A quantum model for the stock market
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1 A quantum mode for the stock market Authors: Chao Zhang a,, Lu Huang b Affiiations: a Schoo of Physics and Engineering, Sun Yat-sen University, Guangzhou 5175, China b Schoo of Economics and Business Administration, Chongqing University, Chongqing 444, China Contact information for the corresponding author: E-mai: zhang_chao@16.com Abstract: Beginning with severa basic hypotheses of quantum mechanics, we give a new quantum mode in econophysics. In this mode, we define wave functions and operators of the stock market to estabish the Schrödinger equation for the stock price. Based on this theoretica framework, an exampe of a driven infinite quantum we is considered, in which we use a cosine distribution to simuate the state of stock price in equiibrium. After adding an externa fied into the Hamitonian to anayticay cacuate the wave function, the distribution and the average vaue of the rate of return are shown. Keywords: Econophysics; Quantum finance; Stock market; Quantum mode; Stock price; Rate of return
2 1. Introduction The study of econophysics originated in the 199s [1]. Some physicists found that a few modes of statistica physics coud be used to describe the compexity of financia markets [,3]. Nowadays most of the econophysics theories are estabished on the basis of statistica physics. Statistica physics is ony one branch of physics. After severa years deveopment of econophysics, some physicists began to use other physica theories to study economics. Quantum is one of the most important theories in contemporary physics. It was the first time that quantum theory was appied to the financia markets when someone used quantum fied theory to make portfoios as a financia fied [4,5], in which path integra and differentia manifod were introduced as the toos to describe the change of financia markets after the gauge transformation. This idea is the same as the essence of the stochastic anaysis in finance. There are some other interesting quantum modes. For instance, Schaden originay described assets and cash hod by the investor as a wave function to mode the financia markets, which was different from usua financia methods using the change of the asset price to be the description [6]. In addition, peope paid more attention to the quantum game theory and that was usefu in the trading strategies [7,8]. In recent years, an increasing number of quantum modes were appied to finance [9-15], which attracted great attention. In this paper, we start from a new approach to expore the quantum appication to the stock market. In Section we begin from severa basic hypotheses of quantum mechanics to estabish a new quantum finance mode, which can be used to study the dynamics of the stock price. In Section 3 a simpe Hamitonian of a stock is given. By soving the corresponding partia differentia equation, we quantitativey describe the voatiity of the stock in Chinese stock market under the new framework of quantum finance theory. A concusion is iustrated in Section 4.. The quantum mode Quantum mechanics is the theory describing the micro-word. Now this theory is to be appied in the stock market, in which the stock index is based on the statistics of the share prices of many representative stocks. If we regard the index as a macro-scae object, it is reasonabe to take every stock, which constitutes the index, as a micro system. The stock is aways traded at certain prices, which presents its corpuscuar property. Meanwhie, the stock price fuctuates in the market, which presents the wave property. Due to this wave-partice duaism, we suppose the micro-scae stock as a quantum system. Rues are different between the quantum and cassica mechanics. In order to describe the quantum characters of the stock, we are going to buid a price mode on the basic hypotheses of quantum mechanics..1. State vector in the Hibert space
3 In the first hypothesis of quantum mechanics, the vector caed wave function in the Hibert space describes the state of the quantum system. Being different from previous quantum finance mode [6,9], here we take the square moduus of the wave function (, as the price distribution, where denotes the stock price and t is the time. Due to the wave property of the stock, the wave function in the so-caed price representation can be expressed by Dirac notations as where n c n, (1) n n is the possibe state of the stock system and the coefficient c n n. It is exacty the superposition principe of quantum mechanics, which has been studied by Shi [16] and Piotrowski [17] in the stock market. As a resut, the state of the stock price before trading shoud be a wave packet, or rather a distribution, which is the superposition of its various possibe states with different prices. Under the infuence of externa factors, investors buy or se stocks at some price. Such a trading process can be viewed as a physica measurement or an observation. As a resut, the state of the stock turns to be one of the possibe states, which has a certain price, i.e. the trading price. In this case, c n denotes the appearance probabiity of each state. In the statistica interpretation of the wave function, t (, ) is the probabiity density of the stock price at time t, and b P( (, dp () a is the probabiity of the stock price between a and b at time t. Actuay the fuctuation of the stock price can be viewed as the evoution of the wave function (, and we wi present the corresponding Schrödinger equation in the foowing text... Hermitian operator for the stock market In quantum mechanics, physica quantities that are used to describe the system can be written as Hermitian operators in the Hibert space, which determine the observabe states. The vaues of physica quantities shoud be the eigenvaues of corresponding operators. Whie in the stock market, each Hermitian operator represents an economic quantity. For exampe, the price operator ˆ (here we approximate the price as a continuous-variabe) corresponds to the position operator xˆ in quantum mechanics, which has been originay used in the Brownian motion of the stock price [18]. Therefore the fuctuation of the stock price can be viewed as the motion of a partice in the space. Moreover, the energy of the stock, which represents the intensity of the price's movement, can be described by the Hamitonian that
4 pays a key roe in the Schrödinger equation..3. Uncertainty principe The reation of two variabes that do not commute with each other can be demonstrated by the uncertainty principe. For exampe, the position and the momentum are two famiiar conjugate variabes in quantum mechanics. The product of their standard deviation is greater than or equa to a certain constant. This means one cannot simutaneousy get the accurate vaues of both position and momentum. The more precisey one variabe is measured, the ess precisey the other one can be known. As is mentioned above, the stock price corresponds to the position. Meanwhie there shoud be another variabe T corresponding to the momentum. As guidance in quantum theory, the correspondence principe figures out that when the aws within the framework of the micro-word extend to macro scope, the resuts shoud be consistent with the outcomes of the cassica aws. In the macro system, the momentum can be written as the mass times the first-order time derivative of the position in some specia cases. As a resut, in our quantum finance mode d T m, (3) dt where we ca the constant m the mass of stock. T is a variabe denoting the rate of price change, which corresponds to the trend of the price in the stock market. In our mode, the uncertainty principe thus can be written as where T, (4) and T are the standard deviations of the price and the trend respectivey, and is the reduced Panck constant in quantum mechanics. The equaity is achieved when the wave function of the system is a Gaussian distribution function. Meanwhie, in finance, the Gaussian distribution usuay may approximatey describe the rate of return of the asset in the baanced market [18]. Taking Yuan (the currency unit in China) as the unit of price in the rest of text, we may estimate the standard deviation of the price as tota variation of the stock price is sma, the standard deviation of be approximated as 1 3 Yuan [19]. When the d dt in the trend (3) can d dt d dt d dt d dt. (5) Meanwhie, the average rate of stock price change can be evauated as 1 Yuan per ten seconds in Chinese stock market. Via the uncertainty principe (4) we estimate the
5 magnitude of m is about 8 1. Athough the unit of this mass, which contains units of mass, ength and currency, is different from the rea mass, it does not affect the cacuation of the wave function which is non-dimensiona and we sti ca it mass in this paper. It shoud be an intrinsic property and represents the inertia of a stock. When the stock has a bigger mass, its price is more difficut to change. In genera, stocks having arger market capitaizations, aways move sower than the smaer market capitaizations ones. Thus, the mass of the stock may be considered as a quantity representing the market capitaization. The uncertainty principe (4) can be often seen in finance. For exampe, at a certain time someone knows nothing but the exact price of a stock. As a resut, he certainy does not know the rate of price change at next time and the direction of the price s movement. In other words, the uncertainty of the trend seems to be infinite. However in the rea stock market, we know more than the stock price itsef at any time. We can aways get the information about how many buyers and seers there are near the current price (e.g. investors in China are abe to see five or ten bid and ask prices and their voumes on the screen via stock trading software). It is actuay a distribution of the price within a certain range instead of an exact price. As a resut, we can evauate a standard deviation of the price. Thus the trend of the stock price may be party known via the uncertainty principe (4). For exampe, a trader sees the number of buyers is far more than the number of seers near the current price, he may predict that the price wi rise at next time. In finance, the standard deviation of the asset price is usuay an indicator of the financia risks. Introducing the uncertainty principe of quantum theory may be hepfu in the study of the risk management theory..4. Schrödinger Equation With the assumptions of the wave function and the operator above, et us consider a differentia equation to cacuate the evoution of the stock price distribution over time. In quantum theory, it shoud be the Schrödinger equation which describes the evoution of the micro-word. Corresponding to our mode, it can be expressed as i (, Hˆ (,, (6) t where the Hamitonian Hˆ Hˆ (, T, is the function of price, trend and time. When we know the initia state of the price, by soving the partia differentia equation (6), we can get the price distribution at any time in the future. The difficuty here is the construction of the Hamitonian because there are ots of factors impacting the price and the trend of the stock, such as the economic environment, the marketing information, the psychoogy of investors, etc. It is not easy to quantify them and import them into an operator. Next we wi use the theories above to construct a simpe Hamitonian to simuate the fuctuation of the stock price in Chinese market under an idea periodic impact of externa factors.
6 3. The stock in an infinite high square we In Chinese stock market there is a price imit rue, i.e. the rate of return in a trading day cannot be more than 1% comparing with the previous day's cosing price, which is appied to most stocks in China. This eads us to simuate the fuctuation of the stock price between the price imits in a one-dimensiona infinite square we. We consider an infinite we with width d %, where is the previous day's cosing price of a stock. After the transformation of coordinate, (7) the quantum we becomes a symmetric infinite square we with width d, and the stock price is transformed into the absoute return. Go on and et r, (8) thus the variabe of coordinate turns to be the rate of the return. At the same time, the width of the we becomes d %. If we approximatey take the average stock price in China to be 1Yuan, to evauate the mass of the stock again in the new coordinate system, there shoud be a division by 1 Yuan from the eft side of the uncertainty principe (4). Therefore the new mass denoted by m shoud be approximatey evauated as currency vanishes and this is deighted in physics. 3 1, in which the dimension of Usuay, when the market stays in the state of equiibrium, the return distribution can be described approximatey by the Gaussian distribution [18] or more precisey by the Lévy distribution []. Whie in Chinese stock market with the price imit rue, the distribution may be more compicated due to the boundary conditions. However, here we ony want to quantitativey describe the voatiity of the stock return by the quantum mode, so we choose a cosine-square function, whose shape is cose to the Gaussian distribution, to approximatey simuate the return distribution in equiibrium. The reason for such a seection is the ground state of the symmetric infinite we discussed above is a concave cosine function with no zero except for two extreme points [1], which is usuay denoted by r ( r) cos( ) (9) d d with corresponding eigenenergy E. (1) md According to the property of the wave function, the square moduus of the state (9) is the probabiity density of the distribution of the rate of the return. As is shown in Fig. 1 (a), what is the same as the Gaussian distribution or the Lévy distribution is that in the center of the we,
7 which corresponds to zero return, the probabiity density has the maxima vaue and it decreases symmetricay and graduay towards the eft and right sides. The main reason why this distribution is not precise enough is it does not have the fat tais and the sharp peak. The cosine distribution function equas to zero at r d / due to the boundary condition of the infinite square we. Judging from the shape, the cosine distribution is a good approximation for the Gaussian distribution with a arge variance or the Lévy distribution with a great parameter. Fig. 1. (a) In the infinite square we with width d., the probabiity density of the rate of stock return can be approximatey described by the Gaussian distribution (dashed ine). Here we use a cosine-square function (soid ine) to simuate the distribution of the rate of return in equiibrium. (b) When a periodic fied put into the Hamitonian, the bottom of the quantum we begins to sope and it changes periodicay. There is aways a ot of market information affecting the stock price. The tota effect of information appearing at a certain time is usuay conducive either to the stock price s rise or to the stock price s decine. In order to describe the evoution of the rate of return under the information, we consider an ideaized mode, in which we assume two types of information appear periodicay. We may use a cosine function cos t to simuate the fuctuation of the information, where is the appearance frequency of different kind of information and here we assume 1 4 s 1. That means the information fuctuates in a singe cyce of about four trading days in China, which may be reasonabe. The vaue of the function cos t changes between [-1,1] over time, where the information is advantageous to the price s rise when the cosine function is ess than zero, and advantageous to the price s decine when it is greater than zero. The stock here is simiar to a charged partice moving in the eectromagnetic fied, where the difference is that the externa fied of stock market is constructed by the information. The stock price may be infuenced by such a fied. Under the dipoe approximation, the potentia energy of the stock can be simiary expressed as efrcos t, where e is a constant with the same order of magnitude as an eementary charge, and F denotes the ampitude of the externa fied. The Hamitonian of this couped system can be written as Hˆ m r efr cost, (11) where the first term is the kinetic energy of the stock return, which represents some properties of the stock itsef. The second one corresponding to the potentia energy refects the cycica
8 impact the stock fees in the information fied. In order to observe the characters of this system entirey, we make the magnitude difference between the kinetic energy (i.e. ground state energy of the infinite square we without externa fied) and the potentia energy not great, thus estimate F 1 19 here. Because the second term of Hamitonian contains the rate of return r, tit appears at the bottom of the infinite square we, which is shown in Fig. 1 (b). The sope of the bottom fuctuates periodicay due to the change of cos t over time, which makes the we no onger symmetric. This refects the imbaance of the market, and the distribution of the rate of return becomes symmetry breaking from the state (9) in equiibrium. After constructing such an idea Hamitonian (11), we need to get the soution of the corresponding Schrödinger equation i r, t m r (. (1) efrcost ( r, However, the anaytica soution for the simiar equation has been studied before []. Therefore we just have a brief review. At first, et ( r, (, ( r, (13) with variabe substitution ef cost r. (14) m In the wave function (13), et where ( r, iect iefrsin t ie F exp 3 t sin t 8m, (15) E c denotes the energy of the driven system. After both sides of Eqs. (1) are divided by ( r,, we find (, satisfies the time-dependent Schrödinger equation m t, t E, t i, t (16) and its soution can be written as, t A exp exp it i me where the energy may be expressed as, (17) E E. (18) c The centric energy E c is determined by the boundary conditions and cose to the energy (1) of the ground state of the stationary Schrödinger equation without externa impact. The energy spits around E c with the integra mutipe of energy unit and every new energy corresponds to a possibe state of the system. From Eqs. (13), (15) and (17), the wave function of the rate of stock return can be written as the superposition of a those states
9 iect iefrsin t ie F ( r, exp A exp itexp ik where the wave vector is ef cost ( r ) ( ) m t sin t 8 m 3 exp ik ef cost ( r ) m, (19) k / m E. () c In the wave function (19), A denotes the ampitude for each possibe state, which are some constants to be determined. In Eqs. (19), we use Fourier expansion for the exponentia function where kef cost n kef exp i i J n expint m n m kef J m, (1) n is the n th Besse function and et Eqs. (19) satisfy the boundary condition of the infinite quantum we d d (, (,. () According to physics, this must be satisfied at any time. By the orthogonaity of Besse function [3] u J m u m n J n,, (3) the summation over n in Eqs. (1) can be reduced, and Eqs. (19) at the boundary turns to be kef exp J i A ik d / ( ) exp ik d / By defining two parameters v E c kef q m and expanding the wave vector k k 1 v n m. (4) (5) to second order of v, the non-normaized approximate soution of A may be found []
10 A i which meets k d q( q ) v J 64 q v 3 4 9q v 48 q J q J q J q J q 1 q v 1 3 q v 64 1 J q J q J q J q 3 3q v 3 3 J 4 q J 4 q /8 From Eqs. (), (5) and (7),, (6). (7) A can be finay determined. According to Eqs. (19) and (6) we have a numerica cacuation for the wave functions at different points in time and simuate their distributions in Fig.. When t, 1s and 5s, the distributions of the rate of return are potted with the soid ine, the dotted ine and the dashed ine respectivey. At the beginning of t, the distribution of the rate of return is neary symmetric with previous day's cosing price, which corresponds to the initia state of the stock. The zero return point r is the most probabe and the probabiity density decreases towards both sides of the price imits. The average rate of return equas zero at this time. By adding the externa fied of information in, the distribution of the rate of return starts its evoution over time. When t 1 s and t 5s, the maxima of the probabiity density shift from the point but the peak vaues seem not to change. r, Fig.. Numerica simuations of the probabiity density of the rate of return at (a) t (soid ine), (b) t 1 s (dotted ine) and (c) t 5s (dashed ine), in which parameters are respectivey e , m 1 4, 1, F 1 19 and d.. At t the distribution corresponds to the initia state of the system. The externa fied makes the distributions imbaance at t 1 s and t 5s. The change of the distribution refects the imbaance of the market under the infuence of
11 externa information. It aso can be seen from the evoution of the average rate of return over time. In quantum mechanics, the average vaue of the rate of return can be written as 1 d / r( ( r, r ( r, dr C, (8) d / where C is a constant to keep the wave function normaized. The average rate of return fuctuating in one period is shown in Fig. 3. By anayzing Eqs. (19), we obtain that the period of the wave function is /. In the first period, the average rate of return is symmetric with t /, which is due to the biatera symmetry of the Hamitonian (11) or the wave function (19) in one time period. For the parameters seected, the average rate of return vibrates about times within a period, and the ampitude of the fuctuation achieves about 3%. It tes that the stock price dose not definitey rise under the advantageous information, whie with the disadvantageous information, the stock price may not surey decine. The price is aways voatie and Fig. 3 refects that the price fuctuates more frequenty than the tota effect of the market information. Fig. 3. The average rate of return fuctuates in a cyce under the market information with the same parameters as in Fig.. The fuctuation of a singe cyce is given in the figure, in which the symmetry axis appears at ampitude of the fuctuation achieves 3%. t / and the 4. Concusions In this paper, we start from severa fundamenta assumptions of quantum theory to buid a quantum finance mode for stock market. By constructing a simpe Hamitonian, we give an exampe of this quantum mode, in which the square moduus of the ground state in the infinite square we represents the distribution of the stock return in equiibrium. By putting an externa fied in, which simuates the externa factors affecting the stock price, we quantitativey describe and discuss the distribution of the rate of return and the evoution of the average rate of return over time. The mode gives a new theory of quantum finance, which may be hepfu
12 for the deveopment of econophysics. If we can further find some ways to quantify the factors affecting the stock price and write them into the Hamitonian, or use some more accurate quantum states, such as the ground state of the harmonic osciator, which exacty equas to the Gaussian distribution, to describe the price distributions in the equiibrium market, the fuctuation of the stock price is abe to be estimated more precisey. Acknowedgements We woud ike to thank Prof. Wang Xuehua for his encouragement of this work and Dr. Liu Jiaming for hepfu discussions. References [1] R.N. Mantegna, H.E. Staney, An Introduction to Econophysics: Correations and Compexity in Finance, Cambridge University Press, Cambridge, [] S.N. Durauf, Statistica mechanics approaches to socioeconomic behavior, in: W.B. Arthur, S.N. Durauf, D.A. Lane (Eds.), The Economy as an Evoving Compex System II, [3] Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, H.E. Staney, Physica Review E 6 (1999) 139. [4] K. Iinski, Physics of Finance, Wiey, New York, 1. [5] Bea E. Baaquie, Quantum Finance, Cambridge University Press, Cambridge, 4. [6] M. Schaden, Quantum finance, Physica A 316 () [7] D. Meyer, Quantum strategies, Phys. Rev. Lett. 8 (1999) 15. [8] J. Eisert, M. Wikens, M. Lewenstein, Quantum games and quantum strategies, Phys. Rev. Lett. 83 (1999) 377. [9] C. Ye, J.P. Huang, Non-cassica osciator mode for persistent fuctuations in stock markets, Physica A 387 (8) [1] Ai Atauah, Ian Davidson, Mark Tippett, A wave function for stock market returns, Physica A 388 (9) [11] F. Bagareo, Stock markets and quantum dynamics: A second quantized description, Physica A 386 (7) [1] F. Bagareo, An operatoria approach to stock markets, J. Phys. A: Math. Gen. 39 (6) [13] F. Bagareo, The Heisenberg picture in the anaysis of stock markets and in other socioogica contexts, Qua Quant (7) 41: [14] F. Bagareo, A quantum statistica approach to simpified stock markets, Physica A 388 (9) [15] F. Bagareo, Simpified stock markets described by number operators, Rep. Math. Phys. 63 (9),
13 [16] Leiei Shi, Does security transaction voume price behavior resembe a probabiity wave, Physica A 366 (6) [17] E.W. Piotrowski, J. Sadkowski, Quantum diffusion of prices and profits, Physica A 345 (5) [18] P.H. Cootner (Ed.), The Random Character of Stock Market Prices, MIT Press, Cambridge, MA, [19] Mikae Linden, Estimating the distribution of voatiity of reaized stock returns and exchange rate changes, Physica A 35 (5) [] R.N. Mantegna, H.E. Staney, Nature 376 (1995) [1] D.J.Griffiths, Introduction to Quantum Mechanics, Prentice Ha, Upper Sadde River, NJ, [] Mathias Wagner, Strongy driven quantum wes: an anaytica soution to the time-dependent Schrödinger equation, Phys. Rev. Lett. 76 (1996) 41. [3] M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematica Functions, Dover, New York, 197.
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