Security Transaction Differential Equation

Transcription

1 Security Transaction Differential Equation A Transaction Volue/Price Probability Wave Model Shi, Leilei This draft: June 1, 4 Abstract Financial arket is a tyical colex syste because it is an oen trading syste and behaved by a variety of interacting agents. The consequence of the interaction aears quite colex and nonlinear. Therefore, how to observe this syste and find a silified ethodology to describe it is, robably, a key to understand and solve the roble. In this aer, the author observes a stationary transaction volue distribution over a trading rice range, studied the relationshi between the volue and rice of transaction through the aount of it in stock arket. The robability of accuulated trading volue (i.e. actual suly/deand quantity or transaction volue) that distributes over a trading rice range gradually eerges kurtosis near a transaction rice ean value in a transaction body syste when it takes a longer trading tie, regardless of actual trading rice fluctuation ath, tie series, or total transaction volue in the tie interval. The volue and rice behaves a robability wave toward an equilibriu rice, driven by an actual suly/deand quantity restoring or regressive force that can be reresented by a linear otential (an autoregressive ite in atheatics). In ters of hysics, the author derives a tie-indeendent security transaction robability wave differential equation and obtains an exlicit transaction volue distribution function over the rice, the distribution of absolute zero-order Bessel eigenfunctions, when the suly and deand quantity is dynaic in a stable transaction body syste. By fitting and testing the function with intraday real transaction volue distributions over the rice on a considerable nuber of individual stocks in Shanghai 18 Index, the author deonstrates its validation at this early stage, and attets to offer a icro and dynaic robability wave theory on the rice volatility with actual suly/deand quantity (transaction volue) in financial econoics. JEL classification nubers: G1; D5; C51; C5 Keywords: Colex syste; Restoring or regressive force; Linear otential (an autoregressive ite); Probability wave differential equation; The distribution of absolute zero-order Bessel eigenfunctions The author has been a full-tie individual stock trader and researcher fro Beijing, China since He articularly areciates Mada Suhua Liao, his other, for understanding and suorting his career constantly. Eail: Leilei-shi@63.net or Shileilei8@yahoo.co.cn - 1 -

2 The Object, Reference, and Benchark of Study Financial arket is a tyical colex syste because it is an oen trading syste and behaved by a variety of interacting agents, for exale, retail traders, enterrises, bankers, security coanies, insurance cororations, funds, other investent institutions, and even governent agencies etc. The consequence of the interaction aong all of the agents aears quite colex and nonlinear. Therefore, how to observe this syste and find a silified ethodology to describe it is, robably, a key to understand and solve the roble. In his Nobel Meorial Lecture, Arrow (197) wrote: Fro the tie of Ada Sith s Wealth of Nations in 1776, one recurrent thee of econoic analysis has been the rearkable degree of coherence aong the vast nubers of individual and seeingly searate decisions about the buying and selling of coodities. According to a static and qualitative descrition in general equilibriu theory, oversuly quantity and excess deand quantity roduces a driving force toward an equilibriu rice in an econoic syste (see figure 1). Price Equilibriu Force Suly Deand Quantity Figure 1: Equilibriu force toward an equilibriu rice Soros (1987) guessed: In natural sciences, the henoenon ost siilar to that in financial econoics robably exists in quantu hysics, in which scientific observation generated Heisenberg s uncertainty rincile Unfortunately, it is iossible for econoics to becoe a science To date, no aroxiately correct and falsifiable deterinistic odel of arket dynaics exists (McCauley and Küffner, 4). Insired by his guess, the author, however, is engaged in acadeic study on the volue and rice of transaction on individual stocks in ters of hysics (Shi, 1) and has roduced successful outcoes (Shi,, 4). In this aer, the object of study is that the robability of accuulated trading volue (i.e. transaction volue or actual suly/deand quantity) distributes over a rice range in intraday transactions on individual stocks. The reference or the origin of coordinate is the rice zero oint. According to the static odel, there is no trading at all unless the rice is rational or equilibriu (McCauley and Küffner, 4). - -

3 And, the benchark is aount of the transaction, a holonoic constraint (a generalized velocity indeendent constraint, i.e. Φ(, t) = ) between the volue and rice of transaction at a rice coordinate. According to classic dynaics, the nuber of degrees of freedo in generalized coordinates (volue and rice) is equal to that the nuber of the coordinates inus the nuber of indeendent constraint 1. So, the author chooses the rice as an indeendent generalized coordinate to describe the volue/rice behavior as a whole.. Transaction (or Actual Suly/Deand) Energy Hyothesis A transaction body syste is defined as a set of interacting transactions in a given tie interval, for exale, a set of intraday transactions on a stock. Transaction volue v eans accuulated trading volue or actual suly/deand quantity at a rice in a transaction body syste, if not secified. Obviously, it varies over a rice range or there is a kind of distribution. What the rice volatiles uward and downward in reference to an equilibriu rice in a transaction body syste is like to what a body of ass slides back and forth to an equilibriu osition in a restoring force syste (see figure (a)). In addition, the robability of accuulated trading volue (actual suly/deand quantity) that distributes over its rice range gradually eerges the axiu around the rice ean value in intraday transactions on an individual stock regardless of its rice fluctuation ath, tie series, and total transaction volue. This distribution is siilar to that of a large volue of articles in a ground state in a haronic oscillator syste in quantu echanics (see figure (b)). The gradually eerging henoenon observed in the volue and rice behavior is the sae as that observed in double-slit electron interference in hysics. Therefore, the author assues that the volue/rice behaves a robability wave toward an equilibriu rice, driven by a restoring force. In this way, the author finds a silified ethodology to describe the volue/rice behavior in this colex syste. x x Figure (a): A ass slides back and forth in a restoring force syste (It is siilar to what the rice volatiles uward and downward) In this dynaic odel, there is transaction volue (or actual suly/deand quantity) distribution over a rice range (reference to the shaded area in figure 1). Here, an equilibriu rice does not ean a rational rice

4 Figure (b): The distribution of a large volue of articles in the stable (or ground) state of a haronic oscillator Let E PE + ( 1 P) E T + W, (1) where E is transaction energy or liquidity energy at a rice in a transaction body syste, which v is equal to = t v t t = v tt ; P is an accuulated trading volue robability that is equal to the ratio of accuulated trading volue v at a corresondent rice to total accuulated trading volue V over a trading rice range; T is transaction dynaic energy or distribution energy and W is interacting transaction energy or otential energy at the rice (the interaction between the volue v and V v ), resectively, of which the otential energy roduces a driving force toward an equilibriu rice in the syste. They are and Thus, v vt T PE = P( vtt ) = vtt = () V V V v ( V v) v (1 P) E = E =. (3) V Vt W v E + V t + W ( ) =, (4) where the equation (4) is a transaction energy hyothesis in its differential exression. It eans that transaction energy is always equal to the su of its dynaic energy and otential energy over a rice range, which could be analogous to a conserved energy syste in hysics. If v / V = 1, i.e. there is no rice volatility in a given trading tie interval, then, W ( ) = ; If W ( ) >, then, v / V < 1, i.e. there is rice volatility in the tie interval. The otential energy can reresent for the rice volatility or ibalance between the suly quantity and deand quantity (see Section in details). In addition, if ( v / V ) ax ( V v) / V << is satisfied over a - 4 -

5 trading rice range in a transaction body syste, then, its transaction energy ean value or the su of transaction dynaic energy is subject to E = T i << W over the rice range, and i i i becoes an error ite in coarison with i E i or i W i.. Linear Potential (Energy) and Actual Suly/Deand Quantity Restoring Force In Section, the author assues that the volue/rice behaves a robability wave toward an equilibriu rice, driven by a restoring force that is roduced by the interaction between the volue v and V v. Let us study how the force is roduced in a financial econoics ersective, and find what can reresents for it in ters of hysics. According to econoic theory, buying or selling behavior is constrained. The deand quantity is constrained by budget or the aount of cash hold by buyers, while the suly quantity is constrained by the volue of roduct (or stock) hold by sellers. The actual suly/deand quantity or transaction volue is constrained by the aount of transaction and its rice. How does the volue/rice behave in a set of intraday transaction on an individual stock? Assue that the stock initial rice is its equilibriu rice, and the total aount of cash used in buying stock (deand quantity) and the total volue of stock to be sold (suly quantity) are given in this transaction body syste. Reference to initial rice, if the balance is constantly aintained between the suly quantity and deand quantity on intraday transaction, then, the trading rice is always equal to its initial rice or equilibriu rice, and has no volatility at all (Here, the transaction otential energy is equal to zero). Any a variable is exclusively deterined by the other two aong the volue, rice, and aount of transaction. Reference to an equilibriu rice, the suly quantity is frequently not equal to the deand quantity as a consequence of searate individual buying and selling decisions in an actual stock trading scenario. If the deand quantity is greater than the suly quantity reference to its equilibriu rice in a short ter interval, then, the rice volatiles ositively and deviates fro its equilibriu rice. The ore the rice deviates fro it, the higher the rice is. The sae aount of reained cash could buy fewer volue of stock. The deand quantity is reduced. When the deand quantity is less than suly quantity, the rice will volatile negatively and show regressive. It adjusts toward its equilibriu rice. When reached it, the rice volatiles continuously in the sae direction and deviate fro it again because of rice otion inertia. The ore the rice deviates fro it, the lower the rice is. The sae aount of reained cash could buy ore volue of stock. The deand quantity is increased. When the deand quantity is greater than suly quantity once again, the rice will volatile in a ositive direction siultaneously and adjust toward its equilibriu rice again until it returns to its equilibriu rice once a ore tie. Hereafter, the rice will reeatedly deviate fro its equilibriu rice and adjust toward it

6 In a transaction body syste, the rice deviates fro its equilibriu rice because of ibalance between the suly quantity and deand quantity in a short ter. The deviation changes the ibalance and causes the rice regression. When the rice returns to its equilibriu rice, it will continue its volatility in the sae direction because of rice otion inertia until it aears regressive again and then returns to its equilibriu rice. It deonstrates that there exists an actual suly/deand quantity (i.e. transaction volue) regressive force that drives the rice uward and downward toward its equilibriu rice as a result of interaction aong total transaction behavior in a transaction body syste whenever the ibalance between the suly quantity and deand quantity deterines the rice volatility direction reference to resent trading rice as a consequence of seeingly searate individual suly and deand decisions. Fro the equation (1) or (4), we know that the otential energy is in roortion to the rice and is a linear otential. In addition, we assue that the volue/rice behaves a robability wave toward an equilibriu rice in a set of intraday transactions on a stock, driven by an actual suly/deand quantity regressive force. Therefore, the linear otential (an autoregressive ite in atheatics), which can reresent for an actual suly/deand quantity regressive force, is exressed by ( ) W ( ) = A( ) A, (5) where is an equilibriu rice, is its rice ean value, and A is a transaction coefficient that could either be a constant or not over a rice range. Putting the equation (5) into the equation (4) and differentiating it, we then define actual suly/deand quantity (or transaction volue) regressive force quantitatively in this holonoic constraint syste as F W v V = = = A = vtt t 1 v v V where F is the regressive force, which is also called as a restoring force. Thus, we see that the coefficient A is the agnitude of actual suly/deand quantity (transaction volue) restoring force in a transaction body syste. The inus sign eans that the force is always toward to its equilibriu rice (reference to figure 1). tt (6) Fro definition (6), we can conclude that the restoring force is equal to zero in an absolute equilibriu transaction body syste, in which the suly and deand quantity is constantly balanced or v / V = 1 is satisfied.. Probability Wave Differential Equation and Its Solution Suose that the transaction volue/rice robability wave function ψ ( ) (Derbes, 1996) is Reference to resent rice, we define that the rice volatility is ositive if deand quantity is greater than suly quantity, whereas it is negative if deand quantity is less than suly quantity

7 is / B ψ ( ) = R e (7) where R is the wave alitude, S its action or Hailtonian rincial function, and B a constant to ake its hase diensionless. In an interacting transaction body syste, the volue/rice behavior satisfies a Hailton-Jacobi equation S t S + H (, ) =, (8) S where H (, ) is Hailtonian. The action derived fro the Hailton-Jacobi equation (8) is ( v ) β S = α t Et +, (9) where α and β are any constants. In convenience, let α = 1 and β =, then, we define S v Et. (1) t Thus, generalized oentu or transaction oentu in the rice coordinate is S Q = v t. (11) According to security trading regulation, transaction riority is given to or constrained by the rice first and the tie first regardless of articiant s rationality. If current trading rice is then, the next trading riority is offered to iniize the rice volatility with resect to c, c. Therefore, we have its atheatic exression as δ G( ψ, ) d =, (1) where G (,ψ ) is its energy functional in a transaction body syste. The actual transaction rice volatility ath is chosen by its energy functional G (,ψ ) to iniize its wave function ψ with resect to rice variations. Substituting the (11) into the (4) and using the (1), we, therefore, derive a transaction robability wave differential equation as, B V d ψ dψ + + ψ d d [ E W ( ) ] = (13) - 7 -

8 or B V d ψ dψ + + ( ψ d d [ E A )] =. (14) V When E = vtt, we choose a natural unit ( = 1 ) and obtain a transaction volue B distribution function over a rice range fro the equation (14) as ψ ) = C J [ ω ( )], ( =,1, L) (15) ( where J [ ω ( )] are zero-order Bessel eigenfunctions, is the rice, ψ ( ) are the robability of accuulated trading volue at the rice (reference to figure 3), C are noralized constants, and ω are eigenvalues ( ω > ) and satisfy v ω = vtt A = vtt + F = vtt P = vtt = const. (16) V Figure 3: The absolute zero-order Bessel eigenfunctions When transaction energy or the agnitude of restoring force is a constant over a rice range, we V choose a natural unit ( = 1) and have the volue distribution functions over a rice range B fro the equation (14) as A (,1, A ) ψ ( ) = C e F (17) with = E A = const. 1+ > ( =,1, L) (18) where is the order of the eigenfunctions, F,1, A ) is a confluent ( hyergeoetric function or the first Kuer s function, and A is the agnitude of actual - 8 -

9 suly/deand quantity restoring force or eigenvalues. When =, (,1, A ) 1 F, (19) it is an exonent distribution, which is sufficiently covered by the function (15) (Coare figure 3 with figure 4 (a) and see the sale fitness in figure 5 (c)); When = 1 A1, ( ) = C e ( 1 A ) When = 1 ψ ; () 1 A1, ( ) = C e F( 1,1 A ) ψ, (1) 1, it is alost a unifor distribution over a rice range (see figure 4 (d)). The volue/rice behaves a rando characteristic. 1 1 The eigenfunctions (17) are lotted ( =,1, and1 ) in figure ψ b ψ ( ) a ( ) c ( ) ψ d ψ ( ) 1 Figure 4: The eigenfuntions when transaction restoring force is a constant over a rice range. Eirical Results The author uses the function (15) to fit intraday real transaction volue distribution sales over the rice on the first 3 individual stocks in Shanghai 18 Index in June, 3. There are total 63 ( 3 1 ) sales. However, 11 sales are halt sales and one sale data is lost. Used sales are 618. Figure 5 illustrates the fitting results of three tyical sales, figure 6 shows the fitting goodness distribution of used sales, and figure 7 is their significant test result at 95% level

10 (3/6/13) Data: Sheet1_D Model: Probawave Chi^ =.61 R^ = (Volue) Probability P ±.1 P ±.4 P ± Price (yuan) Figure 5 (a): Fitting goodness in sale 1 (Its rice ean value is RMB5.74 yuan) (Volue) Probability Data: Sheet1_D Model: Probawave Chi^ =.1 R^ = P ±.319 P ±.6637 P ± (18) Price (yuan) Figure 5 (b): Fitting goodness in sale (Its rice ean value is RMB6.8 yuan) (369) Data: Sheet1_D Model: Probawave Chi^ =.74 R^ =.8653 (Volue) Probability P1.565 ±.454 P ±.6497 P ± Price (yuan) Figure 5 (c): Fitting goodness in sale 3 (Its rice ean value is RMB1.87 yuan) - 1 -

11 Fitting Goodness Distribution (618 Sales) % 31 5.% % % % R^<..<=R^<.3.3<=R^<.4.4<=R^<.5.5<=R^ Figure 6: Fitting goodness distribution Significance Test at 95% Level % % Significance Lack of Significance Figure 7: Test in significance ( F test). Analysis on the Sales Lacking Significance 35 sales aong 618 lack significance at 95% level. They are characterized by at least two axiu volue robability values over its rice range (see figure 8). The ibalance between suly quantity and deand quantity has been enlarged too uch, referencing to an equilibriu rice, so as to cause their equilibriu rice change in those transaction body systes. The volue/rice waves around fro one equilibriu rice to another one. They are not stable. The volue distribution over the rice is the suerosition of two functions as follow: ( J [ ω ( )] + J [ ( )] ) ψ ( ) = C 1 ω. ()

12 Figure 8 illustrates the fitting results of two sales by the function (), figure 9 shows the fitting goodness distribution aong the 35 sales, and figure 1 is the test result in significance at 95% level. (Volue) Probability (366) Data: Sheet1_D Model: Probawave Chi^ =.63 R^ =.746 P ±.163 P ± P3 5.5 ±.473 P ± Price (yuan) Figure 8 (a): Fitting goodness by the suerosition of two functions with an eigenvalue (Volue) Probability (364) Data: Sheet1_D Model: Probawave3 Chi^ =.69 R^ =.4931 P ±.158 P ± P ±.195 P ± P ± Price (yuan) Figure 8 (b): Fitting goodness by the suerosition of two functions with two eigenvalues - 1 -

13 Fitting Goodness Distribution by Suerosition (35 Sales) % 7.% 57.14% % R^<.3.3<=R^<.4.4<=R^<.5.5<=R^ Figure 9: Fitting goodness distribution in ters of suerosition functions Significance Test by Suerosition at 95% Level % 1.86% Significance Lack of Significance Figure 1: Test in significance at 95% level for suerosition functions

14 The rest one sale lacking significance is an aroxiate rando distribution. It is the case that its transaction energy or the agnitude of restoring force is a constant over a trading rice range, though it is rare (1/618=.16% in this aer). When the sale is fitted by the first order function (), the test shows significance at 95% level (coare figure 11 with figure 1) Data: Sheet1_D (366) Model: Probawave (Volue) Probability Chi^ =.33 R^ =.714 P ±.96 P ± P ±.919 P ± Price (yuan) Figure 11: The sale fitted by the function () lacks significance at 95% level ( R.7 < R =. 9 ) = crit 65 (366).1 Data: Sheet1_D Model: ProbawaveA1 (Volue) Probability Chi^ =.8 R^ =.353 P ±.13 P ± P ± Price (yuan) Figure 1: The sale fitted by the function () shows significance at 95% level ( R.35 > R =. 16 ) = crit

15 . Conclusions The author draws soe ain conclusions fro the volue/rice robability wave odel as follows: 1. The author observes and deonstrates that there exists a stationary volue distribution over a rice range on an individual stock in great ajority intraday transaction situations. The robability of accuulated trading volue (i.e. actual suly/deand quantity or transaction volue) that distributes over its rice range gradually eerges the axiu or kurtosis around the rice ean value in intraday transactions on an individual stock regardless of its rice fluctuation ath, tie series, and total transaction volue. It is that the volue/rice deviates fro an equilibriu rice frequently and shows rice ath uncertainty in a short ter, but adjusts toward it and exhibits a relatively stable volue distribution over a rice range in a longer ter. It behaves a kind of robability wave toward an equilibriu rice, driven by an actual suly/deand quantity restoring force or a suly/deand quantity ibalance restoring force.. A linear otential (energy) is an autoregressive ite in atheatics and can reresent actual suly/deand quantity (i.e. transaction volue) restoring force that is quantitatively and easurably defined in a transaction body syste. If its equilibriu rice is equal to the rice ean value and the agnitude of restoring force is a constant over a trading rice range in a long ter, then, the otential ean value is equal to zero. This could exlains why ost long-ter return anoalies tend to disaear, which was defended by Faa (1998). Moreover, if ( v / V ) ax ( V v) / V << is satisfied over a trading rice range in a transaction body syste, then, its transaction energy ean value or the su of transaction dynaic energy becoes an error ite in coarison with the su of the linear otential energy. The transaction dynaic energy is an exlicit exression for ARCH. 3. Had established and tested the transaction volue/rice robability wave odel or equation (13) or (14), the author concludes that although the volue/rice behaves aarently quite colex and nonlinear as a consequence of interaction aong a variety of agents, it is holonoically constrained by the aount of transaction and governed by a second order linear differential equation (a Stur-Liouville equation). The noralized ψ ( ) is the transaction volue robability at the rice in the equation. Fro this odel, the author obtains a volue distribution function over a rice range, the distribution of absolute zero-order Bessel eigenfunctions (see figure 3). To the author s knowledge, it is a new exlicit distribution function. 4. The function gives a tentative exlanation for shaded area in figure 1. It is the distribution of actual suly/deand quantity (accuulated trading volue or transaction volue) over a rice range in an econoic syste when its suly and deand is in a dynaic state. The robability of total actual suly/deand quantity over the rice range above its equilibriu rice in an econoic syste is total excess deand quantity robability reference to its equilibriu rice, whereas the robability over the rice range below its equilibriu rice is total excess suly quantity robability. And, the trading is ost active, rather than the least,

16 at equilibriu rice. 5. In this odel, the eigenvalue indicates quantitatively that a set of intraday transactions on an individual stock, driven by a variable restoring force toward an equilibriu rice over a rice range, is a relatively stable syste ( v tt + F = const. ) in an intraday interval. Meanwhile, a set of discrete eigenvalues and equilibriu rices in eigenfunctions show intrinsic uncertain and changing characteristics in a transaction body syste, and could be used in analyzing and describing its ossible behavior or state. The volue/rice behaves both relatively stable and frequently changing in stock arket, easured by a set of eigenvalues and equilibriu rices. 6. There exist unstable transaction body systes when the ibalance between suly quantity and deand quantity has been enlarged too uch, referencing to an equilibriu rice, so as to cause equilibriu rice change in the syste. The volue/rice waves around fro one equilibriu rice to another one. Therefore, the volue distribution over the rice is the suerosition of functions (15). In addition, the volue/rice rando walk is characterized by the robability wave odel, though it is rare. 7. Reference to a transaction energy equilibriu rice, the transaction energy in dro is the sae as that in rise, but its volue robability in the dro is greater than that in the rise. It eans that the suly quantity is greater than the deand quantity and its volue distribution over the rice range is skewed in a transaction energy equilibriu syste. This conclusion will hel us to recognize and eventually accet indisensable arket akers or reasonable rice aniulators in stock arket. 8. The asytotic behavior in the robability wave function (15) describes Mandelbrot s early findings (Mandelbrot, 1963) and gives soe reasonable exlanations for heavy tail, cluster, and eory in the rice volatility. The rice volatility has its eory like a robability wave. v The greater an eigenvalue ( ω = vtt = vtt + F ) is in the eigenfunction, the ore the zero V oints exists over a given rice range in the eigenfunction. Therefore, it shows clear cluster and heavy tail. 9. Transaction oentu is critically defined in a rice coordinate. It is transaction volue er given tie interval. It deterines the rice ean value in a transaction body syste through its weight and therefore deterines its rice rise or dro in general. Fro this, the author suggests using the oentu and the ratio of suly quantity to deand quantity to easure and ake early warning on risk in an asset arket. 1. This odel is validated at current stage. The author attets to offer a icro and dynaic robability wave theory on the rice volatility with actual suly/deand quantity in financial econoics. It is a reasonable solution to Soros s guess

17 References Arrow, Kenneth J., 197, General Econoic Equilibriu: Purose, Analytic Techniques, Collective Choice, Nobel Meorial Lecture, Nobel Prize Foundation; Derbes, D., 1996, Feynan s Derivation of the Schrǒdinger Equation, Aerican Journal of Physics 64, ; Faa, Eugen F., Market Efficiency, Long-ter Returns, and Behavioral Finance, Journal of Financial Econoics, 1998, 49, 83-36; Mandelbrot, Benoit, 1963, The Variation of Certain Seculative Prices, Journal of Business, 36(October), ; McCauley, Joseh L. and Cornelia M. Küffner, 4, Econoic Syste Dynaics, Econohysics Foru, htt:// Shi, Leilei, 1, Security Transaction Probability Equation, Working aer; Shi, Leilei,, Security Transaction Wave Equation, Working aer; Shi, Leilei, 4, Security Transaction Probability Wave Equation A Volue/Price Probability Wave Model, Econohysics Foru, htt:// ersion=abs; Soros, George, 1987, The Alchey of Finance: Reading the Mind of the Market, International Publishing Co