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A new analysis of intermittence, scale invariance and characteristic scales applied to the behavior of financial indices near a crash
ARTICLE IN PRESS
Physica A 367 (2006) 345–352 www.elsevier.com/locate/physa
A new analysis of intermittence, scale invariance and characteristic scales applied to the behavior of financial indices near a crash Maria Cristina Mariani, Yang Liu Department of Mathematical Sciences, New Mexico State University, 88003-8001 Las Cruces, NM, USA Received 13 July 2005; received in revised form 29 November 2005 Available online 3 February 2006
Abstract This work is devoted to the study of the relation between intermittence and scale invariance, and applications to the behavior of financial indices near a crash. We developed a numerical analysis that predicts the critical date of a financial index, and we apply the model to the analysis of several financial indices. We were able to obtain optimum values for the critical date, corresponding to the most probable date of the crash. We only used data from before the true crash date in order to obtain the predicted critical date. The good numerical results validate the model. r 2006 Published by Elsevier B.V. Keywords: Econophysics; Scale invariance; Intermittence; Stock market prices; Financial indices; Crashes
1. Introduction During the last years the study of log-periodic structures and characteristic scales and the relation with the concept of scale invariance had grown due to the great amount of physical systems presenting log-periodic structures: fluid turbulence [1,2], diamond Ising model [3], earthquakes [4], materials rupture [5], black holes [6] and gravitational collapses [7] among others. In a mathematical context, we recall constructions as the Cantor fractal [3][8], with a discrete scale changes invariant. The presence of logarithmic periods in physical systems was noted by Novikov in 1966 [9], with the discovery of intermittence effect in turbulent fluids. The relation between both effects has been deeply studied, but it has not been formalized yet. At the same time, the complexity of international finance has grown enormously with the development of new markets and instruments for transferring risks. This growth in complexity has been accompanied by an expanded role for mathematical models to value derivative securities, and to measure their risks. A new discipline Econophysics, has been developed [10]. This discipline was introduced in 1995, see Stanley et al. [11]. It studies the application of mathematical tools that are usually applied to physical models, to the study of Corresponding author.
E-mail addresses: [email protected] (M.C. Mariani), [email protected] (Y. Liu). 0378-4371/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.physa.2005.11.047
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financial models. Simultaneously, there has been a growing literature in financial economics analyzing the behavior of major Stock indices [10,12–15]. The Statistical Mechanics theory, like phase transitions and critical phenomena have been applied by many authors to the study of the speculative bubbles preceding a financial crash (see for example [16,17]). In these works the main assumption is the existence of log-periodic oscillation in the data. The scale invariance in the behavior of financial indices near a crash has been studied in Refs. [18–20]. This work is organized as follows: In Section 2 we give a short introduction to the relation between intermittence and scale invariance, the conditions that a function has to satisfy when both effects are present. We analyze the relation with characteristic scales, and we finally present a method that detects characteristic scales in different systems using the previous results. In Section 3 we present a model that predicts the existence of intermittence and characteristic scales in the behavior of a financial index near a crash [20]. In Section 4 we develop the methods that we will use for our numerical analysis. Finally, we apply the model to the analysis of the behavior of several financial indices: the S&P500 index near the October 1987 crash, and the Argentina MERVAL index as well as the Brazil BOVESPA index and the Mexico MXX index near the October 1997 Asian crash. 2. Scale invariance and intermittence In this section we analyze the relation between intermittence and scale invariance and we introduce definitions and notation that will be used later. For further details see Ref. [20] and its references. A function A, that depends on a variable x, is invariant for the scale change lx when AðxÞ ¼ mAðlxÞ, (1) where m is a constant independent of x. For a detailed discussion about this definition see Ref. [20]. Any observable which remains invariant for the scale change x-lx can be expressed as 1 X AðxÞ ¼ xlogl m an ei2pn log l x .
(2)
n¼1
Now we recall the difference between discrete and continuous invariance. When the observable given by (2) presents continuous scale invariance, for any real number l there exists m such that condition (1) is fulfilled. We can deduce that in this case logl m does not depend on l, and an ¼ a0 don . When the observable A(x) satisfied Eq. (1) only for numerable values of the l’s, it presents a discrete scale invariance. A particular type of scale invariance is the one arising in the existence of intermittences or ‘‘stationary intervals’’, constant in the logarithm of the independent variable. The functions that can be obtained from this analysis are f F ðxÞ ¼ beaF ðloga xÞ ,
(3)
f C ðxÞ ¼ beaC ðloga xÞ ,
(4)
where b and a are real numbers, a is positive and F ðxÞ ¼ IðxÞ and CðxÞ ¼ IðxÞ þ 1 are the Floor and Ceiling functions, respectively. Hence, the value obtained when applying the Floor function to a variable x will be the nearest entire number to x from the left, and the value obtained by the Ceiling function will be the nearest entire number to x from the right. These two functions are discrete scale invariants, and more specifically, they satisfy Eq. (1) only when l ¼ an ;
n 2 Z.
(5)
We recall that the conditions for a function to have discrete scale invariance, after we know that the system has intermittences are the following: (i) The intermittence intervals must be constant in logarithmic scale, i.e., the steps have to be discrete dðln xÞ ¼ K,
(6)
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Fig. 1. Basic scheme showing the relation between a power-law (in solid line) and a function like the one defined in (4) (dash line). The black points are the intersection points of both functions.
where K is a positive number. Let a ¼ eK . Then we have d loga x ¼ 1.
(7)
Then, the interval of time of the intermittence is such that the logarithm of the variable x, in a basis a, has advanced one unit (in that period of time). Hence, we can conclude that due to the longitude of the intermittences there exists a basis in which the logarithm of the variable is equal to one. (ii) The stationary intervals are consecutive, when one finishes, begins the next one. (iii) The function such that its variable is discretized with the rule dðloga xÞ ¼ 1 is a power-law, and the beginning and the end of a stationary interval have both to be a point of the function. We will call this function the basis function, which can be illustrated as follows: Now we apply the tools given by the theory previously developed in order to analyze data presenting characteristics similar to those mentioned above. Therefore, the data basis function will be a power-law (Fig. 1). First, we will find this law; the second step will be to reduce the free parameters to only one: the logarithmic basis. Finally, we will use (3) or (4) in order to find the value of a minimizing the distance with the data. Hence, we obtain the system characteristic scales. The estimation of the power-law is crucial in this method because it will be the basis of the function that will be used for fitting the data. 3. Financial indices prices near a crash The evolution of a financial index represents the changes of a portfolio [21]. There is a simple model that takes account of the evolution of an asset price in the market [22], this model considers two contributions to the percentage variation in the asset price: one deterministic, and one stochastic dS ¼ m dT þ s dX , (8) S where S is the asset price, m is a constant (called drift), dT is the interval of time, s is another constant (called volatility) and dT is a random variable which is assumed following normal probability distribution. We recall that the physical process represented by (8) is Brownian motion with a drift. We focus our attention in the deterministic contribution dS ¼ m dT. (9) S As a financial index can be considered an asset, its (deterministic) financial behavior will be given by (9). Our hypothesis is that near a crash Eq. (9) is modified dS Sc S ¼m , dT Tc T
(10)
where Tc and Sc are respectively, time and price for which the crash takes place. The heuristic analysis is as follows: near a crash there is a factor that produces a considerable increase in the index price, on the other
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hand, when the price is very near to the crash price, there has to exist another factor smoothing those variations, otherwise the crash would take place before the real date (for further details see Ref. [20]). Changing the variables in (10) we obtain dP P ¼m , dt t
(11)
where the variables are not anymore absolute data of the system: t and P are the distance to the critical time and critical price, respectively. The second assumption will be that the temporal steps are discrete; therefore, the index evolution is not continuous and we have the intermittence phenomena. The index evolution is given by dt ¼ Kt.
(12)
Then the frequency in the index price changes is proportional to the distance to the date in which the crash takes place. Eq. (12) implies that d ln t ¼ K.
(13)
From Eqs. (11) and (13) we arrive at functions like (3) or (4). In this case we will work with function (3), due to the fact that intermittences must take into account that the time approaches the critical time from the right, because of the change of variables (11). 4. The data analysis methods: analysis of the parameters and estimation of the critical time We are plotting the t–p figure where t is the time distance from the crashing time and p is the price distance from the crashing price. We can model the market behavior by using the following equation: p0 f ðtÞ ¼ beaF ðloga ðt0 tÞÞ ,
(14)
where p0 is the crashing price, t0 is the crashing time, f(t) is the market price at time t and F is the floor function. Let us remove the floor function at this moment. Consider p0 f ðtÞ ¼ bea log
a ðt0 tÞ
,
(15)
we can rewrite Eq. (15) in log-domain as a . ln p0 f s ðtÞ ¼ ln b þ glnðt0 tÞ; where g ¼ ln a
(16)
We want to remark that the purpose of Eq. (15) is to show the linear relationship in log-domain between tt0 and p0f(t), which is trying to describe the envelop behavior of indices near a crash. The floor function is not really dropped from the model. Thus, ln p0 f s ðtÞ has a linear relationship with lnðt0 tÞ. Suppose p0 and t0 are given (or somehow estimated), we can easily estimate ln b and g by linear regression. However, Eq. (15) only describes the power-law of the system. It is a smooth line while the market will move up and down around it following the model described by (14). We must get back to (14) to describe the dynamic behavior of the market. In other words, we need to find the characteristic scale a. We recall that the parameters a and b have been determined in (16). Rewrite Eq. (14) in log-domain as lnðt0 tÞ ln p0 f ðtÞ ¼ ln b þ aF . (17) ln a
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Then subtract (16) from (17) and multiply by 1/g, we have that lnðt0 tÞ G ðlnðt0 tÞÞ ¼ lnðt0 tÞ ln a F ln a 1 ln p0 f ðtÞ ln p0 f s ðtÞ . ¼ g
ð18Þ
Therefore,
lnðt0 tÞ þ ln a G ðlnðt0 tÞ þ ln aÞ ¼ lnðt0 tÞ þ ln a ln a F ln a lnðt0 tÞ ¼ lnðt0 tÞ þ ln a ln a F þ1 ln a lnðt0 tÞ ¼ lnðt0 tÞ lna F ¼ Gðlnðt0 tÞÞ. ln a
ð19Þ
We can see that G( ) is periodical with a period of ln a. To find a, we need to find the period of G( ). An ideal periodical function will have a spike in frequency domain at its period. So we use Fourier transform and look for the frequency with a high power in the frequency domain. A general discrete Fourier transform is given by F ðk Þ ¼
N X
f ðnÞej2pnk=N .
(20)
n¼0
In our method, we use the following transform: QðkÞ ¼
N=2 X
G ðxn Þej2pðk1Þðn1Þ=ðNK Þ ,
(21)
n¼1
where K is some constant that will determine the resolution of the Fourier transform, and xn, n ¼ 1; 2; ; N are the equal distant points between lnðt0 T 0 Þ and lnðt0 T 1 Þ, where T0 and T1 are respectively, the start and ending points of the observation window. Notice that the original market data is daily price. When we use them in log-domain, the sample points of G( ) are no longer equal distant. However, (21) requires to evaluate G( ) at equal distant points xn. To get the Fourier transform, we use linear interpolation to evaluate G(xn). If we have tm1 oxn ptm ; the interpolation is given by G ym G ym1 G est ðxn Þ ¼ xn ym1 for ym1 oxn pym , (22) ym ym1 where ym ¼ lnðt0 tm Þ. We remark that the characteristic that the market must exhibit for assuming this approximation is that the behavior of the market is ‘smooth’ up to the crashing point. In other words, we assume the index curve is differentiable at each time point except for the crashing point. Therefore, we can use interpolation. While higher-order interpolation could be used, we found that a simple linear interpolation between two points is accurate enough for our purpose. The Fourier transform actually shows us the power at each frequency. Ideally, the power of the dominant frequency will be a local maximum. So we pick the frequency with a local maximum power within a roughly preset range. Once we have the frequency f, we can easily calculate the characteristic scalar a by a ¼ e1=f . Then a is a function of p0 and t0, and Eq. (14) is fully determined.
(23)
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Now we can use the Square Error e2 between Eq. (14) and the real market data to measure how good our model describes the market behavior. Notice that at this stage e2 is a function of only p0 and t0. i.e. e2 ¼ e 2 p0 ; t 0 . (24) P 2 2 Explicitly, e ¼ ð f ðtÞ PðtÞÞ ; where p(t) is the real market closing price. t Therefore, by minimizing e2, we should be able to get the best estimate of t0 and p0. 5. Numerical results In this section we present the numerical results. We plot the function of the error between the model and the real index value against the t and p. The optimal estimation should be corresponding to the area with minimum error. To give a better view, we plot both the surface graphs and the contour graphs. We can see that e2 is not sensitive about p, so it provides little capability to predict the crashing prices (Figs. 2–5). However, the optimal values of t can be clearly identified from the figures. We list the predicted crashing date comparing with the real crashing date in the following table Table 1. We are using daily market closing price for the index value.
Fig. 2. Error function of SP500 in log-domain.
Fig. 3. Error function of BVSP in log-domain.
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Fig. 4. Error function of MXX in log-domain.
Fig. 5. Error function of MERV in log-domain.
Table 1 Comparison between the real crashing date and the estimated date
Real crashing date Real topping date Estimated date
S&P500
BVSP
MXX
MERVAL
10/19/1987 10/06/1987 10/28/1987
10/27/1997 07/11/1997 07/03/1997
10/21/1997 10/21/1997 10/30/1997
10/22/1997 10/22/1997 11/08/1997
We recall that in the case of the BVSP index, the crash happened 75 days after the market top. We also recall that the graphs are plotted in log domain. 6. Conclusions The effects of certain local crisis on various and distant markets have largely been cited. The collapse of the crashes of 1987 (S&P500) dragged the collapse of markets worldwide. However, not every crisis has sufficient
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strength as to drag the fall of leading indices in other countries. In Ref. [16] it has been shown that the crashes of Asian indices had consequences on emergent markets: the Asian crisis had sufficient strength as to drag the fall of leading Latin American indices. Clearly, all these indices crashed in similar dates due to a dragging correlated effect, which most likely started with the instability of the HSI index. This signals the likelihood of the events in different markets and different economic realities which strengthens the hypothesis of imitation and long range correlations among traders. About the stability of the method, we want to remark that, even if the maximal and minimal values for the error are apparently similar, the graphs are plotted in log-domain. Furthermore, we want to remark that we believe the noise in the market can actually change the crashing date, and the change may be significant. We believe the market is a very complex system. The crash might be delayed due to some market force, so it happened sometime after the market top, but that is out the scope of this work. Our goal in this work is to develop a method for finding the market top which leads to a crash. We also want to remark that we only use the data up to a couple of weeks before the crash in all the cases to estimate the critical time. The estimations have errors of around only 10 trading days. The excellent results validate the method. Acknowledgments The authors want to thank the anonymous referees for their careful reading of the manuscript and their fruitful remarks. This work was partially supported by ADVANCE – NSF. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
E.A. Novikov, Phys. Fluids A 1 (1990) 814. A. Johansen, D. Sornette, Phys. D 138 (2000) 302. D. Sornette, Phys. Rep. 297 (1998) 239–270. H. Saleur, C. Sammis, D. Sornette, Nonlin. Proc. Geophys. 3 (1996) 102. A. Johansen, D. Sornette, H. Saleur, Eur. Phys. J. B 15 (2000) 551. M.W. Choptuik, Phys. Rev. Lett. 70 (1993) 9. A.M. Abrahams, Gen. Relat. Gravit. 26 (1994) 379. B.B. Mandelbrot, The Fractal Geometry Of Nature, Freeman, New York. E.A. Novikov, Dokl. Akad. Nauk. SSSR 168 (1996) 1279. R.N. Mantegna, H.E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, Cambridge, 1999. H.E. Stanley, V. Afanasev, L.A.N. Amaral, S.V. Buldyrev, A.L. Golderberger, S. Havlin, H. Leschorn, P. Maass, R.N. Mantegna, C.-K. Peng, P.A. Prince, M.A. Salinger, M.H.R. Stanley, G.M. Viswanathan, Physica A 224 (1996) 302. H.E. Stanley, L.A.N. Amaral, D. Canning, P. Gopikrishnan, Y. Lee, Y. Liu, Physica A 269 (1999) 156. M. Ausloos, N. Vandewalle, Ph. Boveroux, A. Minguet, K. Ivanova, Physica A 274 (1999) 229. J.-Ph. Bouchaud, M. Potters, The´orie des riques financiers, Alea-Saclay/Eyrolles, Paris, 1997. R.N. Mantegna, H.E. Stanley, Nature 376 (1995) 46. A. Johansen, D. Sornette, O. Ledoit, Int. J. Theor. Appl. Finan. 3 (2000) 219. M. Figueroa, M.C. Mariani, M. Ferraro, The effects of the Asian crisis of 1997 on emergent markets through a critical phenomena modl., Int. J. Theor. Appl. Finan. 6 (2003) 605–612. A. Johansen, D. Sornette, Physica A 245 (1997) 411. L. Amaral, Comp. Phys. Commun. 121 (1999) 145–152. M. Ferraro, N. Furman, Y. Liu, M.C. Mariani, D. Rial, Analysis of Intermittence, Scale Invariance and Characteristic Scales in the Behavior of Major Indices near a Crash, Physica A 359 (2006) 576. J.C. Hull, Options, Futures and Other Derivatives, Prentice Hall Inc., NJ, 1997. D. Duffie, Dynamic Pricing Asset, Princeton University Press, NJ, 1996.
Physica A 367 (2006) 345–352 www.elsevier.com/locate/physa
A new analysis of intermittence, scale invariance and characteristic scales applied to the behavior of financial indices near a crash Maria Cristina Mariani, Yang Liu Department of Mathematical Sciences, New Mexico State University, 88003-8001 Las Cruces, NM, USA Received 13 July 2005; received in revised form 29 November 2005 Available online 3 February 2006
Abstract This work is devoted to the study of the relation between intermittence and scale invariance, and applications to the behavior of financial indices near a crash. We developed a numerical analysis that predicts the critical date of a financial index, and we apply the model to the analysis of several financial indices. We were able to obtain optimum values for the critical date, corresponding to the most probable date of the crash. We only used data from before the true crash date in order to obtain the predicted critical date. The good numerical results validate the model. r 2006 Published by Elsevier B.V. Keywords: Econophysics; Scale invariance; Intermittence; Stock market prices; Financial indices; Crashes
1. Introduction During the last years the study of log-periodic structures and characteristic scales and the relation with the concept of scale invariance had grown due to the great amount of physical systems presenting log-periodic structures: fluid turbulence [1,2], diamond Ising model [3], earthquakes [4], materials rupture [5], black holes [6] and gravitational collapses [7] among others. In a mathematical context, we recall constructions as the Cantor fractal [3][8], with a discrete scale changes invariant. The presence of logarithmic periods in physical systems was noted by Novikov in 1966 [9], with the discovery of intermittence effect in turbulent fluids. The relation between both effects has been deeply studied, but it has not been formalized yet. At the same time, the complexity of international finance has grown enormously with the development of new markets and instruments for transferring risks. This growth in complexity has been accompanied by an expanded role for mathematical models to value derivative securities, and to measure their risks. A new discipline Econophysics, has been developed [10]. This discipline was introduced in 1995, see Stanley et al. [11]. It studies the application of mathematical tools that are usually applied to physical models, to the study of Corresponding author.
E-mail addresses: [email protected] (M.C. Mariani), [email protected] (Y. Liu). 0378-4371/$ - see front matter r 2006 Published by Elsevier B.V. doi:10.1016/j.physa.2005.11.047
ARTICLE IN PRESS M.C. Mariani, Y. Liu / Physica A 367 (2006) 345–352
346
financial models. Simultaneously, there has been a growing literature in financial economics analyzing the behavior of major Stock indices [10,12–15]. The Statistical Mechanics theory, like phase transitions and critical phenomena have been applied by many authors to the study of the speculative bubbles preceding a financial crash (see for example [16,17]). In these works the main assumption is the existence of log-periodic oscillation in the data. The scale invariance in the behavior of financial indices near a crash has been studied in Refs. [18–20]. This work is organized as follows: In Section 2 we give a short introduction to the relation between intermittence and scale invariance, the conditions that a function has to satisfy when both effects are present. We analyze the relation with characteristic scales, and we finally present a method that detects characteristic scales in different systems using the previous results. In Section 3 we present a model that predicts the existence of intermittence and characteristic scales in the behavior of a financial index near a crash [20]. In Section 4 we develop the methods that we will use for our numerical analysis. Finally, we apply the model to the analysis of the behavior of several financial indices: the S&P500 index near the October 1987 crash, and the Argentina MERVAL index as well as the Brazil BOVESPA index and the Mexico MXX index near the October 1997 Asian crash. 2. Scale invariance and intermittence In this section we analyze the relation between intermittence and scale invariance and we introduce definitions and notation that will be used later. For further details see Ref. [20] and its references. A function A, that depends on a variable x, is invariant for the scale change lx when AðxÞ ¼ mAðlxÞ, (1) where m is a constant independent of x. For a detailed discussion about this definition see Ref. [20]. Any observable which remains invariant for the scale change x-lx can be expressed as 1 X AðxÞ ¼ xlogl m an ei2pn log l x .
(2)
n¼1
Now we recall the difference between discrete and continuous invariance. When the observable given by (2) presents continuous scale invariance, for any real number l there exists m such that condition (1) is fulfilled. We can deduce that in this case logl m does not depend on l, and an ¼ a0 don . When the observable A(x) satisfied Eq. (1) only for numerable values of the l’s, it presents a discrete scale invariance. A particular type of scale invariance is the one arising in the existence of intermittences or ‘‘stationary intervals’’, constant in the logarithm of the independent variable. The functions that can be obtained from this analysis are f F ðxÞ ¼ beaF ðloga xÞ ,
(3)
f C ðxÞ ¼ beaC ðloga xÞ ,
(4)
where b and a are real numbers, a is positive and F ðxÞ ¼ IðxÞ and CðxÞ ¼ IðxÞ þ 1 are the Floor and Ceiling functions, respectively. Hence, the value obtained when applying the Floor function to a variable x will be the nearest entire number to x from the left, and the value obtained by the Ceiling function will be the nearest entire number to x from the right. These two functions are discrete scale invariants, and more specifically, they satisfy Eq. (1) only when l ¼ an ;
n 2 Z.
(5)
We recall that the conditions for a function to have discrete scale invariance, after we know that the system has intermittences are the following: (i) The intermittence intervals must be constant in logarithmic scale, i.e., the steps have to be discrete dðln xÞ ¼ K,
(6)
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Fig. 1. Basic scheme showing the relation between a power-law (in solid line) and a function like the one defined in (4) (dash line). The black points are the intersection points of both functions.
where K is a positive number. Let a ¼ eK . Then we have d loga x ¼ 1.
(7)
Then, the interval of time of the intermittence is such that the logarithm of the variable x, in a basis a, has advanced one unit (in that period of time). Hence, we can conclude that due to the longitude of the intermittences there exists a basis in which the logarithm of the variable is equal to one. (ii) The stationary intervals are consecutive, when one finishes, begins the next one. (iii) The function such that its variable is discretized with the rule dðloga xÞ ¼ 1 is a power-law, and the beginning and the end of a stationary interval have both to be a point of the function. We will call this function the basis function, which can be illustrated as follows: Now we apply the tools given by the theory previously developed in order to analyze data presenting characteristics similar to those mentioned above. Therefore, the data basis function will be a power-law (Fig. 1). First, we will find this law; the second step will be to reduce the free parameters to only one: the logarithmic basis. Finally, we will use (3) or (4) in order to find the value of a minimizing the distance with the data. Hence, we obtain the system characteristic scales. The estimation of the power-law is crucial in this method because it will be the basis of the function that will be used for fitting the data. 3. Financial indices prices near a crash The evolution of a financial index represents the changes of a portfolio [21]. There is a simple model that takes account of the evolution of an asset price in the market [22], this model considers two contributions to the percentage variation in the asset price: one deterministic, and one stochastic dS ¼ m dT þ s dX , (8) S where S is the asset price, m is a constant (called drift), dT is the interval of time, s is another constant (called volatility) and dT is a random variable which is assumed following normal probability distribution. We recall that the physical process represented by (8) is Brownian motion with a drift. We focus our attention in the deterministic contribution dS ¼ m dT. (9) S As a financial index can be considered an asset, its (deterministic) financial behavior will be given by (9). Our hypothesis is that near a crash Eq. (9) is modified dS Sc S ¼m , dT Tc T
(10)
where Tc and Sc are respectively, time and price for which the crash takes place. The heuristic analysis is as follows: near a crash there is a factor that produces a considerable increase in the index price, on the other
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hand, when the price is very near to the crash price, there has to exist another factor smoothing those variations, otherwise the crash would take place before the real date (for further details see Ref. [20]). Changing the variables in (10) we obtain dP P ¼m , dt t
(11)
where the variables are not anymore absolute data of the system: t and P are the distance to the critical time and critical price, respectively. The second assumption will be that the temporal steps are discrete; therefore, the index evolution is not continuous and we have the intermittence phenomena. The index evolution is given by dt ¼ Kt.
(12)
Then the frequency in the index price changes is proportional to the distance to the date in which the crash takes place. Eq. (12) implies that d ln t ¼ K.
(13)
From Eqs. (11) and (13) we arrive at functions like (3) or (4). In this case we will work with function (3), due to the fact that intermittences must take into account that the time approaches the critical time from the right, because of the change of variables (11). 4. The data analysis methods: analysis of the parameters and estimation of the critical time We are plotting the t–p figure where t is the time distance from the crashing time and p is the price distance from the crashing price. We can model the market behavior by using the following equation: p0 f ðtÞ ¼ beaF ðloga ðt0 tÞÞ ,
(14)
where p0 is the crashing price, t0 is the crashing time, f(t) is the market price at time t and F is the floor function. Let us remove the floor function at this moment. Consider p0 f ðtÞ ¼ bea log
a ðt0 tÞ
,
(15)
we can rewrite Eq. (15) in log-domain as a . ln p0 f s ðtÞ ¼ ln b þ glnðt0 tÞ; where g ¼ ln a
(16)
We want to remark that the purpose of Eq. (15) is to show the linear relationship in log-domain between tt0 and p0f(t), which is trying to describe the envelop behavior of indices near a crash. The floor function is not really dropped from the model. Thus, ln p0 f s ðtÞ has a linear relationship with lnðt0 tÞ. Suppose p0 and t0 are given (or somehow estimated), we can easily estimate ln b and g by linear regression. However, Eq. (15) only describes the power-law of the system. It is a smooth line while the market will move up and down around it following the model described by (14). We must get back to (14) to describe the dynamic behavior of the market. In other words, we need to find the characteristic scale a. We recall that the parameters a and b have been determined in (16). Rewrite Eq. (14) in log-domain as lnðt0 tÞ ln p0 f ðtÞ ¼ ln b þ aF . (17) ln a
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Then subtract (16) from (17) and multiply by 1/g, we have that lnðt0 tÞ G ðlnðt0 tÞÞ ¼ lnðt0 tÞ ln a F ln a 1 ln p0 f ðtÞ ln p0 f s ðtÞ . ¼ g
ð18Þ
Therefore,
lnðt0 tÞ þ ln a G ðlnðt0 tÞ þ ln aÞ ¼ lnðt0 tÞ þ ln a ln a F ln a lnðt0 tÞ ¼ lnðt0 tÞ þ ln a ln a F þ1 ln a lnðt0 tÞ ¼ lnðt0 tÞ lna F ¼ Gðlnðt0 tÞÞ. ln a
ð19Þ
We can see that G( ) is periodical with a period of ln a. To find a, we need to find the period of G( ). An ideal periodical function will have a spike in frequency domain at its period. So we use Fourier transform and look for the frequency with a high power in the frequency domain. A general discrete Fourier transform is given by F ðk Þ ¼
N X
f ðnÞej2pnk=N .
(20)
n¼0
In our method, we use the following transform: QðkÞ ¼
N=2 X
G ðxn Þej2pðk1Þðn1Þ=ðNK Þ ,
(21)
n¼1
where K is some constant that will determine the resolution of the Fourier transform, and xn, n ¼ 1; 2; ; N are the equal distant points between lnðt0 T 0 Þ and lnðt0 T 1 Þ, where T0 and T1 are respectively, the start and ending points of the observation window. Notice that the original market data is daily price. When we use them in log-domain, the sample points of G( ) are no longer equal distant. However, (21) requires to evaluate G( ) at equal distant points xn. To get the Fourier transform, we use linear interpolation to evaluate G(xn). If we have tm1 oxn ptm ; the interpolation is given by G ym G ym1 G est ðxn Þ ¼ xn ym1 for ym1 oxn pym , (22) ym ym1 where ym ¼ lnðt0 tm Þ. We remark that the characteristic that the market must exhibit for assuming this approximation is that the behavior of the market is ‘smooth’ up to the crashing point. In other words, we assume the index curve is differentiable at each time point except for the crashing point. Therefore, we can use interpolation. While higher-order interpolation could be used, we found that a simple linear interpolation between two points is accurate enough for our purpose. The Fourier transform actually shows us the power at each frequency. Ideally, the power of the dominant frequency will be a local maximum. So we pick the frequency with a local maximum power within a roughly preset range. Once we have the frequency f, we can easily calculate the characteristic scalar a by a ¼ e1=f . Then a is a function of p0 and t0, and Eq. (14) is fully determined.
(23)
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Now we can use the Square Error e2 between Eq. (14) and the real market data to measure how good our model describes the market behavior. Notice that at this stage e2 is a function of only p0 and t0. i.e. e2 ¼ e 2 p0 ; t 0 . (24) P 2 2 Explicitly, e ¼ ð f ðtÞ PðtÞÞ ; where p(t) is the real market closing price. t Therefore, by minimizing e2, we should be able to get the best estimate of t0 and p0. 5. Numerical results In this section we present the numerical results. We plot the function of the error between the model and the real index value against the t and p. The optimal estimation should be corresponding to the area with minimum error. To give a better view, we plot both the surface graphs and the contour graphs. We can see that e2 is not sensitive about p, so it provides little capability to predict the crashing prices (Figs. 2–5). However, the optimal values of t can be clearly identified from the figures. We list the predicted crashing date comparing with the real crashing date in the following table Table 1. We are using daily market closing price for the index value.
Fig. 2. Error function of SP500 in log-domain.
Fig. 3. Error function of BVSP in log-domain.
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Fig. 4. Error function of MXX in log-domain.
Fig. 5. Error function of MERV in log-domain.
Table 1 Comparison between the real crashing date and the estimated date
Real crashing date Real topping date Estimated date
S&P500
BVSP
MXX
MERVAL
10/19/1987 10/06/1987 10/28/1987
10/27/1997 07/11/1997 07/03/1997
10/21/1997 10/21/1997 10/30/1997
10/22/1997 10/22/1997 11/08/1997
We recall that in the case of the BVSP index, the crash happened 75 days after the market top. We also recall that the graphs are plotted in log domain. 6. Conclusions The effects of certain local crisis on various and distant markets have largely been cited. The collapse of the crashes of 1987 (S&P500) dragged the collapse of markets worldwide. However, not every crisis has sufficient
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strength as to drag the fall of leading indices in other countries. In Ref. [16] it has been shown that the crashes of Asian indices had consequences on emergent markets: the Asian crisis had sufficient strength as to drag the fall of leading Latin American indices. Clearly, all these indices crashed in similar dates due to a dragging correlated effect, which most likely started with the instability of the HSI index. This signals the likelihood of the events in different markets and different economic realities which strengthens the hypothesis of imitation and long range correlations among traders. About the stability of the method, we want to remark that, even if the maximal and minimal values for the error are apparently similar, the graphs are plotted in log-domain. Furthermore, we want to remark that we believe the noise in the market can actually change the crashing date, and the change may be significant. We believe the market is a very complex system. The crash might be delayed due to some market force, so it happened sometime after the market top, but that is out the scope of this work. Our goal in this work is to develop a method for finding the market top which leads to a crash. We also want to remark that we only use the data up to a couple of weeks before the crash in all the cases to estimate the critical time. The estimations have errors of around only 10 trading days. The excellent results validate the method. Acknowledgments The authors want to thank the anonymous referees for their careful reading of the manuscript and their fruitful remarks. This work was partially supported by ADVANCE – NSF. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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