preset i the time series. C N = 0.25 implies that 25% of the data is iflueced by the past. 6. THE V STATISTIC: ( R/ S) V = The V statistic was origially used by Hurst for testig stability. If the process is a idepedet radom process, the the plot of V versus Log will be flat. If the process is persistet (H > 0.5) the the graph will be upwardly slopig ad if the process is atipersistet (H < 0.5) the graph will be dowward slopig. 7. DATA ANALYSIS: The mothly gold rates (i rupees) required for study was collected from Ja 1971 to Apr 2010 spaig 40 years obtaied from World Gold Coucil. Figure1 Graph of log versus log R/S 8. CONCLUSION Figure 2 Graph of log versus V plot From the study carried out, the Hurst Costat H = 1.047. The fractal dimesio idex D = 2 1.047 = 0.953 which idicates that gold rates are highly persistet resultig i a chart that is less jagged, resemblig a straight lie. The plot of V versus log has upward slope cofirmig its persistece. The correlatio betwee periods C = 2 2H-1-1 =1.1347 which idicates the presece of log term memory. 12
9. BIBLIOGRAPHY 1. Barsley.M. (1988), Fractals everywhere, Bosto M A etc: Academic Press, Ic. 2. Edgar.E.Peters. (1994), Fractal Market Aalysis, Joh Wiley ad Sos, New York, pp189-196. 3. Edgar.E.Peters. (1991), Chaos ad Order i capital markets, Joh Wiley ad Sos, New York. 4. Edgar.E.Peters. (1989), Fractal structure i the capital markets, Fiacial Aalysts Joural (July Aug 1989) pp32-37. 5. Lo, A.W. (1991), Log-Term Memory i Stock Market Price, Ecoometrica, 59, pp 1279 1313. 6. Madelbrot.B. (1982), The Fractal Geometry of Nature, New York, W.H.Freema, 7. Madelbrot.B. (1972), Statistical Methodology for o-periodic cycles from the covariaces of R/S Aalysis, Aals of Ecoomics ad Social measuremet. ABOUT THE AUTHORS First author: Mrs. E. Priyadarshii, M.Sc. M.Phil. P.G.D.C.A. is a seior lecturer (Mathematics) i Sathyabama Uiversity, Cheai. She has 20 years of teachig experiece i various Colleges. At preset she is doig research at Sathyabama Uiversity. Secod author: Dr. A. Chadra Babu is the Head of the Mathematics Departmet, MNM Jai Egieerig College, Cheai-97. He has got rich experiece i teachig ad guidig research studets. He has authored may books i Mathematics ad Computer Sciece. 13