Some politics and statistical physics behind Sri Lankan stock market crashes

Transcription

1 RUHUNA JOURNAL OF SCIENCE Vol., September 2006, pp issn X c 2006 Faclty of Science University of Rhna. Some politics and statistical physics behind Sri Lankan stock market crashes Janak R. Wedagedera Department of Mathematics, University of Rhna, Matara, Sri Lanka, janak@maths.rh.ac.lk, W. Ajith Raveendra de Mel Department of Mathematics, University of Rhna, Matara, Sri Lanka, ajith@maths.rh.ac.lk, We investigate the mechanism of how the so called critical crashes happen in the stock market from a statistical physics point of view. We shall consider a modelling approach proposed in Johansen et al. (2000) to stdy the natre of a possible crash occrred in the Sri Lankan stock market in 994 with a mentioning of the political events of the contry that had been prevailing dring that time which cold be the key force that had driven the market towards the said crash. We shall determine the parameters that governed this crash and fit a periodic fnction for the actal data based on the critical phenomena in statistical mechanics. We also propose a modelling approach via which we illstrate the critical crashes in the market are not nsal phenomena when the market is modeled as a system in statistical physics, where we employ a version of Deridda s Random Energy Model Derrida (980), Derrida (997) applied to the price flctations in the financial market. Key words : financial derivatives, stock market, stochastic differential eqations, Cayley tree models, statistical mechanics. What is a financial crash? A financial crash on the macroscopic level is seen as a sdden drop of the price of a derivative or stock price. Thogh it seems as a drop-down in price which may take place over a few hors or a few days, the events that drive a market towards a crash wold have been bilding-p perhaps over months or years. Pal and Baschnagel (999) illstrated, sing Dow Jones Index data that crashes are otliers which are not explained or modeled by sal statistical flctations occr in the probability distribtion of the index. Ths, identifying and trying to model crashes in standard statistical techniqes is not possible. In fact, Johansen A. (998) and Pal and Baschnagel (999) demonstrated how the Dow Jones Index had decreased by a certain percentage from year 900 to year 994 where they illstrated when the freqency of this decrease N(D) is smaller than 5% (where D stands for the magnitde of the decrease or draw down), the freqency cold reasonably be approximated by an exponential with N(D) = N 0 exp( D/D c ). This behavior is consistent with the price distribtion so can nicely be explained sing standard statistical theory. In contrary, there have been three occasions in which the Dow Jones Index had decreased more than 25%: dring World War I, in

2 34 Rhna Jornal of Science, pp , (2006) and in 987 which do not fit in the above explanation. This sggest the need of a critical phenomena (Dorlas (999), Schwabl (2002), Yeomans (992))to explain these otliers. The se of the word critical in the above in fact refers to sch analogies one observes in complex systems where sch systems dynamically driven ot of eqilibrim. Examples are earthqakes, avalanches, crack propagation etc Johansen et al. (2000). The first proposal for a connection between crashes and critical points was made by Sornette et al. (996). In the first half of this paper we shall concern on to investigate if there has been similar behavior occrred in the Sri Lankan stock market (or commonly known as the Colombo Stock Exchange) along the same lines proposed in Johansen et al. (2000) and Pal and Baschnagel (999). Indeed, we report on, pon a carefl observation and analysis on the past data (from ) of the Milanka price index of the Colombo Stock Market, Sri Lanka, abot a possible critical crash which we claim to have happened most probably on the st March 994. In the second half of the paper we propose a model which has been extensively stdied in the literatre (Derrida (98), Derrida (985), Derrida (997)) in a different context namely, in spin glasses in statistical physics, to be sed to explain how the so called critical crashes cold take place in financial market when we model the stock price process as spin glass or a directed polymer along a Cayley tree (Bffet et al. (993))... March 994 Crash in the Sri Lankan stock market In Figre we show plots of data of the main financial index sed in the Sri Lankan stock market, namely the Milanka Index for the period We observe a crash-like behavior on the st March 2004 which had reslted de to the economical and political factors that prevailed in the contry dring that time. In fact, on the st of March 994, the political pictre of the conty enters into a new regime as then governing party had a defeat at the Provincial Concil Elections that had been held the following day, which had triggered a seqence of dramatic political events that ltimately lead to a change in the government as well as the Presidents office. This has been the key factor for the crash-like behavior of the market observed jst before the date of the election. In Johansen et al. (2000) it is hypothesized that stock market crashes are cased by the slow bilding p of long-range correlations leading to a collapse of the stock market in one critical instant. In fact, Figre illstrates the slow formation of critical crash that happened in the Sri Lankan stock market in 994. One can observe a staircase-like increase with narrowing width in the index which had lead to a peak on the st March 994 at which the stock price hit the maximm At this point the width of the horizontal line segment is seen vanishingly small. Sch a rhythmic behavior with a shrinking period seems to be a characteristic precrsor of crashes similar to the observations made on the Dow Jones Indstrial Average for S&P500 data (Pal and Baschnagel (999)). (So in contrary to the poplar belief,

3 Rhna Jornal of Science, pp , (2006) 35 Milanka Index Jan 85 Mar Mar 03 Date S d daynmber d Figre A sdden drop-down of relatively large magnitde (approximately more than 25%) in a financial index or the stock price is known as a financial crash. Top Left: This data show the evoltion of the stock price of the Colombo Stock Exchange over the period 994 to The fall started on the st March 994 when the Milanka Index was at and the local minima in this dration recorded was on 20 Jne 994 on which date the Index vale was Top Right: The evoltion of stock price from day 900 to day Here we observe the step-like behavior followed by the process (shown in horizontal line segments) whose step size becomes shorter as the market approaches the crash. Bottom: In the model we claim the exact date of the crash as st March 994 corresponding to the day nmber t C = 274 for which the stock price S(t C ) = LKR This claim is statistically tested in Section 2.2. ( Sorce, The Data Disk (2005) - Colombo Stock Exchange.) ever increasing stock price cold be a signal of a possible crash in the ftre so these sitations shold be treated with cation). Statistical flctations govern the normal behavior of the market when spply and demand are well maintained. However, the things start to behave differently

4 36 Rhna Jornal of Science, pp , (2006) when the market is in the imminent of a crash. At sch times, vast majority of traders spontaneosly decide to sell. This nsal cooperativity shakes the liqidity of the market. There is no sfficient demand to defy the exploding spply. The prices drop, and the market falls ot of eqilibrim (Pal and Baschnagel (999)). This sggests that crashers are triggered by a spontaneos development of long-range correlations among the traders. Ths, it is important to incorporate the factors that depend on the behavior of the traders if one is to bild-p a model for crashes. 2. Critical Crashes - Sornette-Johansen Model In Johansen et al. (2000) it has been stdied the similarities between a phase transition that takes place in a physical system (sch as in a magnetic material), and the crashes occr in the financial market. In contrast however, one cannot directly identify a parameter in the financial market which plays the role of a parameter like the temperatre in a physical system which drives the system towards a phase transition. It is also arged that sch a parameter cannot exist; if it were, the traders in the market wold be able to identify it and then shall be able to control it so that a crash woldn t happen. This sggests that the stimlation for a crash shold grow from inside the market. This is a phenomena known as the self-organized criticality (Bak. (996), Kiyono et al. (2006)) in statistical physics in which it is explained how physical systems ndergo phase transitions withot any fine-tning of external control parameters sch as temperatre or external magnetic field (Pal and Baschnagel (999)). In Johansen et al. (2000) it also sggests the imitation as the prime sorce which drives the market towards a crash. When the prices increase, the traders tend to keep their attention towards the market events. Eventally, some decide to by with the hope that the trend of price hike will long last. This frther increases the prices, which in trn, motivates more traders to speclate. Ths, a speclation bbble is created and this bbble will evolve in time in a breathing like fashion. It is inflated by speclation, bt the growth can also stagnate since some traders fear the trend will soon reverse and decide to sell. Even if the nmber of this kind of traders is small the bbble keeps increasing and eventally reaches a level where many traders assme the price will fall in the ftre and place sell orders. This fashion or the imitation by the others tend to spread across varios levels of the traders in the market. This may trigger a crash with some probability for the bbble to deflate smoothly - withot casing a crash. 2.. A microscopic model for crashes Johansen et al. (2000) introdces a fnction h(t) called the hazard rate which measres the probability per nit time that a crash will occr in the next time interval [t, t + t], provided it has not happened before. When the bbble inflates the risk of a crash becomes more likely, ths increases the hazard rate h(t). A possible form for h(t) derived by Sornette et al based on properties of the magnetic

5 Rhna Jornal of Science, pp , (2006) 37 Figre 2 Oscillatory behavior with the increasing amplitde and shortening period of the fnction h(t) are the key featres of the behavior of the financial index as it approaches the crash point. ssceptibility of a certain hierarchical diamond lattice proposed by Derrida et al. (983) is given by h(t) = (t c t) γ [B 0 + B cos (ω ln(t c t) + ψ)] (t t c ). () Here, γ is a critical exponent which shold be niversal, i.e., the same for all crashes, B 0 and B are constants, and the phase ψ can be related by ψ = ωln(/t 0 ) to a time scale t 0. This time scale serves to match the eqation (), which shold be valid only asymptotically close to the crash, on the data of the price evoltion. The motivation behind having the above ansatz for h(t) is in fact the large increasing trend with oscillations with vanishing periods we observe in the data (see the top-right of the Figre ). A typical plot of h(t) near the critical point will immediately reveal this (see Figre 2). In Eqation (), we find two major characteristics: Firstly, the term (t c t) exhibits a power-law arond t = t c. Derrida et al. (983) γ proved that the ssceptibility χ of the system behaves as χ Re [ A 0 (t t c ) γ + A (t c t) γ+iω +... ] (2) A (t c t) γ cos [ω log(t c t) + ψ] +... (3) where Re stands for the real part of its argment. Ths it is reasonable to propose (Johansen et al. (2000)) to consider a form of the eqation (2) the hazard rate fnction. According to (2), we see that in addition

6 38 Rhna Jornal of Science, pp , (2006) to the power-law behavior also is a periodic behavior which is governed by the trigonometric term. Sornette et al frther obtain the link between h(t) and the actal price evoltion. This can also be obtained (Pal and Baschnagel (999)) by noting the fact that h(t) µ(t) where µ(t) is the drift of the stock price process (or the associated geometric Brownian motion). This is obtained as follows: The evoltion of the financial index can be decomposed into a deterministic (increasing fnction of t) and sperimposed stochastic flctations. If we neglect the flctations and only attempt to model the deterministic increase by geometric Brownian motion, we get the differential eqation for the stock price as ds det = µ(t)s det (t)dt, (4) where S det stands for the deterministic component of the stock price. This approximation amonts to assming that the long-term dynamics before the crash is dominated by the drift term with time-dependent µ. Ths it is reasonable to assme Sbstitting (5) in (4) and integrating one finds Inserting () into (6) we obtain τ 0 µ(t) = κ h(t) (κ > 0). (5) ln S det(t c) S det (t) = κ tc t dt h(t ). (6) dx x γ cos(ωlnx + ψ) = γ ω 2 + ( γ) 2 cos ψ τ τ cos(ωlnτ + φ) (0 < γ < ). Here, τ = t c t, tan ϕ = in which ω, φ = ψ ϕ,. Ths we find, γ lns det(t) = lns det (t c ) [C 0 τ β + C τ β cos(ωlnτ + φ)], (7) C 0 = κb 0 β, C = κb cos ϕ β (0 < β = γ < ). (8) ω 2 + β2 It is essential to have γ (0, ) as we mst have a finite price at t c. If t t c, This solves to, ln S det(t) S det (t c ) S det(t) S det (t c ). S det (t) S det (t c ) ( [C 0 τ β + C τ β cos(ωlnτ + φ)] ). (9) For frther details and the derivation of the above we refer the reader to Pal and Baschnagel (999). We shall now match already available data to the average

7 Rhna Jornal of Science, pp , (2006) 39 (a) t C = 264, MSE = Convergence failed. S t Time t days (c) t C = 274, MSE = Convergence sccessfl. S t Time t days (b) t C = 27, MSE = Convergence failed. S t Time t days (d) t C = 277, MSE = Convergence failed. S t Time t days (e) t C = 280, MSE = Convergence sccessfl S t Time t days (f) t C = 283, MSE = Convergence failed S t Time t days Figre 3 Nonlinear fit for the crve given by the Eqation 9. Parameters are estimated for varios a- priory vale for the crash date t C (days) as shown in the bottom of the each plot together with the mean sqare error vales and comments on the convergence of the Levenberg-Marqardt method for nonlinear least-sqares available with Mathematica 5.0 Wolfram (2004). The best fit MSE vale with sccessfl convergence of the nmerical scheme is given in the case (c) which corresponds to the crash date st March 994 we claimed in the caption to Figre. behavior of the stock price via (9). With this we shall be able to qantify the oscillatory behavior which is observable prior to a critical crash from the time series data of the stock market. The main difficlty in handling sch data sing

8 40 Rhna Jornal of Science, pp , (2006) latter methods is the divergence of the parameters and indexes in the proximity of the critical time t c. In conclsion, this techniqe has given s valable information which are otherwise impossible to infer (for example via statistical time series analysis of data) how probable a critical crash cold happen by a given crash date t c. On the other hand, the method can be made available to real-time financial information systems to monitor the price evoltion and to isse forecasts and warnings to the market Statistical prediction of the crash date In Figre 3 we show the periodic fnction we obtained by fitting the eqation (9) for Milanka data for different a posterior known vales for t c. The most probable date for the crash is fond to be the same as the one we conjectred. This gives the least mean sqare error with sccessfl convergence of the nmerical scheme sed as explained in the figre caption. This kind of analysis shall be of extremely sefl in decision making made by the investors and management of the market. 3. How the critical crashes happen? an explanation from statistical mechanics We know that the price of an index (or the stock price as we consider in this section) is a stochastic variable whose vale depends on many parameters and variables. Clearly we can plot the stock price S(t) as a fnction of time, bt it is not the time that actally affect the change in S(t) bt the activities of agents and other economical and political facts in a society. Needless to say (as we have already discssed in the previos section) not all these factors (or parameters) are possible to be qantified in a model. Nevertheless, we know by or experience that when the agents take decisions at random, the market becomes a highly active place. When the agents tend to speclate and delay taking decisions, the market from the srface seen as a cool place whilst there are ice-berg type activities happening. Here we propose the virtal inverse activity fnction (VIAF), denoted β whose vale is inversely proportional to the activity of the market in the sense we discssed in the above for the sole prpose of showing how a critical crash is possible when we model the market via models in statistical physics. In other words, β low β high market becomes relatively active (or hot) and, market becomes less active (or cool). It is obvios that β shold also be proportional to the hazard rate fnction (or be the same as that) as discssed in Section 2.. (See pp. 6 of Johansen et al. (2000)). In the next section we shall show that if we vary β from lower vales to higher vales there is a critical vale of β, say, β c beyond which the market enters into a new phase in which the activities of the agents become frozen. This means at the β = β c, the market ndergoes a crash or a phase transition as in physical systems.

9 Rhna Jornal of Science, pp , (2006) The model Let s consider the evoltion of the stock price S(t) over a day. We also assme that the increments and decrements of S(t) over a small time interval (within a day) prely reslts de to the activities of the agents and that it is these same activities that affect the virtal fnction β(t) to increase or decrease. However, we assme that β(t) is fixed dring a day (or we consider the average vale β(t) as the vale of the VIAF for the given day. Ths, for a given β we let the price evolve in a day over a large nmber of (say, n) small time intervals. We let the p-down behavior of the price controlled by a Bernolli random variable σ j = ± with probability p = /2, dring each time interval considered on a day. If there are n time intervals each of dration t, this will resolve a seqence of increments (decrements) { j } n j= (See Figre 4). What we wold like to investigate is the evoltion of S(t) when β increases - or when the market slowly enters into a less active place from a highly active one. We also assme that j follows a Log-Normal distribtion (in other words this means the log retrns, i.e. log follows a Normal (Gassian) distribtion). Indeed, in financial market terminology, the p-down rates = exp(ν t) where ν is the volatility, and ths log follows a Gassian distribtion scaled by / t when ν is Gassian, i.e. ν N (0, ϑ 2 ). This means that the change in S(t) is given by (t) Stochastic Volatility By allowing ν to assme a distribtion we consider the sitation where the volatility is a stochastic variable rather than a constant. The latter sitation is one of the major assmptions in the Black-Sholes theory (Black (989)). Ths, the evoltion of (t) is conveniently represented in a binary tree as shown in the Figre 4. Now a seqence { j } n j= corresponds to a path ω, which traverses across all levels of the tree. Now the problem is eqivalent to a self-avoiding random walk along a Cayley tree as it has been treated in (Derrida and Gardner (985) and, Derrida and Spohn (988). Going along the same lines as in Bffet et al. (993) to solve this problem, let s define a path at the top of the tree and of length ω = n as a finite seqence {(j, ω j ), j n} sch that ω j+ = 2ω j 2 ( σ j+), where σ j {, }. The prpose of σ s in the model is to realize a random path in the tree. Moreover, we shall keep the set of vales { ij } nchanged (or qenched) for fixed β. Attach independent identically distribted random variables j,k to the bonds of the tree. We assme that the common distribtion of jk obeys E[exp[ β]] < for all β 0.

10 42 Rhna Jornal of Science, pp , (2006) σ= σ= σ = Figre 4 S 2 S S The evoltion of the process (t) over 3 time steps in a day. The thick path shows the price evoltion whose p-down movements are characterize by tossing an nbiased coin for which σ = ±. Ths, if the initial stock price is S 0 the final stock price S(t) = S The microscopic description of the system consists of 2 n (n = 3 here) energy levels with ij obeying some probability distribtion (lg-normal here) S 3.3. The (free) energy density of the pricing process The free energy density of a flctating system plays a role similar to the potential energy in mechanics. It is a qantity which relates the entropy and the internal energy of a system in statistical physics. On the other hand the knowledge of free energy density allows one to estimate of all the other thermodynamic qantities sch as specific heat, entropy etc. The convergence of free energy density to some finite vale for low temperatres implies a phase transition of the system. In the present problem too, we shall proceed to compte the free energy density associated with the process { j }. To do this we force the agents to redce their activity (or to become highly correlated among themselves) which shall enable s to see what happens to the market as β is increased. For this we first need to have a fnction to qantify the energy or the Hamiltonian of the system. This we denote by H and define by n H = log j, (0) where j = jk at the level j and k {, 2,..., 2 j }. This choice for H is natral in the financial market context as it is S 0 i (where S 0 is the initial stock price) which determines the final stock price at the end of the evoltion. (We identify n S 0 j= j as a possible macroscopic order parameter). j=

11 Rhna Jornal of Science, pp , (2006) 43 For fixed β the free energy of this process is defined by f(β, λ, γ) = β lim n n log Z n(β), () where β is the average of the VIAF over the total time the market is being observed (over a day in the present case) and Z n is the partition fnction defined by Z n = ω: ω =n exp [ βh ] = ω: ω =n exp [ β ] n log j. (2) Notice that Z n is nothing bt the sm of exponentially weighted log retrns over all possible paths in the tree. Let s make the transformation v j = ln j in (0) after which the partition fnction is written, Z n = [ exp β ] v j, (3) j ω: ω =n which is exactly the same partition fnction obtained in Bffet et al. (993) Sketch of the derivation of free energy density The derivation of the limit in () has been done in Bffet et al. (993) by noting the fact that the normalized partition fnction M n (β) := Z n (β)/ (2φ(β)) n is a martingale Williams (99) (i.e. E [M n+ V n ] = M n (β) ), with respect to the filtration {V n, n } defined by V n := {v j,k ; k 2 j, j n} - which is the set of all random variables v j,k between the levels and n. Here φ(β) := E[exp( βv)]. Let s assme for the time being that M n (β) is a positive martingale so that it converges to a finite random variable M (β) Williams (99). Moreover, let s also assme that M n (β) < with respect to the distribtion of v j,k. Indeed, these have been proved in Bffet et al. (993) bt for β < β c for some β c which is determined by the parameters of the distribtion of v j,k. Since j= βn log Z n(β) = βn log [(2φ(β))n M n (β)] = β log [2φ(β)] + βn log M n(β), (4) we find that lim sp n βn log Z n(β) log [2φ(β)], (5) β

12 44 Rhna Jornal of Science, pp , (2006) almost srely (a.s.) with respect to the distribtion of v j,k, since the second term on the right hand side of the eqation (4) is vanishing in the limit n. Moreover, it is also proved in Bffet et al. (993) for β < β c that, lim inf n lim n βn log Z n(β) log [2φ(β)] and hence, (6) β βn log Z n(β) = β log [2φ(β)] := g ϑ(β) a.s. (7) However, for β β c there is no garantee that the moments of M n (β) are finite bt since the fnction nβ log Z n(β) is decreasing we have for any ɛ > 0 from which it follows that nβ log Z n(β) n(β c ɛ) log Z n((β c ɛ)) (8) lim sp n Since nβ log Z n(β) is convex in β it also follows that lim inf n nβ log Z n(β) g ϑ (β c ɛ). (9) nβ log Z n(β) g ϑ (β c ɛ) (20) and hence lim n nβ log Z n(β) = g ϑ (β c ) if β β c. (2) What we have jst illstrated was a sketch of the proof of the following theorem in Bffet et al. (993): Theorem. The limit lim n { βn log Z g ϑ (β) n(β) = holds almost srely with respect to the distribtion of v j,k. β β c (22) g ϑ (β c ) β > β c Remark. Let s consider an example to determine the β c as in the above theorem. Let the random variables v follows a Gassian distribtion with zero mean and variance ϑ 2. Then β c = 2 log(2)/ϑ. Figre 5 shows the behavior of the fnction g ϑ (β) plotted for increasing vales of ϑ. There is a niqe minima for the fnction and in case g ϑ (β) is strictly decreasing β c is considered to be tend to infinity. The eqation (22) shows that the free energy density for an active market (i.e. low β) is governed by the random activities of the agents (as it depends on β), bt for β > β c, the internal energy of the system dominates the randomness ths the market behaves as one with frozen activities of the agents (since the free energy is invariant). Figre 6 shows how f(β) behaves against β as we vary the width ϑ of the distribtion of the price change j. The crves are the plots of the eqation (22).

13 Rhna Jornal of Science, pp , (2006) 45 g ϑ Β ϑ increasing Β Figre 5 Either there exists β c > 0 sch that the fnction g ϑ (β) is strictly decreasing on (0, β c) and strictly increasing on (β c, ) or the fnction g ϑ (β) is strictly decreasing on (0, ). This behavior is responsible for the apparent discontinity in the free energy density as it has been proved in Bffet et al. (993). - free energy density Β Figre 6 Left: Negative free energy density f(β) as a fnction of β for varios vales of ϑ =.,.4,.23,.3,.8. ϑ increases from the top crve to the bottom crve respectively over the above vales. Ths when the variation of j is small, free energy settles at a lower vale β increases. The vertical line segments indicate the corresponding transition vale (β c) for each crve. Right: The second derivative of f(β) has a discontinity at β = β c. This implies a crash (phase transition) at β c and moreover, we identify this as a second order phase transition in the statistical physics terminology. By varying β what we illstrate here is how the free energy density of the market behaves p-to the day on which the crash occrs. As it is shown in Figre 6, we notice that the free energy is nchanged for β > β c, which in the context of financial market means that, the long range correlations among the agents dominates in this regime over the random activities of the agents. Another characteristic of physical systems that exhibit phase transitions is the discontinities of thermodynamic qantities sch as density, pressre or free energy density. In the present case we see that the second derivative of the free energy has

14 46 Rhna Jornal of Science, pp , (2006) a discontinity at β = β c : Indeed, we find 2 β 2 f(β) = { 2 log 2/β β β c 0 β > β c. (23) The above discontinity characterizes a so called second order phase transition in statistical mehcanics. 4. Conclsions As far as the athors are concerned, the Sri Lankan stock market has never been stdied for critical crashes sing the methods discssed in this paper or by any other means. As sch, we believe that these reporting will provide with some valable insight to the market specially in careflly monitoring the market for precrsors that sign a possible crash. Using these the decision makers can have an a-priory estimate of the possible crash date - bt as Johansen et al. (2000) points ot the date of the actal crash is prely random. There is no reqirement that this date be coincide with the date estimates sing the above method. The reslts we obtained in Section 3 exhibit how a phase transition or a critical crash takes place in the market when we control a virtal parameter, namely β. It wold be worthwhile to see how β is related to the hazard rate fnction introdced in Section 2.. Or work in this regards is prely pedagogical and illstrative as in practice it is not possible to find a control parameter sch as β for markets (Pal and Baschnagel (999)). Nevertheless, the markets are self-evolved de to the internal activities of the interacting agents towards crashes in which sitations one can see the inverse activity fnction β increase. So there is no harm of varying β to see how the market will respond to these changes which we have qantified by the free energy density fnction f(β). The discontinity in the second derivative of the free energy fnction means the market entering into a new regime.this, when translated into the pricing process wold imply that the trend of the pricing process cold alter. Indeed, one can see jst after the crash-point the overall trend of the pricing process starts to decrease. The reslts clearly show the crashes can be explained and they are not nsal phenomena in the market from the statistical mechanics point of view. Acknowledgments The athors wish to thank the reviewers of the paper for their valable comments and also to the Department of Mathematics, University of Rhna and the Colombo Stock Exchange - Matara Office, Sri Lanka, for their spport given in this work. References Bak P How Natre Works: The Science of Self-Organized Criticality. Springer- Copernics, New York. Black, F How we came p with the option formla. J. Portfolio management 5 4.

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