Download Energy Conservation Theory and the Fundamental Line of DJIA: A Physics Approach and more Lecture notes Advanced Physics in PDF only on Docsity!
Stock mechanics: energy conservation theory and the
fundamental line of DJIA
Çağlar Tuncay Department of Physics, Middle East Technical University 06531 Ankara, Turkey caglart@metu.edu.tr
Abstract
DJIA is tested whether it can be described as a mechanical system conserving total energy with K (=½v 2 ) + U, where U is calculated as the negative of work done by force obtained in terms of the second derivative of price, assuming unit mass.
Keywords : Potential and kinetic energy; Equations of motion; Oscillations; Crashes
PACS numbers: 89.65.Gh
1. Introduction
Market indices as well as share prices, during their time excursions pass through several states; oscillations, rises and falls, crashes (crises), etc. During that time, some quantities may remain as conserved, such as total energy. As a result, time series may be predicted epoch by epoch, in terms of some simple analytical functions resulting from energy conservation, once the potential energy of the current epoch is known. In this work we will exemplify an application of energy conservation theory (e-ct) on DJIA in the “0th^ -order approximation”. Some possible extensions of energy conservation theory will be pronounced in the final section.
2. Energy conservation theory
The whole history of DJIA, utilizing daily real data [1] can be decomposed into five main epochs, each with a different length of duration. Refering to Fig. 1. the first epoch is the transitional period lasted about 3565 days from the establishment of the index on. The main characteristics of this epoch is (almost) triangular squeezing of prices with the climax meeting the beginning of the Sept.1929 crash (“Black Monday”) at a value of 381.17, which ended after about 700 days at a value of 41.22, (Fig. 2.). Then a sharp rise occured till the value of about 200. Afterwards some decrease came reaching its deepest value at 92.92. These two successive local minima, i.e., 41.22 and 92.92 are very important figures in the history of DJIA since, the line passing through them defines the lower rising edge of the pronounced
triangle and moreover the rightward extension of this line plays the role of carrying (supporting) prices (i.e. the line passing through minima) for many decades, till the value increases to nearly 1000. (See, the longest straight line in Fig. 1.) The pronounced line is in fact worth to be considered as the fundamental line of DJIA because of several reasons. First of all, it can be taken as the 0th^ -order approximation for the index, since the corresponding time series is observed to evolve about (below and above of) it. Secondly the fundamental line, has played the role of supporting line till the date of 16.June.1969 (10,172 th^ day). The day after, i.e. at 17.June.1969 the index has broken it down (after many previous hits) and later has never crossed it. Starting in Apr.1999 DJIA has approached it from below many times, (at index values between 11,000 and 11,700) till the end of the same year, but has ceased to override it, and the last log-periodic crash has followed. (For detailed studies of this crash, in a power law formalism, see [2-6].) Slope (β) of the present log-linear increase of the fundamental line is calculated to be β= ln(92.92/41.22)/(3390-946)=0.000333day–1^. Therefore, the equation of line can be written as
χ(t)=χ 0 exp(0.000333 t) , (1)
the constant χ 0 takes care of t=0 value, i.e., the initial price level. χ 0 may very well be taken as 41.22 or 92.92 and the date of the corresponding day will be the origin for time. We have the price velocity and acceleration as v = dχ/dt = βχ, and a = dv/dt = d^2 χ/dt 2 = β^2 χ, respectively, and both increase exponentially in time. Assuming m=1[7], one may write an explicit function for the 0th^ -order force
F zeroth-order^ = a = (0.000333) 2 χ 0 exp(0.000333 t). (2)
Once the force is known, one may calculate the work done by this force, along the path of Eq. (1) that DJIA follows within the 0th^ -order approximation: ∆W=∫Fdx=∫β^2 χ dχ = ½β^2 χ^2. The present indefinite integral may be converted into a definitine one by substituting the initial χ-value for t=0, and the general term for any t; therefore ∆W = ½β^2 (χ^2 – χ 02 ), which is nothing but –∆U, by definition. Therefore,
∆U = –½β^2 (χ^2 – χ 02 ). (3)
Since, v = βχ; then one may calculate the change in kinetic energy in the same journey of DJIA, as ∆K= ½ (v 2 – v0^2 ) = ½β^2 (χ^2 – χ 02 ). So,
∆U + ∆K = 0 = ∆(U + K). (4)
Please note that, Eq. (4) is a result of the exponential growth (Eq. (1)) of DJIA. There might be found some other quantities as being conserved during the same time excursion of DJIA, and within world market. With respect to this possibility, it is better to have a separation between any conservation theory and the current energy conservation theory (e-ct). Energy conservation (Eq. (4)) implies absence of any frictional force or damping etc. on DJIA, along the way under consideration. Yet, DJIA is observed to deviate from its usual exponentially increasing path, and spend about twenty years (1962-1982) below the index value of 1000. During that time, it bounced up and down many times between the levels of 750 and 1000 as displayed in Fig. 3. Inspiring also from the quadratic form of potential energy in Eq. (3), we may approximate this oscillatory mood by a sinusoidal (harmonic) form. Utilizing the observational data, the following expression may be proposed to represent the third epoch as
References
[1] For detailed information about NYSE, URL: http://biz.yahoo.com/i/. [2] J. A. Feigenbaum and P. G.O. Freund, arXiv:cond-mat/9509033. [3] D. Sornette et al. , J. Phys. I Fr. 6 , 167 (1996). Preprint : arXiv:cond-mat/9510036. [4] For many other articles of D. Sornette see also several issues of the journal Eur. Phys. J. B. and search Preprint : arXiv:/cond-mat/ [5] A. Johansen, and D. Sornette, Preprint : arXiv:cond-mat/ [6] M. Ausloos and K. Ivanova, Preprint arXiv:cond-mat/0108013. [7] Ç. Tuncay, Preprint : arXiv:physics/0503163, and arXiv:physics/0506098.
Figure captions
Fig. 1. DJIA, with a logarithmic price axis. Five fundamental epochs composing its history can be selected. The first “transitional” epoch is characterized by its triangular squeezing the price about the first “psychological resistance” of the value of 100. In the second and the fourth epochs log-linear (i.e., exponential) ascending of the price can be distinguished (arrows 1 and 2). In the third epoch, just below the second “psychological resistance” of the value of 1,000 oscillations are dominant. In the fifth, and the final epoch, the evolution of formation (about the third “psychological resistance” of 10,000 index) is not completed yet.
Fig. 2. The first transitional epoch in DJIA. Notice the supporting line, rightward extention of which displays that it is fundamental for DJIA.
Fig. 3. The third, (oscillatory) epoch of DJIA. When the time difference between any successive hits to the supporting level of 750 (i.e., the period T) increases, the amplitude (A) also increases. As a result A/T ratio, and ET remains the same.
Figures
01.10.1927 01.10.1940 01.10.1953 01.10.1966 01.10.1979 01.10.1992 01.10.
3
4
5
6
7
8
9
10
ln DJIA
date (day)
0.000333t*
m+3.^
ln 100
ln 1000
ln 10000
Figure 1.
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
100 DJIA
^1
st^ epoch
day
380
195
110
41,
92,
160
(^245 944 2155 ) <1228 days> <1223 days>
30,201535e3,29482e-4t
Fig. 2.
