BIFURCATION ANALYSIS OF A SINGLE-GROUP ASSET FLOW MODEL

Transcription

1 BIFURCATION ANALYSIS OF A SINGLE-GROUP ASSET FLOW MODEL H. MERDAN, G. CAGINALP y, AND W.C. TROY z Abstract. We stuy the stability an Hopf bifurcation analysis of an asset pricing moel that base on the moel introuce by Caginalp an Balenovich in 999, uner the assumption of a xe amount of cash an stock in the system. First, we stuy stability analysis of equilibrium points. Choosing the momentum coe cient as a bifurcation parameter, we also show that Hopf bifurcation occurs when the bifurcation parameter passes through a critical value. Analytical results are supporte by numerical simulations. A key conclusion for economics an nance is the existence of perioic solutions for an interval of the bifurcation parameter, which is the tren-base (or momentum) coe cient. Key wors. Asset price ynamics, stability of price ynamics, Hopf bifurcation, price tren, momentum, market ynamics, liquiity, perioic solutions. AMS subject classi cation. : 9B5, 9B6, 9B50, 9G80, 9G99, 34D0, 34C60, 37G5, 37N40. Introuction. A central theme in classical nance is that market participants all have access to the same information, an all seek to optimize their returns so that a unique equilibrium price is establishe (see, for example, [3], [8], [0]). The approach to equilibrium is often assume to be a process involving some ranomness or noise, but otherwise smooth an rapi. Asie from noise, one can expect that prices will not overshoot the equilibrium price since the equation governing the change in price, P, is a rst orer in time, i.e., P 0 = F (D=S) where S an D are supply an eman that epen on price but not on the recent price erivative history. As such, there is no mechanism for oscillations or cyclic behavior within this setting. A well known example of cyclic behavior in economics is calle the "cobweb theorem" whereby prices oscillate perioically ue to the time lag between supply an eman ecisions. Agricultural commoities provie a simple example with a elay between planting an harvesting (see [] (pages gives two agricultural examples: rubber an corn) an [36]). In nancial markets, however, the prevailing theory (at least uring latter part of the 0th century), calle the e cient market hypothesis (EMH), maintains the existence of in nite arbitrage capital that woul quickly exploit any eviations between the traing price an the intrinsic or funamental value of the asset, which are necessarily unique since the participants have the same information an calculation of future returns. The absence of any elay in information or traing exclues, mathematically, the possibilities of overshooting the equilibrium price or oscillating about it. While policy makers often iscuss instabilities in asset prices, classical nance tens to treat these as rare occurrences within a stochastic setting. In particular, much of classical nance is base on the concept that an asset s price, P, is governe by Current aress: Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 560, USA. Permanent aress: Department of Mathematics, TOBB University of Economics an Technology, Ankara, TURKEY. H. Meran was supporte by TUBITAK (The Scienti c an Technological Research Council of Turkey) y Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 560, USA z Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 560, USA

2 the equation log P = t + W (.) where log P = P P is the relative change in price, is the expecte return, W is Brownian motion, an is the stanar eviation [37]. This means that any abrupt changes in asset prices, such as the 000 point (9%) rop, an rapi recovery in the Dow.Jones Inustrial Average on May 6, 00 can be attribute to rare probabilistic events. Consequently, little insight can be gaine from this perspective into causes of such market crashes that occur in the absence of changes in valuation. An alternative perspective into asset price ynamics that has been stuie since 990 is the asset ow approach in which in nite arbitrage is not assume (see [4], [5], [6], [7], [9], [], [4], [7], [8] an references therein). These moels, escribe below, stipulate that relative price change epens on supply an eman, but these quantities can epen on other factors such as the price tren. Traing motivations beyon valuation are a main issue for behavioral nance (see, for example, [3], [3], [], [3], [3]). Furthermore, the cash an stock are nite, so that one cannot rely on near perfect arbitrage to enforce an equation such as (.) above. The epenence on tren an the niteness of assets are factors that market practitioners generally assume. There are also statistical stuies that con rm the role of momentum (or tren base) investing in asset markets [30]. In this paper we pursue two relate issues. First, we stuy the onset of instability as parameters such as the tren (or momentum) coe cient is varie. In particular, we can use this as the bifurcation parameter in the system of ODE s. Secon, we emonstrate numerically that there is an interval in the tren parameter for which one has cyclic behavior in price an other variables. Mathematically, the proof of a Hopf bifurcation is further explore by numerical computation. The cyclic behavior within a particular interval of the bifurcation parameter (the tren coe cient) is signi cant in terms of economics an nance since the moel assumes a single valuation (as oes classical nance) but yiels results that show that equilibrium in the classical context oes not exist for a range of parameters. Inee the introuction of tren as a motivation for ecisions, couple with the niteness of assets are the only prerequisites for a raical change in the type of equilibrium that is attaine. In nancial economics, the equilibrium price has generally been assume to be a single value. However, our work suggests that for a range of parameters, the equilibrium must be regare as a limit cycle, i.e., a perioic orbit to which the solution trajectories converge. The solution trajectory starts at some initial value an oscillates until it reaches a cyclic orbit, an remains cyclic ine nitely. Thus the notion of equilibrium is that the variables (incluing price) oscillate perioically, even though there is no new information, strategies or assets introuce into the equations after the initial conitions. The perioicity in the asset ow equations is not a feature of an exogenous factor such as time lag ue to planting/harvesting, etc., as it is for the cobweb theorem. Stuies on Hopf bifurcation an its properties has attracte very much interest in mathematical biology, meicine, ecology, economics, an so on. Many theoreticians an experimentalists have pai great attention to i erential equations having perioic solutions which have signi cant biological an physical meaning. More speci cally, they concentrate on the stability of solutions an Hopf bifurcation occurrence (see, for example, the references [], [7], [4], [5], [9], [35] an the references therein). Stability an bifurcation analysis of the asset ow moels have recently been stuie by several researchers. DeSantis an Caginalp [9] have stuie stability of a

3 multi-group (two-group) asset ow moel of a nancial instrument with one group focuse on price tren, the other one on value. They etermine a curve of equilibrium points of the moel given the basic parameters governing the investor group motivations an assets, an showe that a strong motivation base on price tren is associate with instability. DeSantis, Swigon an Caginalp [4] have stuie stability an bifurcation properties of the same moel, an showe that as key parameters were varie, one move from an unstable point to a nearby stable point along this curve of equilibria. In many cases, however, the numerical computations showe that the trajectory unerwent a large "excursion" prior to converging to the stable point. In this paper, the stability an bifurcation properties were establishe for the curve of equilibria. Duran [6] has also stuie the numerical stability of a single group asset ow moel. In the current work, we use a relate moel to etermine conitions on parameters for which an equilibrium point is stable. Moreover, we show the existence of Hopf bifurcation by choosing the tren-base (or momentum) coe cient is a bifurcation parameter which is an important conclusion in economics an nance points of view. We also support the analytical result numerically by using MATLAB. This paper is organize as follows. Section presents the mathematical moel. In Section 3 we stuy theoretical results such as conitions for equilibrium an stability. Section 4 gives a etaile Hopf bifurcation analysis of the moel. Section 4 presents numerical results incluing a family of perioic solutions at the critical bifurcation value. While the vast majority of analysis in ynamical systems focuses on the initial onset of stability, existence of a perioic solution is of great importance in many applications. Finally, in the conclusion (Section 5) we iscuss the implications of our results in terms of nance, an irections for future research.. The Moel. Caginalp an his collaborators have moele a close market containing N shares of a risky asset, (e.g., a stock), an a total of M ollars istribute arbitrarily among a single group of investors at the outset. The ynamics of the market is etermine by the following system of i erential equations (see [5] an references therein) P P t = F k k B B ; (.) B t = k( B) + (k )B + B( B) P P t ; (.) t = c P q P t c ; (.3) t = c P a P q c P ; (.4) a where. P is the traing price of the asset at time t; P a is the funamental value,. is a time scale, 3. F is an increasing function satisfying F () = 0; such as x or log x; an F 0 () > 0; so prices remain unchange if supply an eman are equal, while prices increase if eman excees supply, 3

4 NP 4. B is the fraction of total funs in the risky asset, i.e. B = NP +M, so M the fraction in cash is B = NP +M, 5. k is the transition rate function e ne in terms of the sentiment functions, an, as k = + tanh ( + ) : Here, is the tren-base component of the investor preference that quanti es the investor group s sentiment towar the price tren, an is the value-base component of it that quanti es that towar the eviation from funamental value. They are e ne mathematically as follows := q c Z t P () P () e c(t ) ; (.5) Z t P a () P () := q c e c(t ) : (.6) P a () Eq (.5) is the mathematical expression of a sum of the impact of all former relative price changes before time t. Eq (.6) represents the funamentalist sentiment, i.e., the tenency to buy, with some nite reaction time, when the stock is unervalue.. In both equations, the parameters q i an c i, i = ;, are the amplitue constants an the inverses of the time scales for the two motivations, respectively. For example, one may consier that c is a measure of the memory length. A linear sum of the functions an gives the sentiment function that quanti es the investor sentiment towar the price tren an eviation from funamental value. The function k may inclue other motivations an behavioral e ects besie the tren an iscount such as high/low liquiity, price an traing history, an inherent behavioral biases (fear/hope). Furthermore, the structure of the sentiment function coul be nonlinear rather than simply linear aitive (see [34], [3], [3] for more iscussions). Di erentiating these equations using the Leibnitz rule one can obtain the i erential equations.3 an.4. As in [5], let the liquiity be e ne as the ratio of cash to asset quantity, L := M=N. It was observe there that liquiity is a thir variable (in aition to traing price an funamental value) that has the units of ollars per share, proviing a natural unit to measure price (see scaling below). With this e nition one also has a simple relation for the ratio between the fraction of total funs in the risky asset, B; an that in cash, B; namely, B B = N M P = P L an B = P L + P : (.7) If M an N are xe in the system, then L is a constant so that the rate of change of B can be obtaine from that of P by using the equation in (.7). Therefore, we will consier a system of the equations involving only the equations (.), (.3) an (.4) instea of the equations (.)-(.4) above as long as as M an N are xe. Next, we scale these equations as follows. Let where L is constant. Then we can write Eq (.) as P e t = P e k F k ep := P L an e Pa := P a L ; (.8) 4 ep : (.9)

5 Similarly, Eqs (.3) an (.4) can be written as t = c P q e ep t c ; (.0) t = c Pa e P e q c ; (.) ep a where all parameters are the same as in the system of equations (.)-(.4). 3. Local stability analysis of equilibrium points. We stuy the stability analysis of the rescale moels moel e ne by the equations (.9), (.0) an (.) uner the following constraints: i. F (x) = x ; ii. k + ( + ) for small values of + ; iii. j + j < "; where " is a small positive number, iv. e Pa = e P a > 0; e P a is constant, v. c ; c ; q an q are all positive parameters. If we rewrite the moel uner the constraints (i)-(v), then we have the following equations e P t = ( + + ) e P ; (3.) t = c q ( + + ) ep ( ) c q c ; (3.) t = c q! ep ep a c : (3.3) As a vector equation form one can write them as where X =( e P ; ; ) t, F = (f ; f ; f 3 ) t an X 0 = F(X); (3.4) f ( e P ; ; ; q ; q ; e P a ; c ; c ) : = ( + + ) e P ; (3.5) f ( P e ; ; ; q ; q ; P e a ; c ; c ) : = c q ( + + ) c q c ; (3.6) ep ( )! f 3 ( P e ; ; ; q ; q ; P e ep a ; c ; c ) : = c q c : (3.7) ep a The equilibrium points of the system (3.4) can be obtaine by solving the following equation for e P ; an F(X) = 0: 5

6 f = 0 yiels that ep = ( + + ) : (3.8) From f = 0 together with (3.8) we obtain eq = 0 yieling i.e., ep eq = + eq eq, (3.9) eq = e P eq ep eq + (3.0) Finally, from f 3 = 0 we have eq = q ep a e Peq ep a : (3.) Now combining Eqs (3.9) an (3.) yiels the following quaratic equation in e P eq ep eq + e Pa + e P a q that has the following positive root ep eq = epa! ep eq ( + q ) e P a q = 0 (3.) r Pa e q + Pa e P + e a q + 4( + P q ) e a q : (3.3) From a market point of view, only the positive equilibrium points are relevant, so we ignore P e eq = (:) p : ; which is negative. Uner the constraints (i)-(v) the equilibrium points have the following forms epeq ; eq + eq ; eq = eq ; 0; eq = e P a + q ( e P a e Peq ) ep a q ( e P a e Peq ) ; 0; q = e P eq ; 0; q e Pa e Peq ep a (3.4)! ep a Peq e (3.5) ep a! = P e P eq ; 0; e! eq ; (3.6) ep eq + where P e eq enotes the scale equilibrium price etermine by (3.3). Notice that the system has in nitely many equilibrium points, an the form of the equilibrium in (3.6) shows that each equilibrium point epens only the parameters: Pa e an q. Notice also that (3.5) inicates that if Pa e = P e eq ; then the equilibrium point will be (; 0; 0); or equivalently, (P eq ; eq ; eq ) = (L; 0; 0) so that P eq = P a = L at this equilibrium. 6

7 Next we shift the equilibrium point epeq ; eq ; eq to the origin. Let y := ep Peq e, y := eq an y 3 := eq be new variables. Then, the system (3.4) can be written as follows Y 0 = JY + h:o:t; (3.7) where Y =(y ; y ; y 3 ) t, h:o:t represents higher orer terms in (y ; y ; y 3 ) an J is the Jacobian matrix of the system at these equilibria with the form J = 6 4 (+ P e eq) (+ P e eq) c q c q (+ P e eq) c ep eq P e c q (+ P e eq) eq P e eq c q ep a 0 c : (3.8) in which e P eq is etermine by (3.3). The nonlinear system (3.7) is linearly equivalent to the following linear system locally Its characteristic polynomial is where Y 0 = JY: (3.9) Q() := 3 + a + a + a 0 ; (3.0) a : = trace(j) = + c + c c q ( + e P eq ) a : = 3X J kk = Sum of the principle minors of J k= = (c + c ) + c c + c ( + e P eq ) a 0 : = et(j) = c c + q P e ( + P e eq ) a e P eq ; (3.) q ep a c q ep eq! ; (3.) : (3.3) Note that a 0 is always positive uner the constraints (iv)-(v). Theorem 3.. Equilibria of the system of equations (3.)-(3.3)) are asymptotically stable if the following conition hols (C) either q (0; K ) \ (0; K ) or q (0; K ) \ (K 3 ; ), where 8 < K : = min : (+c +c ) P e eq c (+ P e ; (c+c) P e eq eq) c c (+ P e eq) + P e eq (+ P e + q P e eq eq) c Pa e 9 = ; ; (3.4) K : = B p B 4AC ; (3.5) A K 3 : = B + p B 4AC ; (3.6) A 7

8 in which A : = c c ( + P e eq ) 4 4P e ; (3.7) eq B : = c c ( + c )( + e P eq ) ep eq + c (c + c )( + e P eq ) e P eq + c c q ( + e P eq ) 4 4 e P a e Peq ; (3.8) C : = (c + c ) + (c + c ) + c c (c + c ) + c q ( + c )( + e P eq ) e P a : (3.9) Proof. The Routh-Hurwitz criteria gives necessary an su cient conitions for all of the roots of a polynomial (with real coe cients) to lie in the left half of the complex plane (see [], page 50). Thus, all of the roots of the characteristic polynomial Q() (see (3.0)) are negative or have negative real part if an only if. all coe cients a 0 ; a an a are strictly positive,. a a a 0 > 0: From (3.3) together with the constraints (iv)-(v) one can see that a 0 is always positive. From (3.), the coe cient a is positive if Similarly, From (3.), a is positive if q < (c + c ) P e eq c c ( + P e eq ) + P e eq ( + P e eq ) + q P e eq : c Pa e q < ( + c + c ) e P eq c ( + e P eq ) : Let K enote the minimum of the right han sies of the inequalities above, i.e., ( ( + c + c ) P K := min e eq c ( + P e ; (c + c ) P e eq eq ) c c ( + P e eq ) + P e eq ( + P e eq ) + q e ) P eq : c Pa e Thus, both coe cients a an a are positive if q < K : Next we etermine the conitions on the parameters such that a a a 0 > 0: Using (3.), (3.) an (3.3) one obtains that a a a 0 > 0 i Aq Bq + C > 0; where the coe cients A; B an C of the quaratic polynomial in q are all positive an e ne by (3.7)-(3.9). The roots of the polynomial Aq Bq + C are (q ) ; = B p B 4AC : A 8

9 Note that q is a (real) positive parameter. Then, the polynomial has real roots if B 4AC 0: Thus, the inequality Aq Bq + C > 0 is satis e i where either q < K or q > K 3 ; K : = B p B 4AC ; A K 3 : = B + p B 4AC A which are both positive. Finally, both conitions () an () of the Routh-Hurwitz criteria are satis e together if either q (0; K ) \ (0; K ) or q (0; K ) \ (K 3 ; ). This completes the proof. 4. Bifurcation analysis of the moel. Lemma 4.. The (characteristic) polynomial Q() = 3 + a + a + a 0 has a pair of pure imaginary roots if an only if a a = a 0 an a > 0: Proof. Let = iw be a root of Q() where w R an w > 0: Substituting = iw into the equation 3 + a + a + a 0 = 0 one obtains the following (a 0 a w ) + iw(a w ) = 0: The left han sie of the equation above is zero if an only if both a 0 w = 0 which lea to the following equality a a w = 0 an a 0 a = w = a ) a a = a 0 : (4.) Now using the rst ientity, one can etermine w = p a where a > 0: Let us now assume that a a = a 0 an a > 0: Then, the polynomial Q() can be factore an the equation 3 + a + a + a 0 = 0 can be written as follows 3 + a + a + a a = ( + a ) + a ( + a ) = ( + a )( + a ) = 0 which yiels the following roots = a an = i p a : (4.) It completes the proof. Remark 4.. In Lemma 4., if the two conitions a a = a 0 an a > 0 are satis e, then both a an a 0 have the same sign. In other wors, if Q() has a pair of pure imaginary roots, then the coe cients a an a 0 must have the same sign. Furthermore, since a 0 is always positive, a must be positive in orer to have pure imaginary roots. Theorem 4.3. The characteristic polynomial Q() has pure imaginary roots, namely ; = i p a, if the following conition hols (C) either q = K < K or q = K 3 < K, where K, K an K are e ne by (3.4)-(3.9). 9

10 Proof. If q is equal to either K or K 3, then a a = a 0. On the other han, both a an a are positive if q < K. Then the proof follows from Lemma 4.. Next we etermine the conitions on parameters such that Q() has a pair of complex conjugate roots of the form ; () = m() in(); (4.3) where m() an n() are real numbers, an n() is nonzero. Here, represents the bifurcation parameter. In orer to show that the system unergoes a Hopf bifurcation at the equilibrium point we rst nee to n a critical value, c ; of at which m( c ) = 0 an n( c ) = n 0 > 0: (4.4) Notice that all coe cients of Q() epen on the parameters q ; q ; c ; c an e P a. Therefore, can be chosen as one of them to stuy Hopf bifurcation analysis. In this work, we will choose q to be a bifurcation parameter an x the others. As a result, the equilibrium point will be unique an also isolate for these xe parameters since it is inepenent of the parameter q : In other wors, varying q oes not change the equilibrium point. In this case, we can apply the existence theory, i.e. the general Hopf bifurcation theorem (see the references [9] an [6]), in literature to show the existence of Hopf bifurcation. Let us now assume that Q() has roots of the form in (4.3). Then, substituting it into Q() yiels the following two equations n m a m a = 0; (4.5) a m + a m + a a a 0 = 0: (4.6) One can see that if a was zero, then a 0 ha to be zero ue to Eq (4.6). However, we know that a 0 is always positive (see (3.3)) so that a cannot be zero. Furthermore, from Remark 4., a cannot be negative either in orer to have pure imaginary roots. Then, the roots of the quaratic equation (4.6) have the following forms m : = m () = a + p (a ) a (a a a 0 ) ; (4.7) a p m : = m () = a (a ) a (a a a 0 ) ; (4.8) a where a 0 ; a an a are all functions of : Notice rst that the quantity a a a 0 in the raican etermines the sign of the roots m ; so that it also etermines the critical value of the bifurcation parameter, namely c. Notice also that the raican (a ) a (a a a 0 ) cannot be negative since we assume that m is a real number! (see (4.3)). A complete analysis of the roots m an m together with that of the characteristic polynomial Q() are given below: Case. if a > 0 an a a a 0 = 0 in (4.7), then m ( c ) = a + p (a ) a = 0 so that Q() has a pair of pure imaginary roots, namely ; = i p a ; where a = a0 a an a 0 is always positive (see (3.3)) (see also (4.3) an (4.5)). The thir root of Q(), which is enote by 3 in the rest of the work, will be 3 = a which is negative (see Lemma 4.). 0

11 Case. if a > 0 an a a a 0 > 0 in (4.7), then m < 0 so that Q() has a pair of complex conjugate roots with negative real parts. On the other han, since we assume that a > 0 an a a a 0 > 0 an also know that a 0 is always positive (see (3.3)), a must be positive. Thus, by Theorem 3., the thir root of Q() is either a negative real number or a complex number with negative real part, i.e., Re( 3 ) < 0: In this case, the equilibrium point is asymptotically stable. Case 3. if a > 0 an a a a 0 < 0 in (4.7), then m > 0 so that Q() has a pair of complex conjugate roots with positive real parts. Furthermore, since we assume that a a a 0 < 0; Re( 3 ) > 0 by Theorem 3.. Therefore, the equilibrium point is unstable for this case. Case 4. if a > 0 an a a a 0 = 0 in (4.8), then m ( c ) = a p (a ) a = a so that Q() has a pair of complex conjugate roots whose real parts are a which is negative. However, since we assume that a a a 0 = 0, Theorem 3. leas to result that Re( 3 ) cannot be negative so that the equilibrium point is unstable in this case. Case 5. if a > 0 an a a a 0 > 0 in (4.8), then m < 0 so that Q() has a pair of complex conjugate roots with negative real parts. On the other han, since we assume that a > 0 an a a a 0 > 0 an also know that a 0 is always positive (see (3.3)), a must be positive. Thus, Re( 3 ) < 0 by Theorem 3.: In this case, the equilibrium point is asymptotically stable. Case 6. if a > 0 an a a a 0 < 0 in (4.8), then m < 0 so that Q() has a pair of complex conjugate roots with negative real parts. However, since we assume that a a a 0 < 0, Re( 3 ) cannot be negative by Theorem 3.. Thus, the equilibrium point is unstable for this case. Case 7. if a < 0 an a a a 0 = 0 in (4.7), then m ( c ) = 0: However, since a = a0 a < 0, n = p a = R (see also (4.3) an (4.5)). Thus, there is no solution for the parameters satisfying the conitions in this case. Case 8. if a < 0 an a a a 0 > 0 in (4.7), then m < 0 so that Q() has a pair of complex conjugate roots with negative real parts. But, since a < 0; Re( 3 ) > 0 by Theorem 3.: In this case, the equilibrium point is unstable. Case 9. if a < 0 an a a a 0 < 0 in (4.7), then m > 0 so that Q() has a pair of complex conjugate roots with positive real parts, so the equilibrium point is unstable. Case 0. if a < 0 an a a a 0 = 0 in (4.8), then m ( c ) = a so that Q() has a pair of complex conjugate roots whose real parts are a that is positive. In this case, the equilibrium point is unstable. Case. if a < 0 an a a a 0 > 0 in (4.8), then m > 0 so that Q() has a pair of complex conjugate roots with positive real parts. In this case, the equilibrium point is unstable. Case. if a < 0 an a a a 0 < 0 in (4.8), then m > 0 so that Q() has a pair of complex conjugate roots with positive real parts. In this case, the equilibrium point is again unstable. We conclue from the analysis above that Cases, an 3 all lea to a Hopf a Hopf bifurcation so long as one has a particular transversality conition (speci e below) is satis e.at the critical bifurcation value. Next we start by etermining c an then check the transversality conition. Once again, q will be taken as the Hopf bifurcation parameter, i.e., we set = q in this work. Notice that q or P e a

12 can be also chosen as a bifurcation parameter to stuy the bifurcation analysis of the system as well, however, the equilibrium point is no longer isolate for these parameters. As we mentione earlier, the quantity a a a 0 etermines the critical value of the bifurcation parameter. Therefore, solving the equation a a a 0 = 0 for q we etermine the the critical value of the bifurcation parameter as c = q = K ; K 3 (4.9) where K an K 3 are e ne by (3.5)-(3.9). Thus, from the analysis above, (see Case ), we have m(q ) = 0 an n(q ) = p a ) ; = i p a ; 3 = a (4.0) when the conition (C) hols: In other wors, Q() has a pair of pure imaginary roots uner the conition (C) (see also Lemma 4.). Furthermore, when either q < K K or K 3 < q < K (i.e. the conition (C) hols), the equilibrium point is asymptotically stable (see Case ) However, when K < q < K 3 the equilibrium point is unstable (see Case 3). Lemma 4.4. If the following conition hols (a a a 0 ) 6= 0; then m (c; P e eq; eq ;eq ) (c; P e eq; eq ;eq ) where a 0 ; a ; a an m are all functions of : Proof. Since m () in (4.8) oes not give a pair of pure imaginary roots for any choice of the parameters, we will take m() := m () = a + q (a a ) a (a a a 0 ) 6= 0; as in (4.7) for the bifurcation analysis. Therefore, it is enough to show that m 6= 0 (c; P e eq; eq ;eq ) for the proof. Now i erentiating m () with respect to the bifurcation parameter one obtains m = q (a ) (a ) a (a ) a (a a a 0 ) + 4a 3 (a ) (a) (a (a a a 0 ) a a a 0) a p : (a ) a (a a a 0 ) Remember that when = c, a a a 0 = 0: Then, evaluating the erivative of m at c ; P e eq ; eq ; eq yiels that m = (a ) (c; P e eq; eq ;eq ) + 4a 3 4a 3 (a ) (a a a 0 ) a! (a a a 0 ) = : a (c; P e eq; eq ;eq )

13 Since a cannot be zero, m (c; P e eq; eq ;eq ) 0: Corollary 4.5. If h() := a ()a () the transversality conition is satis e. Proof. One can show that () Re ( c; e P eq; eq ;eq ) 6= 0 as long as (aa a0) (c; e P eq; eq ;eq ) 6= a 0 () oes not have a ouble root, then = m (c; P e eq; eq ;eq ) The proof follows from Lemma 4.4. Next lemma proves that the roots of m() are inee simple. Lemma 4.6. The two roots of m() are simple at c ; P e eq ; eq ; eq Proof. Suppose that m() ha ouble roots. Then, from (4.7) an (4.8) one can see that (a ) a (a a a 0 ) = 0: However, when = c, a a a 0 = 0 so that (a ) = 0; that is, a ( c ) = 0. But, this forces a 0 ( c ) to be zero ue to Eq (4.6). This contraicts the fact that a 0 () is always nonzero (see (3.3)). This completes the proof. Lemma 4.7. The following transversality conition hol () Re 6= 0: ( c; P e eq; eq ;eq ) Proof. One can show that () Re ( c; e P eq; eq ;eq ) = m (c; P e eq; eq ;eq ) Then the proof follows from Lemma 4.6. From the analysis above an the general Hopf bifurcation theorem (see the references [9] an [6] for hypothesis of the theorem), we can euce the following result. Theorem 4.8. For the system of the equations (3.)-(3.3) the followings hol. If the conition (C) is satis e, then the equilibria of the system are asymptotically stable,. If K < q < K 3, then the equilibria of the system are unstable, 3. The system unergoes a Hopf bifurcation at these positive equilibria when the conition (C) is satis e : :. 5. Numerical simulations. In this section, we perform some numerical simulations in orer to support an exten the analytical results that we have obtaine in the former sections. As an example, we consier a close market involving 500 ollars cash an 00 units of a particular stock istribute arbitrarily among a single group of investors at the outset. The system is conserve, that is, there is no cash or stock ow in an out from the system. We assume that the investor group values the stock as P a = 4, so the scale funamental value is e P a = P a =L = 0:8 which is xe for all time t. For simplicity, the time scale is taken as : Also, we x magnitue for the valuation motivation as q = 45 an also take large time scales (c = 0:00; c = 0:00) for 3

14 both tren an valuation motivations, respectively. Here, the liquiity value is L = 5. Our aim is to stuy numerically the ynamics of the market by using the nonlinear system (3.4) with the following parameters = ; c = c = 0:00; q = 45 an (5.) M = 500; N = 00; L = M N = 5; P a = 4; e Pa = P a L = 0:8: (5.) We vary the tren-base (or momentum) coe cient, q ; which is the bifurcation parameter. With respect to the parameters above the system has only one positive equilibrium point, namely E := epeq ; eq ; eq = (0:800; 0; 0:099). K, K an K 3 are calculate as follows K = 494:948; K = 494:4543 an K 3 = 46098:848: Note that (0; K ) \ (0; K ) = (0; 494:4543) an (0; K ) \ (K 3 ; ) = ;. Therefore, by Theorem 3., the equilibrium point E is asymptotically stable when q (0; K ) = (0; 494:4543) an unstable when q > 494:4543: Moreover, the Hopf bifurcation occurs at q = K = 494:4543 < K (see the conition (C) in Theorem 4.3) as it is illustrate by computer simulations below. For each simulation, we use the ODE package (oe45) in MATLAB (7.6.0). All simulations have been one in a time interval [0; 4000] with increment 0:0: In Figure 5. we take P e (0) = 0:8000; (0) = 0 an (0) = 0:099 as an initial conition an plot the graph of each component of the solution for q = 493:5 < q. Following this, we plot the graph of the trajectory (P ; ; ) as follows. These graphs illustrate that the equilibrium point is stable for the q values that are smaller than the critical bifurcation value q: In Figure 5. we use the same initial conitions as in Figure 5. an plot again the graph of each component of the solution for q = 495: > q. Following this, we also plot the graph of the trajectory (P ; ; ). The graphs in this gure show only the parts of whole graphs for t [3500; 4000]. These graphs illustrate that the equilibrium point is unstable for the q values that are larger than the critical bifurcation value q: In Figure 5.3 we use the same initial conitions as in Figure 5. an plot again the graph of each component of the solution for q = 494:4543 = q. Following this, we plot the graph of the trajectory (P ; ; ). Figure 5.3 shows only the parts of whole graphs for t [3500; 4000]. Figure 5.4 illustrates the graphs of the trajectories for the i erent bifurcation values, namely q = 494:46 (inner), 494:48 (mile) an 494:50 (outer), respectively, which lie just above the critical bifurcation value. This gure unerlines that there exists a small interval [q; q ) on which perioic solutions occur. Note that the amplitue of each perioic solution gets larger as the critical value q gets larger from q to q. These graphs support Theorem 4.8. In aition to numerical simulations above, Figure 5.5 shows the graphs of the trajectories for the i erent initial values, namely P e (0) = 0:805; (0) = 0 an (0) = 0:099 (inner), P e (0) = 0:800; (0) = 0 an (0) = 0:099 (mile) an P e (0) = 0:8000; (0) = 0 an (0) = 0:099 (outer), respectively. for q = 494:4543 = q: We also observe that we have a similar picture when we vary (0) (or (0)) xing other components. Figure 5.5 unerlines that there exists a family of perioic solutions when q = 494:454 = q: In other wors, the nonlinear system may have a pair of pure imaginary eigenvalues. 4

15 Zeta Zeta Zeta P G raph of the trajectory t 6 3 x t t 3 x 0 Z eta P Fig. 5.. Graphs of P ; an for q = 493:5 < q c are on the left. Graph of the trajectory is on the right. The star enotes the equilibrium point. 6. Discussion an Conclusion. Classical nance moels of asset price ynamics are largely base on equation (.). An immeiate consequence of this moel is the exclusion of any perioic behavior. Furthermore, any large, rapi eviation in price is attribute to a rare probabilistic event that is possible within Brownian motion. When such a large rapi change (sometimes calle a " ash crash") occurs, there is usually a quest for a cause, an few believe the event is a purely ranom one. Unfortunately, using (.) as a starting point o ers no hope of resolving this issue. In other wors, the moel starts with the assumption that prices behave accoring to Brownina motion (or in more generalize moels, a Levy process in which the price path can inclue iscontinuities). Thus, there is no possibility of analyzing market microstructure or participant strategies. In other wors, any sharp rop in prices woul be attribute to a rapi change in the funamental value of the asset. If it is clear from an assessment of value (e.g., a calculation of potential stream of earnings) that there has been no funamental change then the " ash crash" can only be attribute to a very low probability ranom event. Using the asset ow approach evelope by Caginalp an his collaborators using a ynamical system approach since 990 (see [4]-[], [4], [7], [8] an the references 5

16 Zeta Zeta Zeta P G raph of the trajectory t t t Z eta P Fig. 5.. Graphs of P ; an for q = 493:5 < q c are on the left. Graph of the trajectory is on the right. The star enotes the equilibrium point. therein) one can examine the extent to which the phenomenon of instability arises enogenously. Earlier stuies have explore this possibility in the context of two groups of investors with i erent motivations, namely, tren base an value base (see [4], an references therein). In this stuy we have examine the issue in a more basic context in which there is a single group that is motivate by both tren an value. The mathematical explanations are consistent with the way many practitioners have viewe major crashes. If one examines the news analyses of market professionals after the broa market crashes of October 9, 99 an October 9, 987, in which the major US stock ineces fell by 0% in two ays an one ay, respectively, the mechanisms for the crashes can be attribute to a combination of the niteness of assets an strategies that epen on the tren. In particular, a major factor of the 99 crash is believe to be the use of large amount of "margin" relate selling. This means that an investor who has bought shares largely with borrowe money, is force to sell some of it (or provie more capital) when the ratio of the investors capital to the borrowe money falls below a set fraction, an the broker issues a "margin call." In this way, tren base selling is a key factor as prices are falling. Classical nance woul counter this by saying that there is a nearly in nite amount of cash that is available for arbitrage, an this arbitrage capital woul be use to pro t by buying the shares that others are compelle to sell, thereby implicitly remeying the situation. While the arbitrage capital may be aequate uner normal circumstances, 6

17 Zeta Zeta Zeta P G raph of the trajectory t x t t 3 x 0 Z eta P Fig Graphs of P ; an for q = 494:4543 = q c are on the left. Graph of the trajectory is on the right. The star enotes the equilibrium point. uring major crashes much of this arbitrage capital is usually investe utilizing similar strategies, so that the capital that is suppose to come to the rescue is actually a major part of the problem. In fact, if one examines the buil up to these crashes, incluing more recent ones, the problem often arises as an asset class makes steay gains that are far in excess of the prevailing interest rates. Large investors then observe that borrowing at low rates an buying these assets (sometimes calle "carry trae") appears very pro table. Within the competitive environment in nance, it woul be i cult for an asset manager to forgo such pro ts with the iea of waiting until prices crash. While more stringent margin requirments were impose after the 99 crash, other mechanisms were create, over the ensuing ecaes, incluing erivatives an "portfolio insurance" that were suppose to protect a portfolio with the iea of selling options as the market fell. In essence this amounte to selling when there was a owntren. The central iea here was that there woul be a buyer for every seller at prevailing prices (as one woul expect from the classical theory). Once again the absence of large amounts of capital, reay to buy at prevailing prices, to rectify the situation allowe prices to fall about 0% for the major ineces an more for the smaller capitalize stocks. Ultimately, in each of these crashes, prices foun some support at the much lower prices than previously existe. From our perspective, these woul be the value base investors who ha been awaiting a bargain without much regar for price tren. 7

18 Zeta Graphs of the trajectories x Zeta P Fig Graphs of the trajectories for q = 494:46 (inner), 494:48 (mile) an 494:50 (outer), respectively. Initial conitions are P e (0) = 0:800; (0) = 0 an (0) = 0:099 for each trajectory. Using the asset ow moel introuce by Caginalp an Balenovich in 999 [5] we have examine these phenomena from a ynamical systems approach. In this paper we have stuie the stability an Hopf bifurcation properties of one of the most basic moels. We etermine a curve of equilibria of the moel an etermine the require conitions for the stability of the equilibrium point by utilizing the Routh- Hurwitz criteria. Following this, we give a etail Hopf bifurcation analysis of the moel by choosing the tren-base (or momentum) coe cient as a bifurcation parameter. Theorem 4.8 shows that the Hopf bifurcation occurs as the bifurcation parameter, q, passes through a critical value, q : The stability analysis an bifurcation properties of other asset ow moels (see [9], [4], [6]) have also been stuie. Another interesting aspect of markets involves the issue of perioicity. As iscusse in the introuction, there is no mechanism for cyclic behavior in the absence of some exogenous perioicity, such as the seasons in agricultural proucts. In the context of the asset ow moel, we show numerically that there exist an interval of the bifurcation parameter on which the system has perioic solutions. The existence of enogenous perioic solutions is highly signi cant in terms of altering the notion of equilibrium in economics an nance. In classical economics an 8

19 Zeta Graphs of the trajectories x Zeta P Fig Graphs of the trajectories for the initial values P e (0) = 0:805 (inner), 0:800 (mile) an 0:8000 (outer) for q = 494:4543 = q c: Other components of initial values are xe as (0) = 0 an (0) = 0:099 in each plot. nance, an equilibrium is a xe point to which prices converge. Using assumptions that are compatible with those of market practioners, we show the existence of limit cycles in which prices approach perioic behavior, renering cyclic solutions the new equlibrium concept. Once again, if one ha in nite arbitrage together with complete insight into the market behavior, investors woul seek to capitalize on the perioicity by buying low an selling high, thereby graually eliminating the cyclic behavior in favor of a single point equilibrium. The issue of enogenous perioic behavior has not garnere as much attention as instability in market stuies. There are stuies inicating that some months feature worse performance of the markets, for example late September an early October, though these are sometimes attribute to exogenous events uring several years with very negative returns, or some regular cyclic agricultural events relate to agriculture in the early part of the 0th century. The results above show that as investors focus their strategies more on tren (thereby increasing the aggregate q value), the equilibrium varies from a stable point to a perioic orbit to an unstable trajectory. One can also choose the valuation coef- cient, q ; as a bifurcation parameter to stuy the bifurcation properties of the same moel. Comparing these two results may give a better unerstaning of the nan- 9

20 cial markets by proviing a similar unerstaning in terms of the e ect of increase attention to valuation. This will be the topic of a future stuy. REFERENCES [] L. J. S. ALLEN, An Introuction to Mathematical Biology, Pearson/Prentice Hall, 007. [] J. BELAIR AND S. A. CAMPBELL, Stability an bifurcations of equilibrium in a multipleelaye i erential equation, SIAM J. Appl. Math. 94 (994), pp [3] Z. BODIE, A. KANE AND A. J. MARCUS, Investments, 7th eition, McGraw-Hill Eucation, Boston, 008. [4] G. CAGINALP AND D. BALENOVICH, Market oscillations inuce by the competition between value-base an tren-base investment strategies, Appl. Math.Finance., (994), pp [5] G. CAGINALP AND D. BALENOVICH, Asset ow an momentum: eterministic an stochastic equations, Phil. Trans. R. Soc. Lon. A, 357 (999) pp [6] G. CAGINALP AND G. B. ERMENTROUT, A kinetic thermoynamics approach to the psychology of uctuations in nancial markets, Appl. Math. Lett., 3 (990), pp [7] G. CAGINALP AND G. B. ERMENTROUT, Numerical stuies of i erential equation relate to theoritical nancial markets, Appl. Math. Lett., 4 (99), pp [8] G. CAGINALP AND M. DESANTIS, Nonlinearity in the ynamics of nancial markets, Nonlinear Analysis: Real Worl Applications, (0), pp [9] G. CAGINALP AND M. DESANTIS, Multi-group asset ow equations an stability, Discrete an Contin. Dyn. Syst. Ser B, 6 (0), pp [0] G. CAGINALP AND M. DESANTIS, Stock price ynamics: Nonlinear tren, volume, volatility, resistance an money supply, Quant. Finance, (0), pp [] G. CAGINALP AND H. MERDAN, Asset price ynamics with heterogenous groups, Physica D 5 (007), pp [] G. CAGINALP, D. PORTER AND V. L. SMITH, Initial cash/asset ratio an asset prices: An experimental stuy, Proceeings of the National Acaemy of Sciences, 95 (998), pp [3] K. D. DANIEL, D. HIRSHLEIFER AND A. SUBRAHMANYAM, Investor psychology an security market uner an overreaction, Journal of Finance, 53 (998), pp [4] M. DESANTIS, D. SWIGON AND G. CAGINALP, Nonlinear ynamics an stability in a multi-group asset ow moel, SIAM J. Applie Dynamical Systems, (0), pp [5] R. C. DORF AND R. H. BISHOP, Moern Control Systems, th eition, Pearson Prentice- Hall, Upper Sale River, NJ, 008. [6] A. DURAN, Stability analysis of asset ow i erential equations, Applie Mathematics Letters, 4 (0), pp [7] R. FRISCH AND H. HOLME, The characteristic solutions of a mixe i erence an i erential equation occurring in economic ynamics, Econometrica 3 (935), pp [8] D. FUDENBERG AND J. TIROLE, Game Theory, Massachusetts Institute of Technology, Cambrige, 99. [9] J. HALE AND H.KOCAK, Dynamics an Bifurcations, Springer-Verlag, New York, N.Y., 996. [0] J. M. HENDERSON AND R. E. QUANDT, Microeconomic Theory, A Mathematical Approach, 3r eition, McGraw-Hill, 980. [] N. KALDOR, A classi catory note on the eterminateness of equilibrium, The Review of Economic Stuies, (934), pp [] D. KAHNEMAN AND A. TVERSKY, Prospect theory: an analysis of ecision making uner risk, Econometrica 47 (979), pp [3] L. LOPES, Between hope an fear: the psychology of risk, Av. Exp. Soc. Psychol. 0 (987), pp [4] X. LI, S. RUAN AND J. WEI, Stability an bifurcation in elay-i erential equations with two elays, J. Math. Anal. Appl. 36 (999), pp [5] M. MACKEY AND L. GLASS, Oscillations an chaos in physiological control systems, Science 97 (977), pp [6] R. MAY, Stability an Complexity in Moel Ecosystem, Princeton University Press, Princeton, NJ, 973 [7] H. MERDAN AND M. ALISEN, A mathematical moel for asset pricing, Applie Mathematics an Computation, 8 (0), pp [8] H. MERDAN AND H. CAKMAK, Liquiity e ect on the asset price forecasting, Journal of 0

21 Nonlinear Systems an Applications, (0), pp [9] H. MERDAN, H. AKKOCAOGLU AND C. CELIK, Hopf bifurcation analysis of a general nonlinear i erential equation with elay, J. Comput. Appl. Math. 37 (03), pp [30] J. M. POTERBA AND L. H. SUMMERS, Mean reversion in stock prices: Evience an implications, Journal of Financial Economics, (988), pp [3] H. SHEFRIN, A Behavioral Approach to Asset Pricing, Elsevier, Lonon, 005. [3] H. SHEFRIN AND M. STATMAN, The isposition to sell winners too early an rie losers too long:theory an evience, Journal of Finance, 40 (985), pp [33] V. L. SMITH, G. L. SUCHANEK AND A. W. WILLIAMS, Bubbles, crashes an enogenous expecta-tions in experimental spot asset markets, Econometrica, 56 (988), pp [34] A. TVERSKY AND D. KAHNEMAN, Jugment uner uncertainty: Heuristics an biases, Science,85 (974), pp [35] R. YAFIA, Hopf bifurcation in i erential equations with elay for tumor immune system competition moel, SIAM J. Appl. Math., 67 (007), pp [36] D. S. WATSON AND M. GETZ, Price Theory an Its Uses, 5th eition, University Press ofamerica, Lanham, MD, 993. [37] P. WILMOTT, Paul Wilmott introuces Quantitative Finance, John Wiley & Sons, Appenix. Positivity of the traing price, P, an the equilibrium price, P eq. A. Consier the basic moel (without approximation an normalization by L) P t = k L P: (7.) k We assume of course that L > 0 (an, at this stage, a xe constant). As P becomes very small, the rst term on the RHS of (7.) will ominate. In particular, since 0 < k < ; when t is such that P = 0; we have that k ( k) L > 0; so that the RHS is strictly positive, an therefore, P =t > 0: Hence there is no possibility of negative P in this basic moel. B. Next, suppose that we approximate k = + tanh for small by k ~= + : This means that k k ~= + = + ~= : Whether or not we use the secon approximate equality above, we see that for small that + > 0: Thus substituting this in (7.) above, we have P t = + L P (7.) an the same conclusion follows in exactly the same way. C. Now suppose we are using the moel (7.) above, without any restriction that is small. In this case we know that with := + the value sentiment,, must be positive when P = 0 since P a > 0 an P a P > 0: However, epening on the tren up to time t; we may have < 0 an this may be larger in magnitue than : Thus, ( + ) = ( ) coul be negative, an for P = 0; we coul have P =t < 0; so that prices can become negative. Of course, when > ; the approximation k = + tanh ~= + is not vali at the outset, so it is not surprising that the moel can yiel negative prices.

22 D. The question of whether P eq is necessarily positive. From (3.3) we have with a := P ~ a Pa ~ =q ; q P ~ eq = a + a + 4 ( + q ) P ~ a =q : Since the secon term in the raical, 4 ( + q ) ~ P a =q, is strictly positive, we know that ~P eq > 0: Hence, we know even in the approximate moel that this is true. For the purpose of unerstaning stability, we have P that is near ~ P eq so this allows us to assume that P > 0: