Discrete Dynamics in Nature and Society Volume, Article ID 3957, paes doi:.55//3957 Research Article Analysis of a Heteroeneous Trader Model for Asset Price Dynamics Andrew Foster and Natasha Kirby Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John s, NL, Canada AC 5S7 Department of Applied Mathematics, University of Western Ontario, London, ON, Canada N6A 5B7 Correspondence should be addressed to Andrew Foster, afoster@mun.ca Received 7 June ; Accepted July Academic Editor: Uurhan Muan Copyriht q A. Foster and N. Kirby. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the oriinal work is properly cited. We examine an asset pricin model of Westerhoff 5. The model incorporates heteroeneous beliefs amon traders, specifically fundamentalists and trend-chasin chartists. The form of the model is shown here to be a nonlinear planar map. Since it contains a sinle parameter, the model may be considered the simplest effective model yet derived for financial asset pricin with heteroeneous tradin. Analysis of the map yields results for stability and bifurcations of fixed points and periodic orbits. The model has intricate attractor basin behavior and lobal bifurcations to chaos: symmetric homoclinic bifurcation and boundary crisis.. Introduction The notion that the interaction of investor classes can be expressed as discrete dynamical systems is not new. Followin the seminal models of Brock and Hommes,, a number of influential models, includin 3 7, have been formulated and analyzed usin a dynamical system approach. See 8 for informative recent surveys on this flourishin line of research. The basis of all such models is the empirical evidence that traders are heteroeneous, tendin to form roups relyin on different but simple and fundamental tradin rules. Because the models are naturally formulated to consider the interactions of various identifiable tradin roups and because the observable variables such as asset price and tradin volume are essentially discrete, the models fall into the branch of mathematics known as discrete dynamical systems, maps, or difference equation systems. The study of nonlinear maps has been a very active area of mathematics for more than 3 years due to its wide application and astonishin rane of behavior.
Discrete Dynamics in Nature and Society Westerhoff 7 developed a simple asset pricin model takin into account fundamentalists and trend-chasin chartists. Fundamentalists, or smart money traders, base their decisions on the belief that prices eventually tend to return to their fundamental value. Chartists use technical tradin rules, past trends, and extrapolation of data to predict future prices. In Section, the basic assumptions of Westerhoff s asset pricin model are discussed, and we reformulate the model in the standard form of a nonlinear planar map. In Section 3, we prove results for stability and bifurcations of fixed points and period- cycles in this map and use raphical techniques with illustrations enerated by idmc to investiate its lobal bifurcations involvin chaos and local attractor basin structures. This contrasts with 7, where purely numerical simulations are performed. Because of its functional form containin a sinle parameter, we consider the Westerhoff model 7 to be the simplest effective model yet derived for financial asset pricin based on heteroeneous tradin. As such, its thorouh analysis in this paper will help to uide the formulation and analysis of more detailed models of markets with interactin heteroeneous aents.. The Model Followin 7, we assume the price of an asset at time t to depend upon the demand of the speculators in the previous period. If there is excess demand, then the price increases. Otherwise, the price remains the same or decreases. Let P be the loarithm of the asset price. The chane in P at time t is proportional to the sum of the orders enerated by fundamentalists and chartists, resultin in a map P t P t N Dt F D C t.. Here N>isameasure of the strenth of the demand, the aressiveness of speculators toward the particular asset. The quantity N Dt F DC t is the total excess demand for the asset at time t. Expressions can be iven for the orders enerated by each trader type. Since fundamentalists trust that prices convere to their perceived fundamental value over time, Dt F can be expressed as D F t F P t,. where F is the loarithm of the fundamental value of the asset. This value is considered to be constant and known. If the current price of the asset is larer than the perceived fundamental value then fundamentalists assume that the asset is overpriced, and hence the excess demand for the asset decreases. Likewise, if the current price is smaller than the fundamental price, then fundamentalists assume that the asset is underpriced, and the demand increases. If this were the only roup of traders present, the asset price in the next period would coincide with this increase or decrease in demand for the asset. However, there exists a different roup of traders called trend-chasin chartists who must also be considered.
Discrete Dynamics in Nature and Society 3 as The orders enerated for the asset by trend chasers at time t, denoted by Dt C, are iven D C t P t F V t,.3 where V t N D F t N D C t..4 Here, V t represents the tradin volume at time t. Trend chasers buy when the price is hih and sell when the price is low, assumin that prices will continue the upward or downward trend. Chartists consider V t to provide clues about how reliable their extrapolations may be. More specifically, a hih tradin volume when current prices exceed the fundamental price causes trend chasers to purchase more of the asset, whereas a low tradin volume under the same condition would cause chartists to purchase less of the asset. The total excess demand for the asset at time t can be written as E t N Dt F Dt C.5 N F P t V t P t F, and hence the asset price at time t can be written as P t P t N F P t V t P t F..6 Since the deviation from the fundamental value is important and the actual fundamental value of the asset is not, we can assume the fundamental value is unity i.e., F without loss of enerality. Thus, from the above equations, we obtain the recurrence relation P t P t N NV t, V t N P t V t.7 which is somewhat novel in that it allows us to predict asset price from the current lo asset price P t and the tradin volume from the previous tradin period V t. Writin this system in the standard form of a planar map, we arrive at our final model: P P N NV, V N P V..8 Since the model contains only a sinle parameter N, we believe it may be the simplest effective model yet derived for financial asset pricin with heteroeneous tradin. The model as presented in 7 does not contain system.7, and the stated final model in that publication is in error. However, we wish to emphasize that we have used
4 Discrete Dynamics in Nature and Society the same assumptions as in 7 for formulatin the model, and these assumptions are fundamentally sound. Also, the numerical simulations in 7 were calculated from the definin conditions..4, not the stated final model in that paper, and are correct. Our oal here is to show that by properly reformulatin the model as a standard planar map we can brin a considerable theory to be applied and can ain a better understandin of the pricin behavior. 3. Analysis We refer the reader to, 3 for eneral theory of fixed points, stability, and bifurcations of discrete and continuous dynamical systems. We determine the fixed points of the map.8 by solvin the followin alebraic system: P P N NV, V N P V. 3. This map has three fixed points: P,V,, 3. P,V N,, 3.3 P,V N,. 3.4 The local stability of the fixed points can be determined from eienvalue analysis. Since the derivatives of.8 involve absolute values, we consider the two cases: [ ] [ ] N NV NP N NV NP J P>, J P<. 3.5 N NV NP N NV NP In the limit as P, V,, the Jacobian is [ ] N J,. 3.6 ±N Solvin the characteristic equation det J λi yields the eienvalues λ andλ N. Fixed points of planar maps are asymptotically stable for λ < and λ <. Hence, 3. is asymptotically stable for <N< and is unstable for N>. Recall that a fixed point of a map is nonhyperbolic if at least one eienvalue is on the unit circle. In parametrized systems, nonhyperbolic states are associated with possible chanes in invariant subspaces and yield the possible local bifurcation points of the system. At N, the fixed point 3. is nonhyperbolic with λ, and at N, it is nonhyperbolic with λ. We can identify the bifurcations occurrin at these parameter values. Since λ <
Discrete Dynamics in Nature and Society 5 and λ chanes smoothly from λ < toλ < asn increases throuh N, a perioddoublin bifurcation occurs at λ. It can be shown that the bifurcation at N is transcritical; however, this is not pertinent due to the practical restriction N>inthis model. A similar process is carried out for fixed point 3.3. Substitutin this fixed point into J, notinthatp /N is positive, ives J N, N. 3.7 The eienvalues are found to be λ, 3/4 ± /6 N. Hereλ > for all N>, so 3.3 is unstable. As N increases throuh N 3, λ decreases throuh λ, resultin in the fixed point to chane type from unstable saddle to unstable node. Finally, fixed point 3.4 is examined. Substitutin this value into J, notinthatp /N is neative, yields J N, N. 3.8 The eienvalues are identical to the case of 3.3, and the same stability results are obtained. Fiure is the orbit diaram correspondin to.7. From this fiure it is evident that prices convere to their fundamental value if < N <. If N >, then prices alternate between two values, one that is lower and another that is hiher than the fundamental value. At N.8, a eneric Neimark-Sacker bifurcation occurs for each of the points of the period- cycle. Instead of prices alternatin between two points, they now alternate between values on the two limit cycles Fiure. Each of the two limit cycles is locally stable in period-. The patchwork basin of attraction of each limit cycle is shown in Fiure. The basin boundaries can be found analytically. They are the preiterates of the critical curves 4, the locus of points mappin to the fixed point P,V,.See 5 for another example of the Neimark-Sacker bifurcation in a nonlinear financial system. The existence of the period- Neimark-Sacker bifurcation can be inferred usin the second iterate of the map: P P N NV N N P V, 3.9 V N P N NV N P V. Fixed points of the second iterate correspond to fixed points or to components of period- cycles in the oriinal map. The period- fixed points are the solutions of the alebraic system: P P N NV N N P V, V N P N NV N P V. 3.
6 Discrete Dynamics in Nature and Society.5..5.5..5.5.5.5 3 3.5 Fiure : Orbit diaram: Lo of asset price P for varyin tradin aressiveness N. V.3.3 P Fiure : Phase diaram illustratin attractor basins for symmetric limit cycles N.9. Since system 3. involves absolute values, we solve it for the four cases resultin from the possible sins of P and N NV. We recover the fixed points 3., 3.3, and 3.4 as solutions and obtain three period- cycles N P,V : N N, N N : P P,V V, V N, 3. N N, N N P V, V, 3. P,V : P V, V P V, V, 3.3 where V, N ± N N N 3, P V i N N NV i V i. 3.4 N V i N NV i
Discrete Dynamics in Nature and Society 7 V V.3.3 P.3.3 P a b Fiure 3: Phase diarams illustratin symmetric homoclinic bifurcation. Diaram on riht N 3. includes the period- attractor basins for the two limit cycles prior to the bifurcation. Diaram at left N 3. shows the attractor formed from the merin limit cycles and resultin loss of local attractor basin structure. Period- orbit 3. exists for N and is created by supercritical period doublin 6 of fixed point 3., as shown in Fiure. The orbits 3. and 3.3 exist for N 3and oriinate from saddle period doublins 6 of the fixed points 3.3 and 3.4, respectively. Local stability analysis see the appendix proves eneric Neimark-Sacker bifurcations occur at N.78, creatin stable limit cycles surroundin the unstable spiral points 3. in period-. Period- cycles 3. and 3.3 are found to possess one eienvalue reater than unity for all N 3. Thus, these period- cycles are saddles. The limit cycle behavior exists until N 3., where a symmetric homoclinic bifurcation occurs due to the collision of the two limit cycles formed by the Neimark-Sacker bifurcations with the unstable saddle point at the oriin. This is illustrated in Fiures and 3. As N increases throuh N 3., the system becomes chaotic from the break-up of the limit cycles. The mechanism for transition to chaos via homoclinic connection tanency is similar to that of the delayed loistic map. See Aronson et al. 7 for a thorouh exposition of this lobal bifurcation in the delayed loistic map. A complicatin factor here is that there is a symmetric confiuration. The two homoclinic orbits simultaneously approach the saddle point at the oriin alon the same side in butterfly confiuration, rather than on opposite sides i.e., fiure-eiht confiuration, of the oriin s stable manifold. It is well known 8 that such confiurations lead to interestin and chaotic dynamics. Since price dynamics are chaotic for 3. <N<3.3, the asset price is not predictable for N in this rane Fiure 4. As confirmed by numerical calculation of the maximal Lyapunov exponent, prices are purely chaotic for N>3. At N 3.4, the system underoes a boundary crisis 9. A boundary crisis occurs when the boundary of a chaotic set collides with an unstable fixed point or unstable periodic trajectory. This causes annihilation of the chaotic set and its basin of attraction. The boundary crisis is illustrated in Fiure 5, as the boundary of the chaotic attractor collides with unstable fixed points P, V /N, and P, V /N,. The former attractor becomes leaky, and a typical trajectory will follow the attractor reion as a transient but eventually escape to the attractor at infinity.
8 Discrete Dynamics in Nature and Society V.3.3 P Fiure 4: Phase diaram showin chaotic attractor N 3.. V.3.3 P Fiure 5: Boundary crisis N 3.378. 4. Conclusion Heteroeneous trader models are a viable alternative to the usual stochastic calculus-based models of market behavior. By viewin markets from a different perspective, these models provide new insihts for prediction and underlyin mechanisms. The model studied here is the simplest known of heteroeneous tradin type, in that it is restricted to the behavior of only two trader roups and incorporates just a sinle parameter. As such, the behavior described in this paper is expected to be eneric behaviour common in all heteroeneous trader models. These behaviors are shown to be an approach to fundamental price, period- price oscillation, orbits on limit cycles or orbits alternatin between limit cycles, and chaos. With increasin strenth of demand, the markets become more volatile and less predictable. However, chartists and fundamentalists tend to take opposin actions. For example, chartist orders will enerally counteract a stron reaction of fundamentalists to perceived mispricin of an asset. This interplay of the two roups can lead to reduced strenth of demand and consequent calmin of the markets.
Discrete Dynamics in Nature and Society 9 Appendix The Neimark-Sacker Bifurcation The Neimark-Sacker bifurcation is the discrete equivalent of the Hopf bifurcation in continuous dynamical systems. A theorem for the Neimark-Sacker bifurcation can be iven as follows see. Theorem A.. Let F : R R R ;let μ, x F μ, x be a C 4 map dependin on μ and satisfyin the followin conditions: i F μ, for μ near some fixed μ, ii J F μ, has two nonreal eienvalues λ μ and λ μ for μ near μ with λ μ, iii d/dμ λ μ > at μ μ, iv λ k μ / for k,, 3, 4. Then, there is a smooth μ-dependent chane of coordinates brinin F into the form F μ, x G μ, x O x 5, as well as smooth functions a μ, b μ, and ω μ so that the function G μ, x is iven in polar coordinates by r λ μ r a μ r 3 θ θ ω μ b μ r. A. If a μ >, then there is a neihborhood U of the oriin and δ> such that, for μ μ <δand x U, theω-limit set of x is the oriin if μ<μ and belons to a closed invariant C curve Γ μ encirclin the oriin if μ>μ.ifa μ <, then there is a neihborhood U of the oriin and a δ> such that, for μ μ <δand x U, theα-limit set of x is the oriin if μ>μ and belons to a closed invariant C curve Γ μ encirclin the oriin if μ<μ. The sin of the coefficient a μ characterizes the type of the Neimark-Sacker bifurcation. If a μ >, the bifurcation is supercritical and yields a stable limit cycle for μ>μ.ifa μ <, it is subcritical, yieldin an unstable limit cycle for values μ<μ. With the map at N N in the Jordan normal form: x f x α β β α x x x,x x,x A. a N can be calculated explicitly from the followin : a Re λ λ ξ ξ λ ξ ξ Re λξ, ξ [ xx xx 8 x x i xx ] xx, xx ξ [ xx 4 x x i ] xx, xx
Discrete Dynamics in Nature and Society ξ [ xx xx 8 x x i xx ] xx, xx ξ [ xxx xxx xxx 6 x x x i xxx xxx ] xxx. xxx A.3 For example, the second iterate of the Westerhoff map with p> yields P P N NV N N P V, V NP N NV NP V, A.4 where absolute values are not necessary because the local attractor basins are nonfractal, and a fixed point P,V N /N N, N /N. i We shift this fixed point to the oriin, usin X P P, Y V V : N 3N 4 X 3 N X Y O X, Y, 4 N Y 3N 4 X N 4N 9N 6 Y O X, Y. 4 N A.5 The eienvalues are λ, N 8N3 37N 5N 4 8 N 5 48N 348N 536N ± 3 969N 4 44N 5. 8 N A.6 ii The eienvalues are nonreal for N >.87, and λ N uniquely at N 7 7 /4. iii d/dn λ N 97 467 7/34 > at N N. iv Testin the nonresonance conditions, we find that λ N is not a small root of unity: λ N 33 9 7 594 7 ± i.57 ±.966i, 6 6 λ N.878 ±.5i, λ 3 N.7 ±.7i, λ 4 N.5 ±.86i. A.7
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