Multi-Group Asset Flow Equations and Stability

We consider a two-group asset flow model of a financial instrument with one group focused on price trend, the other on value. We prove the existence of both stable and unstable regions for the system of differential equations and show that a strong motivation based on (particularly recent) price trend is associated with instability. Numerical computations using a set of typical parameters describe precise regions of stability and instability. A precise limiting connection between the discrete and differential equations is also established.


1.
Introduction. Recent events in the world's financial markets have demonstrated the need for developing mathematical models that are capable of addressing issues of market dynamics and stability. Standard theories of asset management and options assume near equilibrium conditions. In particular, asset and option valuation theories are based on the equation where P is the asset price at time t, while dX is a normal random variable (mean 0 and variance dt), σ 2 is the variance, and µ is the drift, so that µdt is the expected return on the investment in time dt. This formalism is based upon the idea that a sufficiently large proportion of investors are informed of the realistic value of the asset and act solely on that basis. It assumes that there is an infinite amount of arbitrage capital that is ever-present to quickly exploit any deviation from fundamental value. However, it has been noted (see [26]) that there exist practical limitations to arbitrage in real markets. Thus, while an arbitrage opportunity might exist, arbitrageurs might not be able to act on it due to risk of losses and the need to liquidate the portfolio due to pressure from investors. Furthermore, on a larger time scale the assumption of rapid assimilation of information leads to an inverse relationship between volatility, defined as σ and mean expected return, µ. A standard optimization result ( [2]) is that all investors should invest in a mixture of a risk-free asset (e.g., Treasury bills) and a single portfolio of risky assets consisting of stocks and bonds. The fraction of wealth invested in risky assets will differ among investors who differ in their utility (or preference) functions, and consequently, risk tolerances.

GUNDUZ CAGINALP AND MARK DESANTIS
The classical theories are idealizations that are limiting cases. In particular, the capital owned by knowledgeable investors focused on value is assumed to be infinite. Starting from a theory in which a crucial quantity is infinite, it is often difficult to design a unique generalization or modification in which it is finite. The expectation, or hope, in the research community is that more refined theories with smaller effects will augment an existing, established theory. Thus it is difficult, if not impossible, to build on a theory that does not recognize a key finite quantity. One must model such effects from first principles. Yet utilizing a perspective of basic modeling that does not emanate from a familiar theory encounters difficulties as a result of the sociology of scientific research.
Two perspectives in addressing this issue are noted below. (i) One can consider first the modeling of an experimental asset market, pioneered by Vernon Smith (2002 Economics Nobel Laureate) and collaborators, where one or two dozen participants trade an asset defined by the experimenter with the objective of earning real money based on their trades. Within this setting one knows precisely the quantity of cash, asset, payouts etc. Thus, one can develop and test mathematical models of these experiments. (ii) Upon generalizing such a model to several groups with varying motivations and assuming some randomness one can try to recover the classical models as a limit of the value oriented group having a preponderance of the assets. This is an issue that we consider in Appendix A of this paper.
We summarize an approach that has been developed by Caginalp and collaborators since 1990 (see www.ssrn.com). This model is flexible enough to account for a variety of factors that may affect investor sentiment, allow for multiple investor groups, and incorporate concepts such as the gradual diffusion of information across the investor population. If we focus on a simple experimental asset market setting, there are a set number of traders who trade a single asset whose worth is defined by its payouts or dividends. The experimenter has the option of allowing these dividends to be traded during the experiment, or deferred to the end of the experiment. Thus we may define M as the constant representing the total cash in the system (assuming for the time being that dividends are deferred), and N as the (constant) number of shares of the asset. We let The liquidity, L := M/N , is a key variable introduced in [4] that has units of dollars per share. Thus, the quantities L , P , and the fundamental value of the asset, P a (t), determined by the definition of the asset, all have these units, and one can measure price in natural units of liquidity. The modeling of asset price dynamics starts by using a simple equation that stipulates that price moves in proportion to the imbalance between demand, D, and supply, S: where τ is a time scale ( [29]). Within a closed system one can express the demand for the asset as D = k(1 − B) and the supply as S = (1 − k) B, where k is a transition rate. One can interpret k as the probability of a unit of cash being submitted as a buy order per unit time. Classical game theory ( [22], [14]) would suggest that k should depend solely on the valuation, since each trader is aware of the same information. However, the dependence of k on other quantities such as the trend in price (price trend or trend) can be determined experimentally ( [8]) and empirically (in the case of world markets)( [5], [23]). Given a particular motivation, one can use the discrete version of these differential equations in conjunction with statistical methods to determine whether there is statistical support for it ( [6]).
With these definitions, one can rewrite 1.4 as The key issue is the dependence of k on investor strategy and behavior. The investor sentiment can be described by a function ζ (t) with range in R, and can be written in terms of a finite set of components: ζ (t) = ζ 1 (t) + ζ 2 (t) + ... + ζ n . Since k must have values in [0, 1] a simple way to map values of ζ into k is through a function such as k := {1 + tanh(ζ 1 + ζ 2 + ... + ζ n )} /2. (1.6) Considering valuation, P a (t), and just one other motivation, namely trend, we have where q i and c i characterize the magnitudes and time scales of the two motivations for investing [3]. The first of these indicates that investors are influenced by the change in price but are more strongly influenced by recent changes. A small value for the time scale 1/c 1 indicates that investors are focused on short term changes in the trend. The ζ 2 (t) parameter represents investors' focus on the deviation between the asset price and its fundamental value. Again, more recent changes have a greater effect on investor decisions. A large value for 1/c 2 indicates that investors take action slowly when there is an over-or undervaluation. An alternative explanation for the 1/c i parameters is that they regulate the rate at which information diffuses to the investor population. The magnitudes of q 1 and q 2 represent the significance of these motivations in trading decisions for the aggregate (homogeneous) investor population. Note that other motivations, ζ 3 (t), ζ 4 (t), etc. are possible. For example, statistical studies ( [5]) have shown that the M2 Money Supply and asset price volatility also impact investor decisions to buy and sell. One can also show (see equations 2.6 and 3.5 in [4]) that the change in the fraction of wealth invested in the asset is given by (1.8) The system of equations 1.5 -1.8 is a complete set of ODEs that have been studied in a number of papers. We will study generalizations of these equations to two or more groups. One can simplify these equations by taking the limit of short time scale and using the approximation tanh (x) x, leading to the approximation One of the concepts that has emerged from this analysis is that the equilibrium price depends not only on P a (which would be the case in classical economics) but also on the liquidity, L. The equilibrium price was found to be (1.10) When q 2 is large (i.e. there is a great deal of importance attached to valuation), the equilibrium price, P eq , is close to P a . However, if q 2 is small, the equilibrium price is close to L, as the valuation is marginalized. This approach to modeling incorporates the issues of trend, overreaction, underreaction, disparate information and motivation of distinct groups as well as the effect of finiteness of assets within a single mathematical model. Thus it is evident that these effects arise from the dynamics of trading when the assumptions are more realistic than the classical idealizations. In the years since 1990 several works have implemented various aspects of this approach. In particular, [17] models overreaction, underreaction, and momentum via the interaction between two heterogeneous investor groups: news watchers and momentum traders. In other words, members of the fundamental group are called "news watchers" since they update the fundamental value via news reports. Similarly, an asset price's tendency to follow a recent trend in price has been studied ( [18], [19]) along with the competing concepts of overreaction and underreaction.
In addition several models including [9] attempt to explain momentum profits by investors' inherent biases in interpreting information. They consider overconfident investors who overreact to private information. Within our models, this corresponds to one group setting a higher price on the fundamental value. The Barberis, Shleifer, and Vishny [1] model incorporates the psychological concepts of representativeness ( [28]) and conservatism ( [13]). Grinblatt and Han [15] explained momentum via the concept of market disposition, i.e. investors are more likely to sell to realize a gain than a loss ( [25], [24]).
Note that trend effects need not be exclusively due to overreaction or underreaction. For example, due to the finiteness of assets, a large buyer attempting to build a position without drastically pushing up prices could cause an upward trend in price. Another possible explanation could be the observation that others are profiting, which causes even the value-based investor to act ( [6]). Duran and Caginalp [12] utilized optimization methods to estimate the parameters of the asset flow differential equations using market data. These equations were applied to typical secondary stock offerings by Caginalp and Merdan [7], who obtained quantitative results relating an influx of shares with lower prices within this model. A sensitivity analysis of parameters was performed in [11]. Recently, liquidity issues were further explored in [20] and [21].
In this paper we present a detailed derivation of the discrete and continuum equations for asset dynamics with disparate groups with differing assessments of valuation and motivational characteristics. We determine the region of equilibrium points given the basic parameters governing the investor group motivations and assets (see Theorem 4.1 in Section 4). Theorem 4.2 (see Section 4) establishes regions of stability and instability. In Section 5 numerical computations for typical parameters display precise regions of stability and instability.
2. The discrete equations. We consider two groups of investors (that can easily be generalized to an arbitrary number of groups) so that group i (i = 1, 2) is endowed at time T j with N (i) (T j ) shares and M (i) (T j ) in cash. The probability that a unit of cash is submitted by investor group i for purchase during the time interval (T j , T j+1 ) of the single equity in the system is given by k (i) * and may depend on many factors including the price history, as discussed below. Similarly,k (i) * is the probability that a unit of stock will be submitted for sale. The demand (in terms of dollars) at time interval δT := T j+1 − T j (assume a uniform spacing so that δT is independent of j) is given by (2) . An equilibrium trading price is one for which the supply,S (T j ), and demand, D (T j ), balance, i.e.
and, for brevity, we have replaced k (i) * (P (T 0 ) , ...P (T j )) by k (i) * on the right hand side. We define τ as a relaxation time that determines the rate at which a nonequilibrium situation returns to equilibrium. For simplicity we can assumek (i) * := 1 − k (i) * though the modeling is similar in the more general case. Next, we describe the dynamics of price. Similar to [16] we postulate that the change in price in a unit time is proportional to the extent to which it is away from equilibrium: where we have suppressed the dependence of F on the other variables. As in earlier papers (see [7] and references therein) we consider a limiting form of this microeconomic equation, i.e. 3.2 below. In particular, we derive this limiting form precisely in Section 3. Note that by using the definition of F , we see that the right hand side is the excess demand, (D − S) /S. Hence, this equation is compatible with basic price theory ( [29]) as well as observations in experimental asset markets ( [27]). The time scale. Since δT is simply the time interval on which we choose to perform the computations in the discrete case, a natural choice for δT is given by δT := τ since the imbalance is restored on this time scale. Then 2.2 is simply 3) This indicates that if the supply and demand do not balance, then a new price, defined as P (T j+1 ), is discovered. This price restores the balance between supply and demand, i.e.
within a time frame that is the natural relaxation time for the problem. Equation 2.2 states that if the price is not at an equilibrium value then it moves toward this price on a time scale given by τ . We can regard δT := T j+1 − T j as the time interval between trades in a discrete system, e.g. a market in which bids and asks are submitted at noon each day or an experimental setup with discrete periods.
The choice δT = τ is perhaps the most natural and closest to an efficient market since it suggests that the basic time scale for readjustment to equilibrium is the one period time scale defined by the market institution.
Prior to focusing on δT = τ we consider the implications of choosing δT := aτ for a = 1. In this case 2.2 can be written as For −1 < δ < 0, i.e. 0 < a < 1, when F (T j ) = P (T j ) the price at time T j+1 does not attain F (T j ) as it does for δ = 0. For example, if δ = −1/2, then so that P (T j+1 ) is only the average of the old price, P (T j ), and the "target equilibrium price," F (T j ). Interpreting δ = 0 within the context of experimental markets, one can consider τ = 3 and δT = 1 so that a = 1/3 and δ = −2/3. This means that the time scale of adjustment to equilibrium is 3 units while the trading time scale is 1. Hence by 2.6 the price adjustment at time T j+1 attains 1/3 of the deviation, F (T j ) − P (T j ), from equilibrium.
Unless specified otherwise, we assume δT = τ below. For δT = τ the equations 2.7 and 2.8 below must be modified by replacing P (T j+1 ) by F (T j ).
The equation 2.2 remains valid if there is an influx or outflow of cash or shares during the time period between T j and T j+1 with the assumption that this new cash or shares cannot be submitted until time T j+1 . At this point we distinguish between a totally conserved system in which no shares or cash are added or subtracted versus one in which there is an influx or outflow of shares or cash. In the latter case, we can define m (i) * (T j )δT and n (i) * (T j ) δT as the net inflows of cash and shares during this time period between T j and T j+1 . Note that negative values for either of these variables denote outflows. Thus we can write the basic conservation law as . In other words, during the time period from T j to T j+1 the cash decreases due to the fraction of existing cash, k (i) * δT , that is submitted for purchase of the equity, and increases due to the fraction of shares,k (i) * δT , that is submitted for sale. Setting m (i) * = 0 for all time and each group, i, specifies the special case in which we have MULTI-GROUP ASSET FLOW AND STABILITY 115 no additional cash entering or leaving the system. Similarly, for each group i, the change in the number of shares owned must satisfy: For each group i, the change in the value of stock owned must be balanced by the flow plus the influx of cash or shares. In fact, comparing the right hand sides of 2.7 and 2.8 we observe: The system of equations 2.2, 2.7, and 2.8 can be studied as a system of difference equations upon specifying initial conditions together with k (i) * as a function of (P (T 0 ), ..., P (T j )) and possibly other variables. In other words, these equations can be solved algebraically for P (T j+1 ), M (i) (T j+1 ) , N (i) (T j+1 ) in terms of the variables at earlier times. The choice of k for n = 2, i.e. two investor groups, using P (T j+1 ) = F (T j ) and the definition of F .

Remark 2.
The relationship between the discrete asset flow equations 2.2, 2.7, and 2.8 and the stochastic asset pricing equation 1.1 can be understood formally in the following sense. One considers two groups, one of which has the preponderance of assets and focuses on value, buying slightly below and selling slightly above the valuation, P a (t) = P a (0) e µt , where µ is the risk-free rate. The other group employs a variety of strategies that cancel out one another. The additional funds entering into the system are a random process leading to a price dynamics that is stochastic and similar to 1.1. Further details are provided in Appendix A.
3. The continuum limit. In deriving the continuum limit it is essential to examine the roles of δT and τ . The relaxation parameter τ is intrinsic to the system and describes the time scale on which the traders react to the changes in price and other variables. On the other hand, δT is a computational time scale so that we can consider the limit δT → 0. In practice, δT is small compared to the overall time period, e.g., daily prices within a year of trading. We can regard the discrete equations (particularly with δT = τ ) as the exact equations, with the continuum as an approximation. Thus, we can approximate the ratio {P (T j+1 ) − P (T j )} /δT by P (T j ). More precisely, for a smooth function, P , we have which we can write as Hence we replace the discrete derivative on the left hand side of 2.2 with a very small error and obtain the continuum limit where we have replaced the discrete variables T j with the continuum variable t in F (see below) and P . Note that the discarded term is τ O [δT ]. Another perspective on this is that in equation 2.2 the discrete derivative is O [1] so that one must have F/P −1 = O [τ ] in order to maintain equality in this equation. Hence, in the discrete equation with δT := τ as a small parameter, we write 2.2 as Then each term is of order τ except for the second term τ O [δT ] which is order τ 2 and can be neglected, leading to 3.2.
To obtain the continuum approximation for the remaining equations we first define We treat the m (i) and n (i) as known functions that represent the inflow/outflow of cash and shares, respectively, due to macroscopic factors. Following a similar argument as above the equations describing the cash and stock position of each group can be written as We define the continuous version of the function F as Thus, equations 3.2 -3.4 can be studied numerically subject to initial conditions upon specifying k (i) . By rescaling time we can set τ to unity in each of the three equations. Equilibrium Conditions. In the totally conserved case (no shares or cash added or withdrawn from the system) setting the time derivatives in 3.2 -3.4 to zero yields provided we definek (i) := 1 − k (i) , so that a necessary and sufficient condition for equilibrium (which we denote now by P eq ) is Note that 3.5 and 3.6 imply the identities establishing a compatibility relationship between preferences (determined by k (i) ) and asset positions of the two groups. Thus, given a compatible set (k (1) , k (2) , M (1) , M (2) , N (1) , N (2) ) there is a unique equilibrium price P eq . This differs from classical theory where P eq would be uniquely determined by P a alone. If we regard the k (i) as known, then we have two nonlinear equations, i.e. equations 3.6, for three unknowns, namely M (1) , N (1) , and P .
Remark 3. The equations 3.2 -3.4 reduce to the single group model of [4] when the assets of one group are reduced to zero. To verify this, let M (2) = N (2) = 0 and drop the subscript 1 on the first group. Thus, Group 1 controls the total amount of cash, M , and total number of shares, N , in the system; and any trading must be among traders within Group 1. Hence, the definition of F , i.e.
The concept of a liquidity value, L := M/N , was introduced in this paper. With this substitution we have the equation In the limit of a single group without influx of additional shares or cash, the variables, M , N , and L are all constants in time. Nevertheless, the fraction of funds in the asset, B, defined by varies in time and satisfies Hence, for any definition of k (i) the multi-group model defined above reduces to the single group model.

4.
Analysis of equilibrium and stability with arbitrary parameter values.
As noted above, the transition rate functions, k (i) * , account for the different motivations investors have to buy or sell an asset. Through statistical modeling and regression analysis on market data ( [5]), it has been shown that two such motivations are (i) the asset's recent trend in price and (ii) the asset price's deviation from its fundamental value. These motivations are modeled by equations 1.7. To account for multiple groups, these equations are modified as follows: where the superscript i = 1, 2, ..., n represents the investor group. The parameters, now group-specific, are discussed in Section 1. For example, ζ (i) 2 (t) represents group i's motivation arising from the deviation between the asset's price and group i's assessment of the asset's fundamental value, P (i) a (t), when trading ( [7]). Thus, a quantification of an investor group's motivation to buy/sell is given by Differentiating equations 4.1 and 4.2 yields the ordinary differential equations Adding these to equations 3.2 -3.4 gives the following system where i = 1, 2, ..., n and Thus we have a system of 4n + 1 equations with 5n + 1 parameters. For the remainder of Section 4 and Section 5 we consider the two group system with conserved cash and shares, i.e. M (1) (t)+M (2) where M 0 and N 0 are fixed parameters representing the total amount of cash and the total number of shares in the system. In addition we make the following assumptions: Group 1 is focused solely on the recent trend in price and Group 2 is focused solely on the deviation from its perception of the fundamental value, P (A3) To simplify calculations we use the Taylor series approximation tanh(x) x. Reasonable realistic parameter values will typically result in the argument, ζ (i) i (t), i = 1, 2, lying in the interval (−1, 1), so that differences between the two models, i.e. with or without this approximation, should be negligible. Assumptions (A2) and (A3) imply k (i) (A6) As discussed in Sections 2 and 3, we set τ = 1, i.e. one period. As noted in Section 3, through a rescaling of time we may set τ = 1, i.e. one period, which implies where we have suppressed the dependence of M (1) , N (1) , P, ζ 1 , and ζ We define equilibrium in terms of all time derivatives set to zero. Note that equations 4.3' and 4.4' indicate that existence of equilibria is independent of the parameters c 2 , and q (1) 1 . Setting the time derivatives in equations 3.2', 3.3', and 3.4' to zero yields the following relationship betweenN (1) andM (1) , where the hat notation is used to denote the equilibrium value, Thus, at equilibrium the algebraic system of equations admits a one-dimensional curve of equilibrium points that may be parameterized by one of the dynamic variables, i.e.N (1) ,M (1) , P eq ,ζ 1 , orζ 2 . Caginalp and Balenovich [4] proved that the equilibrium price for the one group model is between the liquidity value, L, and the fundamental value, P a . We prove a similar result (see Theorem 4.1) for the two group model with conserved cash and shares.
As noted above, the parameter P N0 is the liquidity value and P a is Group 2's assessment of the fundamental value of the asset.
(i) Suppose L = P a . Then for any P eq ∈ R + , P eq ∈ (min [P a , L] , max [P a , L]) the system has a unique equilibrium point of the form Consider the limiting case, L = P eq = P a . For anyζ 2 ∈ (−1, 1) \ {0, q 2 } the system has a unique equilibrium point of the form (ii) Suppose P eq = P a . Then P eq = P a = L, and for anyN (1) ∈ R + , 0 ≤N (1) ≤ N 0 , the system has a unique equilibrium point of the form Moreover, under the above assumptions any equilibrium point of this system must lie in the interval [min(P a , L), max(P a , L)].
Remark 4. Classical finance suggests that, because (1) all traders have the same information and (2) any advantage is quickly exploited by completely rational, well informed traders with ample capital, there exists a unique equilibrium price for a given time t. For example, the basic Capital Asset Pricing Model (CAPM) assumes all traders are rational mean-variance optimizers and share a common economic view of the world, i.e. the investor population has homogeneous expectations ( [2]). The 1 For notational convenience we set Pa = P 1 , and for the remainder of the paper.
CAPM theory then provides a method for calculating the unique equilibrium price, which corresponds to what we have labeled P a , the fundamental value of the asset. As noted in Appendix A, if Group 2 has the vast majority of the assets, then the formal limit of the solution(s) of the difference equations, 2.2, 2.7, and 2.8, is the solution of the classical price equation 1.1. This suggests the price, P , is a small fluctuation about this unique equilibrium price. However, if neither group has an infinite amount of capital, then our model, as shown in Theorem 4.1, suggests that there exists a range of equilibrium prices. Thus, our model encompasses the classical theory as a subset, and one may determine from market data and optimization of parameters which assumptions are applicable at any given time.
Remark 5. It is appropriate to consider the scenario L = P eq = P a as a distinguished limit of Case (i). Indeed, in Case (i) for given M 0 , N 0 , and P a the variablê N (1) can vary, forming a one-dimensional curve of equilibrium points. However, in the limit L → P eq , i.e the case L = P eq = P a , for given M 0 , N 0 , and P a , theζ 2 variable is uniquely determined.
Proof. Let L, P a ∈ R + be arbitrary.

GUNDUZ CAGINALP AND MARK DESANTIS
Using equation 4.6 we obtain the form of an arbitrary equilibrium point for any P eq ∈ R, P eq = 0 and P eq = P a . Substitution of the above point into system 3.2' -3.4', 4.3', and 4.4' shows that the point given by equation 4.10 is indeed an equilibrium point of the system. Although the equilibrium point given by equation 4.10 holds for all P eq ∈ R, P eq = 0 and P eq = P a , we must restrict P eq to the interval [min (P a , L) , max (P a , L)] in order to satisfy assumptions (A1)-(A7). Indeed, suppose the contrary, i.e. P eq > max (P a , L) and consider equation 4.9: Notice that we haveN (1) ≥ 0 as needed. However, we also requireN (1) ≤ N 0 by assumption (A5). For this latter condition we require The left inequality is satisfied. Thus we verify the right inequality: Hence, we obtain a contradiction with assumption (A3) as we require −1 <ζ 2 = Similarly, one may show P eq < min (P a , L) is not permissible.
We have proven that we cannot have P eq > max (P a , L) or P eq < min (P a , L). Thus, min (P a , L) ≤ P eq ≤ max (P a , L). An arbitrary equilibrium point is given by equation 4.10 provided min (P a , L) < P eq < max (P a , L). The cases where P eq is constant, P eq = P a and P eq = L, are considered in (ii) and (iii) below. These cases also exhibit a one dimensional curve of equilibrium points; however, as P eq is constant, we parameterize this curve by variablesN (1) andζ 2 , respectively.
We next consider the limiting case P eq = L = P a and show the existence of an equilibrium satisfying for anyζ 2 ∈ (−1, 1) \ {0, q 2 }.
N0 . Cross multiplying gives: which implies (1)ζ 2 = 0 or (2)M (1) = M 0 andN (1) = N 0 , i.e. group 1 has all of the system's cash and shares. Indeed, assumingζ 2 = 0 and dividing both sides of As 0 ≤M Ifζ 2 = 0, then P eq = P a = L. However, ifζ 2 = 0, then we must haveM Note this result may also be obtained by taking the limit as P eq → L of the arbitrary equilibrium point given by 4.10.
(ii) We show that if the equilibrium price equals group 2's estimation of the fundamental value of the asset, i.e. P eq = P a , then it must also equal the liquidity value, i.e. P eq = P a = L = M0 N0 . Indeed, suppose P eq = P a . Then equation 4.8 yieldsζ 2 = 0 which implies P eq = M0 N0 = L by considering equation 3.2'. Thus, if P eq = P a , then we must have P eq = P a = L. By assumption (A4) P a and thus P eq is a constant. Therefore, we parameterize the curve of equilibrium points byN (1) . It is clear from equations 4.3' and 4.4' thatζ 1 =ζ 2 = 0. Using equation 4.6 an arbitrary equilibrium point is of the formx = (N (1) ,N (1) P eq , P eq , 0, 0) (4.13) for anyN (1) ∈ R, 0 ≤N (1) ≤ N 0 and N 0 > 0. As in the above case, a simple calculation shows that the pointx given by equation 4.13 is an equilibrium point of the system 3. Combining assumptions (A1)-(A7) with the results from Theorem 4.1 allows us to rewrite the criteria for equilibrium in Corollary 1. As noted above the parameters c 1 , c 2 , and q 1 do not affect the existence of equilibrium points. Case (1) L < P eq < P a (a) P a > P eq > 0, (b) 0 < q 2 < P a P a − P eq , and In the limit P eq → L we have the subcase P a > P eq = L.

GUNDUZ CAGINALP AND MARK DESANTIS
Case (2) P a < P eq < L (a) 0 < P a < P eq , (b) 0 < q 2 < − P a P a − P eq , and In the limit P eq → L we have the subcase P a < P eq = L.
Proof. Case (1) L < P eq < P a By assumption (A3) and the condition L < P eq < P a , we have 0 <ζ 2 < 1. Aŝ ζ 2 = q 2 Pa−Peq Pa by equation 4.10 we have P a P a − P eq > q 2 > 0, (4.14) which is (b). Also, P a > P eq and N 0 > M0 Peq imply (a) and the left inequality in (c). Thus, it remains to show: From assumption (A5) we have Using the definition ofN (1) from equation 4.10 yields 0 ≤ [P a − q 2 (P a − P eq )] N 0 P eq − [P a + q 2 (P a − P eq )] M 0 2P eq q 2 (P eq − P a ) ≤ N 0 .
Multiplying the inequality by 2P eq q 2 (P eq − P a ) gives 0 ≥ [P a − q 2 (P a − P eq )] N 0 P eq − [P a + q 2 (P a − P eq )] M 0 ≥ 2N 0 P eq q 2 (P eq − P a ) , which is equivalent to Dividing by − [P a − q 2 (P a − P eq )] P eq gives the desired result Note that from equation 4.14 we have 0 < P a − q 2 (P a − P eq ) < P a .
This justifies reversing the inequalities when we divided by −[P a −q 2 (P a − P eq )]P eq above. We next consider the limiting case P a > P eq = L. The criterion P a > P eq = L along with assumption (A3) implies (a). Condition (c) holds as equation 4.12 gives L = P eq = P a (1 −ζ 2 q2 ). Condition (b) is necessary to ensure P eq > 0. Indeed, by equation 4.12 ).
Multiplying the inequality by 2P eq q 2 (P eq − P a ) gives 0 ≤ [P a − q 2 (P a − P eq )] N 0 P eq − [P a + q 2 (P a − P eq )] M 0 ≤ 2N 0 P eq q 2 (P eq − P a ) , which is equivalent to Dividing by − [P a − q 2 (P a − P eq )] P eq gives the desired result Note that from equation 4.15 we have 2P a > P a − q 2 (P a − P eq ) > P a > 0 This justifies reversing the inequalities when we divided by −[P a −q 2 (P a − P eq )]P eq above. We next consider the limiting case P a < P eq = L. It is easily seen that the criterion P a < P eq = L along with (A3) implies (a). Condition (b) holds because in this case L = P eq = P a (1 −ζ 2 q2 ) by equation 4.12. Case (3) P a = P eq = L It is clear from the arbitrary form of an equilibrium point, equation 4.13, that criteria (a)-(c) hold. Now that criteria for equilibria have been established, we consider the stability of an arbitrary equilibrium point for each of the four cases identified in Corollary 1. For stability, we linearize about an arbitrary equilibrium point and consider the eigenvalues of the corresponding Jacobian matrix evaluated at the equilibrium point. As noted above, this system admits a curve of equilibria. As such, this system will always have a zero eigenvalue with corresponding eigenvector tangent to this curve of equilibria. In addition, we will see that −1 is also a root of the characteristic polynomial in all four cases. Thus, the equilibrium point is considered stable if the remaining three eigenvalues have real parts strictly less than zero and unstable if any one of these eigenvalues has a real part greater than zero.
Theorem 4.2 utilizes the Routh-Hurwitz criterion (see Appendix B) to prove the existence of stable and unstable regions of equilibrium points within the appropriate (parameter) spaces for each of the four cases. As 0 and −1 are always roots, we focus on the cubic factor of the characteristic polynomial. Routh-Hurwitz applied to a cubic polynomial (α 3 λ 3 +α 2 λ 2 +α 1 λ 1 +α 0 ) can be stated as: The real parts of all roots are strictly negative if and only if α i > 0 for i = 0, 1, 2, 3 and α 2 α 1 − α 3 α 0 > 0 ( [10]).
We define the following constants which will be used in the statement and proof of Theorem 4.2: a (−1 + q2) − P 2 eq q 2 + P a (P eq + 2P eq q 2 ) C := M 0 [P a + q 2 (P a − P eq )] [−P eq q 2 + P a (−2q 1 + q 2 )] − N 0 P eq [P a − q 2 (P a − P eq )] [P eq q 2 − P a (2q 1 + q 2 )] Recall that q 1 > 0 by assumption (A7). This agrees with empirical studies on both experimental ( [6]) and world market ( [5]) data which show that if the trend is positive (alternatively negative) then the price continues to rise (alternatively fall) due to momentum traders. (1) L < P eq < P a : (q 1 < 1 2 ) and (c 1 < 1 + c 2 ) ; (2) P a < P eq < L: (q 1 < 1 2 ) and (c 1 < 1 + c 2 ) ; (3) P a = P eq = L: (ii) There exists an unstable region of equilibrium points. In each of the three cases one has the following sufficient criteria for instability: (1) L < P eq < P a : −M 0 + N 0 P eq M 0 + N 0 P eq P a P a − P eq < q 2 < P a P a − P eq and (2) P a < P eq < L: −P a P a − P eq < q 2 < −P a P a − P eq and (3) P a = P eq = L: In addition, for L = P eq = P a , a distinguished limit of cases (1) and (2), there exists a stable region of equilibrium points for and an unstable region of equilibrium points for Remark 6. The conditions for stability are consistent with the intuition of the model, as a small trend coefficient, q 1 , and a large time scale for trend, 1/c 1 , favor stability (see case (1), (2) of (i)). The stable condition for the distinguished limit case (L = P eq = P a ) shows that one can have stability for large values of q 1 provided the trend time scale is large. Furthermore, in case (3) stability can also be maintained for larger values of q 1 if the number of shares of the trend-based group is small compared with the total number of shares in the system.
Proof. Letx = (N (1) ,M (1) , P eq ,ζ 1 ,ζ 2 ) be an arbitrary equilibrium point. Case (1) L < P eq < P a (i) We first prove existence of a stable region of equilibrium points. Linearizing aboutx , P eqN (1) , P eq , 0, q 2 P (2) a − P eq P (2) a ) yields the following characteristic polynomial λ (λ + 1) α 3 λ 3 + α 2 λ 2 + α 1 λ 1 + α 0 (4.16) where α 3 = 1 The zero eigenvalue is due to the curve of equilibria. As such, it does not impact the stability. Thus,x is stable if the roots of the cubic factor of the characteristic polynomial have strictly negative real parts. By the Routh-Hurwitz criterion these roots lie in the left half of the complex plane if and only if α i > 0 for i = 0, 1, 2, 3 and α 2 α 1 − α 3 α 0 > 0. Note that Any equilibrium point must satisfy the criteria given by Corollary 1, which for this case is (a) P a > P eq > 0, (b) 0 < q 2 < P a P a − P eq , and (c) M 0 P eq < N 0 ≤ M 0 [P a + q 2 (P a − P eq )] P eq [P a − q 2 (P a − P eq )] .

GUNDUZ CAGINALP AND MARK DESANTIS
A is positive. This implies that both the denominator and the first term in the numerator are positive. Thus C must be negative.
which holds for Recall equilibrium conditions (b) and (c) for this case Note equation 4.18 is the strict version of the right inequality in (c). For equation 4.18 and subsequently 4.17 to hold we require q 2 = Pa Pa−Peq . Thus, using condition (b), for C < 0 we restrict q 2 to the interval −M 0 + N 0 P eq M 0 + N 0 P eq P a P a − P eq < q 2 < P a P a − P eq and require Choosing gives α 2 < 0. Since α 3 = 1 > 0 there is at least one sign change in the first column of the Routh array. By the Routh-Hurwitz criterion there is at least one eigenvalue with positive real part. Thus, the equilibrium is unstable. Case (2) P a < P eq < L (i) We prove existence of a stable region of equilibrium points. The critical point, x, and characteristic polynomial both have the same form as in Case (1). Similar to the proof of Case (1), we utilize the equilibrium conditions given by Corollary 1 to show that A > 0, B < 0, and C < 0. These criteria are: (a) 0 < P a < P eq , (b) 0 < q 2 < − P a P a − P eq , and (c) Note that Peq[Pa−q2(Pa−Peq)] > 0. Indeed, [P a − q 2 (P a − P eq )] > 0 by (a). In addition, [P a + q 2 (P a − P eq )] > 0 by (b).

GUNDUZ CAGINALP AND MARK DESANTIS
where Notice that each term on the right side of this inequality is positive for this case. Further, the first term on the right side of the inequality is greater than the term on the left side of the inequality provided c 1 < 1 + c 2 . Thus, if c 1 < 1 + c 2 , then α 2 α 1 − α 3 α 0 > 0. Therefore, if q 1 < 1 2 and c 1 < 1 + c 2 , then the equilibrium is stable. (ii) Using the Routh-Hurwitz criterion we show that an unstable region exists by showing α 2 < 0. Indeed, A is positive. This implies that both the denominator and the first term in the numerator are negative. Thus, we need C to be positive. Consider which holds for M 0 [P a + q 2 (P a − P eq )] P eq [P a − q 2 (P a − P eq )] < N 0 . (4.20) Recall equilibrium conditions (b) and (c) for this case (b) 0 < q 2 < − P a P a − P eq and (c) M 0 [P a + q 2 (P a − P eq )] P eq [P a − q 2 (P a − P eq )]

−Pa
Pa−Peq . Thus, using condition (b), for C > 0 we restrict q 2 to the interval M 0 − N 0 P eq M 0 + N 0 P eq −P a P a − P eq < q 2 < −P a P a − P eq and require Choosing gives α 2 < 0. Since α 3 = 1 > 0 there is at least one sign change in the first column of the Routh array. By the Routh-Hurwitz criterion there is at least one eigenvalue with positive real part. Thus, the equilibrium is unstable.
Case (3) L = P eq = P a (i) Linearizing aboutx = (N (1) ,N (1) P eq , P eq , 0, 0) yields the following characteristic polynomial The zero eigenvalue is due to the curve of equilibria. As such, it does not impact the stability. Thus,x is stable if the roots of the cubic factor of the characteristic polynomial have strictly negative real parts. By the Routh-Hurwitz criterion these roots lie in the left half of the complex plane if and only if α i > 0 for i = 0, 1, 2, 3 and α 2 α 1 − α 3 α 0 > 0. Note that 0 ≤N (1) N0 ≤ 1 by assumption (A5). Consider α 2 > 0, i.e.
By the Routh-Hurwitz criterion there is at least one eigenvalue with positive real part. Thus, the equilibrium is unstable.
Finally, we consider the limiting case of cases (1) and (2): L = P eq = P a . Linearizing aboutx yields the following characteristic polynomial The zero eigenvalue is due to the curve of equilibria. As such, it does not impact the stability. Thus,x is stable if the roots of the quadratic factor of the characteristic polynomial have strictly negative real parts. Note that stability only depends on the q 1 and c 1 parameters. The quadratic formula yields Note that As such, there are two cases to consider.
If these criteria are met then we have λ +,− ∈ R and λ +,− < 0. These criteria are met when . Indeed, consider So, for the above criteria to be met we need q 1 < min If these criteria are met then we have Im (λ +,− ) ≥ 0. The second criteria ensures Re (λ +,− ) < 0. These criteria are met when And the second criterion gives Thus, Re (λ +,− ) < 0 provided Further, this also indicates that if q 1 > 1+c1 2c1 , then the equilibrium is unstable.

5.
Equilibrium and Stability for Specific Parameter Values. The system 3.2' -3.4', 4.3', and 4.4' has eight parameters (τ, c 1 , c 2 , q 1 , q 2 , M 0 , N 0 , P a ) which may assume any real and positive value. Now that the existence of stable and unstable regions has been established, we compute specific regions of stability and instability. To this end we set some of these parameters to specific values in order to facilitate the analysis of equilibrium and stability. We set c 1 = 1 10 and c 2 = 1 indicating that Group 1 focuses on the price trend over the past two weeks, while Group 2 considers the current day's deviation from its assessment of the asset's fundamental value. The values for M 0 and N 0 are set at 30, 000 and 3, 000 so that L = 10. P a is set to 12, 8, or 10 depending upon the case. These are typical values for L and P a in that the fundamental value of the asset is set to 120% or 80% of the liquidity value. Note that for L, P a , and P it is their relative magnitudes rather than absolute values that are essential.
We consider the effect of the q 2 parameter on the existence of equilibrium points, and the effects of the q 1 and q 2 parameters on the stability of these equilibrium points. It is clear from equations 4.3' and 4.4' that the existence of equilibria is independent of the values for the q 1 , c 1 , and c 2 parameters. Given Theorem 4.1, we determine the conditions for the existence and stability of equilibria for the following four cases: (1) L < P eq < P a , (2) P a < P eq < L, (3) P eq = P a = L, and (4) L = P eq = P a , a distinguished limit of cases (1) and (2). Note that assumptions (A1)-(A7) are still applicable to all results in this section. Case 1. 10 = L < P eq < P a = 12 Equilibrium. Let P eq ∈ (10, 12) be arbitrary. Then equation 4.10 gives the form of an equilibrium point dependent upon the value of q 2 . All P eq between L and P a yield an equilibrium point of the system 3.2' -3.4', 4.3', and 4.4'; however, only some (P eq , q 2 ) pairs yield an equilibrium satisfying assumptions (A1)-(A7). The region of (P eq , q 2 ) pairs that yield permissible equilibrium points is derived from Corollary 1 and given by L < P eq < P a and (P eq − L) P a (L + P eq ) (P a − P eq ) ≤ q 2 < P a P a − P eq .
The shape of this region is analogous to the Caginalp and Balenovich [4] finding for the single group model. Specifically, a large value for q 2 implies P eq is close to P a , while a small q 2 means that P eq must be close to L. Indeed, suppose Group 2 is strongly influenced by the price's deviation from P a . If the price is above (below) this value, then Group 2 will sell (buy) the asset; thereby, lowering (raising) the price. Thus, it is reasonable to expect the equilibrium price, P eq , to be close to P a .
Stability. Linearization about an arbitrary equilibrium point yields a Jacobian matrix, which is included in Appendix C. The characteristic polynomial can be factored into the form: Thus, as noted above, two of the jacobian's five eigenvalues are 0 and −1. We are interested in the signs of the real parts of the remaining three eigenvalues. These eigenvalues correspond to the roots of the cubic factor of the characteristic polynomial which is given by equation 4.16.
We utilize the Routh-Hurwitz criterion to determine the regions of stability in the (P eq , q 2 )-plane. To determine the effect q 1 has on stability, we set 2 q 1 = 0.447, 2 Using linear regressions on experimental data, Caginalp and Ilieva [6] found a q 1 of 0.447 and a q 10, 20. These regions, which are nested subsets (the larger subset corresponding to the smaller q 1 value), are shown in Figure 2.  Figure 2. Region of (P eq , q 2 ) pairs for L = 10 and P a = 12 corresponding to q 1 = 0.447 (Blue, Red, and Yellow), 10 (Red and Yellow), and 20 (Yellow) that yield stability. Areas outside of the designated colored regions but inside the dashed lines correspond to non-stable equilibria.
The union of the Blue, Red, and Yellow regions, which corresponds to q 1 = 0.447, is identical to the equilibrium region plotted in Figure 1. Thus, we see that all equilibrium points are stable for q 1 = 0.447. As previously noted, the region of equilibria does not change as the value of q 1 changes. However, the value of q 1 does affect the regions of stable and non-stable equilibria. For example, for q 1 = 20 all equilibrium points inside the Yellow region in Figure 2 are stable, while the equilibrium points in the Blue and Red regions are not stable.
The q 2 parameter controls the existence of equilibria, while the q 1 parameter is a key factor in the stability of the equilibria (at least for this parameter region). For example, if q 2 is large, then the equilibrium price, P eq , must be close to P a for stability. The magnitude of the q 1 parameter determines how close P eq must be to P a for this equilibrium to be stable. Alternatively, if Group 1 is very strongly focused on the price trend while Group 2 is not strongly affected by the asset price's deviation from the fundamental value (i.e., large q 1 and small q 2 ), then P eq must be very close to L for stability. Indeed, from Figure 2 it is clear in this scenario that the equilibrium is most likely not stable. Case 2. 8 = P a < P eq < L = 10 Equilibrium. This case contains the same curve of equilibrium points (prior to substituting any parameter values) as in Case 1 where P eq = P a may assume any value between P a and L. Requiring these equilibrium points to satisfy assumptions (A1)-(A7) restricts the set of (P eq , q 2 )-pairs that yield equilibrium to the following region P a < P eq < L and (P eq − L) P a (L + P eq ) (P a − P eq ) ≤ q 2 < −P a P a − P eq which is derived from Corollary 1. For the specific parameter values given above the region is described by 8 < P eq < 10 and (P eq − 10) 8 (10 + P eq ) (8 − P eq ) ≤ q 2 < −8 8 − P eq and shown in Figure 3.  Figure 3. Region of (P eq , q 2 )-pairs that yield permissible, i.e. satisfy assumptions (A1)-(A7), equilibrium points (8 = P a < L = 10).
Large q 2 implies P eq is close to P a , while small values for q 2 mean P eq must be close to L. Again, this result is analogous to that of [4] for the single group model.
Stability. The Jacobian matrix and characteristic polynomial for this case are identical (prior to setting parameter values) to those of Case 1. Refer to Appendix C for the Jacobian matrix and equation 4.16 for the characteristic polynomial.
Using the Routh-Hurwitz criterion, we determine regions in the (P eq , q 2 )-plane that yield stable equilibrium points. To understand the effect of the q 1 parameter on stability, we produce plots for q 1 = 0.447, 10, and 20. These plots are displayed in Figure 4. As q 1 increases, each corresponding region is nested within the prior region.  Figure 4. Region of (P eq , q 2 ) pairs for P a = 8 and L = 10 corresponding to q 1 = 0.447 (Blue, Red, and Yellow), 10 (Red and Yellow), and 20 (Yellow) that yield stability. Areas outside of the designated colored regions but inside the dashed lines correspond to non-stable equilibria.
The union of the Blue, Red, and Yellow regions corresponds to q 1 = 0.447 and is identical to the region of permissible equilibrium points in Figure 3. Thus, we see that all equilibrium points are stable for q 1 = 0.447. As in Case 1, the q 2 parameter controls the existence of equilibria, while q 1 is a key factor in determining stability. As the value of the q 1 parameter increases, the region of equilibria remains unchanged, while the region of (P eq , q 2 )-pairs that yield stable equilibria gets smaller. Thus, if q 1 = 20, then the Yellow region corresponds to stable equilibria, while the Blue and Red regions corresponds to non-stable equilibria. Note that if Group 2 is strongly focused on the asset price's deviation from the fundamental value and Group 1 is not strongly motivated by the recent price trend (i.e., q 1 is small and q 2 is large), then P eq must remain close to P a for stability.
It has been noted previously through experimental ( [8]), theoretical ( [4]), and empirical ( [5]) studies that an increase in the amount of cash available to purchase an asset boosts the asset's price. We increase the value of L (by increasing M 0 while holding N 0 fixed) to study the effect of an increase of cash on the system. Figure 5 contains the plots of the stability regions corresponding to L = 20 and 30, respectively.  Figure 5. Region of (P eq , q 2 ) pairs for P a = 8 corresponding to q 1 = 0.447 (Blue, Red, and Yellow), 10 (Red and Yellow), and 20 (Yellow) that yield stability. The left plot corresponds to L = 20 and the right L = 30. Compare these with Figure 4.
As shown in Figure 5, the region of permissible equilibrium points (dashed lines) becomes smaller as L increases, though the overall shape is similar. Note that a similar phenomenon occurs as L is reduced with P a fixed. Case 3. P a = P eq = L = 10 Equilibrium. For anyN (1)  Stability. The Jacobian matrix evaluated at an arbitrary equilibrium point is given in Appendix C while the characteristic polynomial is given by equation 4.21.
From the diagram on the left in Figure 6 we see that the region of stable equilibrium points increases in size as the value of the q 2 parameter increases. Note that from this diagram we see that for large q 1 ifN (1) ≈ N 0 , then the equilibrium point is not stable. The diagram on the right in Figure 6 shows that for small q 1 all equilibrium points are stable and as q 1 increases the region of stable equilibrium points gets smaller. Case 4. P eq = L = 10 = P a Equilibrium. For anyζ 2 ∈ (−1, 1) \ {0, q 2 } an equilibrium point is given by equation 4.12. Note that P a may be less than or greater than P eq = L. As such, we consider two scenarios: 0 < P a < L and P a > L.
Note that Figure 7 shows that equilibrium may exist for values of P a that are drastically different from P eq .
q2−ζ2 (found by solving the P eq critical point equation for P a ) the Jacobian matrix evaluated at an arbitrary equilibrium point is given by: with characteristic equation: This system has a zero eigenvalue and a negative one eigenvalue with algebraic multiplicity two. Thus, the sign of the real parts of the remaining two eigenvalues will determine the stability of the equilibrium point. Note that the stability is independent of q 2 and only depends upon the value of parameter q 1 .
The remaining eigenvalues are of the form: The real parts of these eigenvalues are negative for q 1 < 11 2 , equal to zero for q 1 = 11 2 , and positive for q 1 > 11 2 . So, the equilibrium point is stable for q 1 ∈ −∞, 11 2 and unstable for q 1 ∈ 11 2 , ∞ . It is interesting to note that the point of transition from stability to instability, q 1 = 11 2 , is also significant in Case 3, P eq = P a = L, in that this is the point where parameter q 1 begins to influence the stability. 6. Conclusion. The stability of financial markets is of crucial practical importance; however, studying stability is not feasible within classical models that are idealizations based on near equilibrium conditions. A key assumption in classical finance is the existence of an infinite amount of capital controlled by experts, free of bias, and unanimous in their assessment of true value of the asset. Thus any study of price dynamics within this setting leads to prices that evolve quickly and smoothly to the equilibrium price with some random fluctuations, or noise, along the way. The equations developed by Caginalp and collaborators since 1990 model asset price dynamics by utilizing basic microeconomics of supply and demand but also allowing these functions to depend on motivations beyond valuation. Also inherent in this approach is the finiteness of assets and the possibility that different investor groups (controlling parts of the total asset pool) may also differ in their assessments of the value of an asset. We sketch an argument in Appendix A that the formal limit of solutions of this model corresponds to solutions of the classical price equation 1.1. Thus, this model encompasses the classical model upon removal of the generalized features.
After a precise derivation of the link between the discrete (difference equations) and the continuum (differential equations) for the multi-group models, we study the stability properties with the objective of determining the parameter regions for equilibrium, stability, and instability. An important difference between the neoclassical dynamics and our asset flow models is that equilibrium is not uniquely determined by the valuation in the latter. Instead it is determined by a set of algebraic equations incorporating the characteristics of each group. In other words, given a particular valuation (held by Group 2 that focuses on value) there are a spectrum of equilibrium prices. One has a unique price given the other parameters such as the cash and share position of each group, etc.
Specifically, for the two group system with conserved cash and shares we prove (Section 4) the existence of a 1-dimensional curve of equilibrium points under various parameter regimes. In addition, this equilibrium price, P eq , must lie between the liquidity value, L := (total amount of cash) / (total number of shares) , and Group 2's assessment of the fundamental value, P a . In Theorem 4.2 the existence of local stable and unstable regions for an arbitrary equilbrium point is established. We note that existence of equilibrium is independent of the q 1 parameter, i.e. the magnitude of trend based investing does not impact the existence of equilibrium.
Numerical computations (Section 5) with specific parameter values establish the exact stable regions for an equilibrium point. In general, we showed that if Group 2 is strongly focused on valuation, i.e. q 2 is large, then the equilibrium price is close to P a . The strength of Group 1's motivation due to the recent price trend determines how close P eq must be to either P a or L to ensure stability. For instance, if q 1 is large, i.e. Group 1 is very strongly focused on the recent trend in price, and q 2 is small, then P eq must be close to L for stability (see Figure 2).
The results show that in Cases 1, 2, and 4 the parameter q 2 controls the existence of equilibria, while in Case 3 existence of equilibria is independent of the q 2 parameter. In all four cases, the q 1 parameter plays a key role in determining the stability. In Cases 1, 2, and 3 as q 1 increases, the region of (P eq , q 2 ) pairs that yield stable equilibria gets smaller (see Figures 2,4,and 6). In Case 4 the equilibrium point is stable for q 1 < 11/2 and unstable for q 1 > 11/2. In Case 3 as the value for q 2 increases, the stable region also grows larger. Figure 6 also shows that if Group 1 is strongly focused on the trend, i.e. q 1 large, and owns a vast majority of the shares, then the equilibrium is most likely not stable. Our results regarding the influence of the q 1 parameter are analogous to those of Duran [11] where he finds that the market price is more sensitive to small changes in the q 1 parameter than the q 2 parameter.
Thus the asset flow models are capable of addressing stability issues. The remaining challenge is to establish a stronger connection with world markets by estimating the assets controlled by different groups characterized by their trading strategy. This requires not only estimation of the q 1 and q 2 parameters but also an estimate of the magnitudes of the assets of each group. With these estimates one can develop criteria that lead to instability in markets.
considering. Since all investors have the same publicly available information, the expectation for the return of any asset is given by P a (t) = P a (0) e µt . (A.1) We consider a small time frame (compared to one year), and denote by P a the fundamental value of the stock during that time. With the assumption of non-zero trading costs we assume that k (2) is given by if P < P a − δ 1/2 if P a − δ < P < P a + δ 0 if P > P a + δ for some small, positive ε.
We make the classical assumption that the knowledgeable investors, namely Group 2, represent the vast majority of wealth, expressed by the following. Assumption A. At the initial time, the endowments to the two groups satisfy the following: The assumption that Group 1 consists of "noise traders" implies that the net flow of orders by this group does not depend on P a or P . Assumption C. The transition rate of Group 1, k (1) , is a constant independent of P a and P and 0 < k (1) < 1.
Under Assumptions A and B, for P < P a − δ, one has that k (2) = 1 from A.2, so we can write 2.1b as (1) (A.6) which is large and positive. Hence, by 2.2 the price quickly moves into the range P > P a − δ. Similarly, when the stock is overvalued, i.e., P > P a + δ, one has k (2) = 0, yielding so that the price falls back into the range P < P a + δ where k (2) no longer vanishes. Finally, we consider the range P a − δ < P < P a + δ where the knowledgeable investors are essentially inactive since it is within their trading costs. In this region of price, the "noise traders" dominate due to the absence of Group 2 investors. Using Assumptions A and B for prices P (T j ) near P a , we have, from 2.1b and A.2, 1 + 2k (1) M (1) /M (2) 1 + 2 1 − k (1) N (1) /N (2) yielding, from 2.2, the identity, Indeed, as P a − δ < P < P a + δ, we approximate P a /P (T j )=1. In addition, we use a geometric series argument and drop the 4k (1) M (1) /M (2) 1 − k (1) N (1) /N (2) as it is much smaller in magnitude than the other terms due to Assumption A.
The system A.8, 2.7, and 2.8 is not yet close to 1.1, since M (1) and N (1) are not constant.
We now examine M (1) (t) and assume that there is a normal random variable, X(T ), governing the influx/outflow of investors and their cash. The parameter M 0 represents the total amount of cash in the system, i.e. M (1) + M (2) = M 0 . Assumption E. The cash supply M (1) (t) in 2.7 includes an inflow/outflow term m (1) (T j ) :=σX (T j ) M 0 that dominates the right hand side of A.8.
Thus, we obtain where σ incorporates the additional constants multiplyingσ. In other words, the existing "noise traders" effectively cancel out one another as they employ a variety of strategies. More significant, however, is the stochastic influx and exit of new investors that becomes the main stochastic contribution to price change. For example, if there is a net inflow of investor cash (as in the late 1990's) due to macroscopic or demographic changes, then the larger supply of cash leads to higher prices. This holds even assuming that investors, on average, are just as likely to sell as they are to buy. Formally taking the continuum limit of A.9 yields dP = σP dX. (A.10) Since, P (t) and P a (t) are essentially identical in classical finance, we assume P (t) satisfies the same differential equation, namely dP = µP dt. The main difference is that we are assuming a deterministic motivation once the price has moved beyond δ away from the fundamental value.