Faculty Economics and Business, Amsterdam School of Economics
Influence of Beliefs on Price
Stability in an Asset Pricing Model
Jesse Treur - 10736735 Supervisor: dr. D´avid Kop´anyi
University of Amsterdam
Econometrics and Operational Research June - 2018
Statement of Originality
This document is written by Student Jesse Treur who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the su-pervision of completion of the work, not for the contents.
Contents
1 Introduction 1
2 Literature Review 3
2.1 Belief Types . . . 3
3 Theoretical Background 4
3.1 The Asset Pricing Model . . . 5 3.2 Dynamical Systems . . . 7
4 Theoretical Analysis 8
5 Simulations 12
6 Comparison to Brock and Hommes (1998) 18
7 Conclusion 20
1
Introduction
During the last financial crisis, it became clear that the current eco-nomic models are not sufficient for decision makers in the government or at banks to base their decisions on (Farmer et al., 2012). Farmer et al. (2012) explain that there is a clear need to describe financial markets in more detail. In the mainstream models, economists generally assume that markets and agents behave rationally. But Simon (1972) describes the idea of bounded rationality, where the information processing ca-pacities of agents are limited. Some of the arguments he gives for this bounded rationality are risk and insecurity avoidance, incomplete infor-mation about alternatives, and that decision making objectives can be different from what has usually been assumed. He explains that it has generally been assumed that agents want to maximize their profit, but it is also possible that agents only want to make a sufficiently high, but not necessarily optimal profit. It is also possible that the investors want to help society as well and do not just focus on making profit. Furthermore, Frankel and Froot (1990) describe that the bubbles in the exchange rates of the dollar cannot be explained by macroeconomic fundamentals. This leads them to believe that there are bubbles that are not characterized by rational expectations. As a result they state that there is evidence that investors have heterogeneous expectations.
A good way to investigate the consequences of bounded rationality is to look at the prices of financial assets. Brock and Hommes (1998) introduce a model where the price of a risky asset is based on hetero-geneous expectations about today’s or tomorrow’s price. They say that these expectations are influenced by the different strategies of investors and the switching costs associated with changing strategies. In their pa-per they describe how the expectations influence the demand for the risky asset, which in turn influences the price of the asset. Brock and Hommes (1998) investigate the interaction between the non-rational expectation rules and fundamentalists. Fundamentalists base their expectations of the price solely on the net present value (NPV) of the future dividends and always assume that the asset’s price will return to this NPV.
In-vestors with non-rational expectation rules bases their expectation of the price on the last price and the trend in the previous periods. It is important to study the relationship between the different rules because it can help to explain or predict the realized price by looking at the composition of the population of traders.
This paper investigates the robustness of the results that Brock and Hommes (1998) derive from their model. The model is slightly adjusted in two ways. Firstly, it no longer looks at the number of risky assets purchased, but at the amount of money invested in risky assets. Secondly, the new model will use both today’s and tomorrow’s price of the asset instead of either one of these prices. For this new model the stability of the system is investigated when it is assumed that all investors have the same beliefs. The main focus of the paper is to check the robustness of the results found by Brock and Hommes (1998) by comparing these to results of the new model.
This paper confirms the robustness of the results found by Brock and Hommes (1998), since I find that in the new model period-doubling-and Hopf bifurcations occur. These are also the bifurcations that occur when the model of Brock and Hommes (1998) is investigated when it is assumed that all investors have the same expectations for the price in the next period. Nevertheless, my thesis also shows that more research into the model by Brock and Hommes (1998) is needed because I clearly show that there are more factors that influence the stability of the system. For example the expectations of both the price in this period and the price in the next period influence the stability.
The plan of the paper is as follows. In Chapter 2 findings and theo-ries of the paper by Brock and Hommes (1998) are discussed. The model that is used in this thesis is introduced in Chapter 3. Chapter 4 shows the results of the theoretical analysis, and these results are discussed. In Chapter 5 the results of the simulations are shown and discussed. A theoretical analysis of the model by Brock and Hommes (1998), when it is assumed that all investors have the same expectations for the price, is done in Chapter 6. Finally, the last chapter concludes and discusses future research that can be done on the topic.
2
Literature Review
As mentioned before, Brock and Hommes (1998) investigate a more com-plex Asset Pricing Model in which two or more belief types can be incor-porated. In their model investors can switch, the switching behaviour is determined by the intensity of choice. They base most of their research into the stability of their model on this intensity of choice. Nevertheless, my model can be compared to theirs because it is also about the stability of the price and the influence that the beliefs have on this.
Brock and Hommes (1998) find that when there are fundamentalists and or trend followers or contrarians in a system, the system is in a steady state until the intensity of choice reaches a certain level. After that, the presence of either the trend followers or contrarians can cause chaotic asset price fluctuations.
They state that a pitchfork bifurcation is caused by the presence of trend followers. Because of this two additional stable nonfundamental steady states emerge. Where one is above the fundamental price and one below. Contrarians in the system initiate a period doubling bifurcation which causes a 2-cycle. This 2-cycle has one point above the fundamen-tal price and one below. On top of that, they also mention that both contrarians and trend followers are sensitive to initial states.
2.1
Belief Types
In their model Brock and Hommes (1998) state that the price of the risky asset is determined by the sum of the expectations of all the investors. They define three different belief types, fundamentalists, trend followers, and contrarians. Brock and Hommes (1998) define the expectation for the price in the next period, for investors of believe type h, as: pet+1 = pt−1+ gh(pt−1− pt−2).
If gh > 0 the investor is a trend follower, who believes that the price
will follow the trend. So, when the price is above the fundamental price, the investor expects the price to continue to move even further away from the fundamental price. Farmer and Joshi (2002) state that trend
followers believe that price changes have inertia. They explain that trend followers go long when prices have been going up, and short if prices have been going down.
On the other hand, when gh < 0 the agent is a contrarian. These
contrarians always expect the trend to be reversed. Farmer and Joshi (2002) state that the contrarians believe the opposite of trend followers. Therefore, contrarians go long when prices have been going down and go short when prices have been going up.
Since fundamentalists always believe that the price will be equal to the fundamental price, fh,t = 0 for fundamentalists. Because I do not
look at deviations from the fundamental price in this thesis, when gh = 0
we call this naive expectations. My thesis looks at the different belief types separately, so there is no possibility to switch strategies and all agents have the same beliefs. This is different from the paper by Brock and Hommes (1998), where they actually focus on the switching costs and how this causes stability or bifurcations.
Brock and Hommes (1998) also discuss a bias in their model. They say that investors can be positively or negatively biased. Investors with a positive bias always expect the price to be a little higher than investors without a bias. Contrarily, agents with a negative bias always expect the price to be a little lower. In this thesis I assume that the investors do not have a bias.
3
Theoretical Background
As discussed in the introduction, this paper investigates the robustness of the results that Brock and Hommes (1998) derive from their model. To do this, a simplified and adjusted version of their model is used. This section starts by introducing the model that is investigated in my thesis. After that, the theory about dynamical systems needed to understand this paper is discussed.
3.1
The Asset Pricing Model
The model for asset pricing that is used in this paper assumes that an agent has an amount of money Wt that he can invest in a risk free and
a risky asset. The risk free asset has a constant interest r %. The risky asset pays a dividend of yt (with an expected value of y) in period t and
can be sold in the next period for price pt+1.
The investor decides which amount of money he wants to invest in the risky asset. When he invests an amount of Wa, then his wealth in
the next period, t+1, will be given by: Wt+1= (1 + r)(Wt− Wa) +
Wa pt
(pt+1+ yt+1). (1)
The investor has mean-variance preferences, which means that he wants Et(Wt+1) − a2Vt(Wt+1) to be maximal. When he makes his decision, the
values of pt, pt+1, and yt+1 are unknown.
When the formula for Wt+1is plugged in into the objective function
this gives (1 + r)(Wt− Wa) + Wa pe t (pet+1+ y) −a 2(W a)2V t pt+1+ yt+1 pt . (2)
In the formula above pet denotes the expectation of the price in period t, and a is a measure of ’risk aversion’. The FOC of the optimalization problem is −(1 + r) + p e t+1+ y pe t − aWaσ2 = 0, (3)
where I assume that Vt(pt+1p+yt t+1) = σ2. Solving the FOC yields
Wa= p e t+1+ y − (1 + r)pet aσ2pe t . (4)
There are I identical agents in the model, who all have the same mean-variance preferences. The market clearing price pt will be determined on
a value that causes the sum of the demand by the investors to equal the total supply: PI
i=1 Wa
i
pt = z
This leads to the formula for the price in period t pt = 1 zs I X i=1 Wia = I X i=1 pe i,t+1+ y − (1 + r)pei,t aσ2zspe i,t . (5)
The expectations of the investors for price in period t and period t + 1 are given by:
pet = pt−1+ g(pt−1− pt−2) (6)
pet+1 = pt−1+ gα(pt−1− pt−2). (7)
In this thesis I assume that all investors have the same beliefs. The investors base their expectations on the price of the risky asset in the last known period and on the price trend. How they expect the difference in price between the last two periods to affect the price of the asset is given by a factor g1, and if they believe the same or something else for the next
period is given by α.
The effect of α is best explained by a few examples. When α = 2 investors have the same expectations with respect to the trend for the next period as for this period. On the other hand, when α = 1 investors think that the price will be the same in the next period as they expect it to be this period. If investors have an α < 0 they have opposite expectations with respect to the trend compared to their expectations in this period.
Plugging the expectation rules for pe
t and pet+1into the price formula
yields pt = I X i=1 pt−1+ αg(pt−1− pt−2) + y − (1 + r)(pt−1+ g(pt−1− pt−2)) aσ2zs(p t−1+ g(pt−1− pt−2)) . (8) Since all investors have the same beliefs this simplifies to
pt= θ
(α − 1 − r)gpt−1− rpt−1+ (1 + r − α)gpt−2+ y
pt−1+ g(pt−1− pt−2)
, (9)
where θ = aσI2zs. Because of which, the parameter θ can be used to look
at what the influence is of one of the factors I,a, σ2, and zs going up or
down.
The main focus of this paper is on the interaction between the vari-ables α and g. I investigate for what combinations of these varivari-ables the system reaches a steady state and for what values bifurcations oc-cur. To do this it is necessary to have a basic knowledge of dynamical systems. Therefore, the following section discusses the necessary theory about dynamical systems to understand this paper.
3.2
Dynamical Systems
In order to better understand this paper, a basic understanding of namical systems is needed. This section briefly explains the theory of dy-namical systems, steady states and some examples of bifurcations that occur in the paper. For simplicity reasons and because it is what is mostly used in the paper, this section investigates two dimensional dy-namical systems. For more details, theoretical background, and infor-mation about dynamical systems see Kuznetsov (2013), and Hommes (2013).
In the dynamical system that is discussed here, there are two state variables zt+1 and xt+1. The law of motion Fλ is a R2 → R2 function
that has zt+1 and xt+1 as outputs when zt and xt are input arguments
and λ is a parameter. So this gives
(zt+1, xt+1) = Fλ(zt, xt) (10)
A system is in a steady state when the state variables do not change over time. In the dynamical system this can be expressed as
(z∗, x∗) = Fλ(z∗, x∗) (11)
This paper aims to find out when the steady state of the dynamical system is locally stable and how the different parameters affect sta-bility. The stability of the system can be determined by looking at
the eigenvalues of the Jacobian, evaluated at the steady state. When Fλ = (f1(zt, xt), f2(zt, xt)) the Jacobian is denoted by:
J = ∂f1 ∂zt ∂f1 ∂xt ∂f2 ∂zt ∂f2 ∂xt ! (12)
If the absolute values of the eigenvalues are smaller than one (||λi|| < 1),
the system is stable. However, as stated before, this paper looks at when the system moves from a stable steady state to an unstable one. This happens as the eigenvalues cross the unit circle.
When λi = 1 a saddle-node bifurcation occurs, one steady state
loses stability and two new steady states are created. Of these steady states one is stable and one is unstable. A period-doubling bifurcation occurs when λi = −1 and it causes a steady state to turn into a
two-cycle. Finally, when ||λi|| = 1 and the eigenvalues are complex, a Hopf
bifurcation occurs. A Hopf bifurcation causes the steady state to lose stability and an invariant circle is created, with quasi-periodic dynamics.
4
Theoretical Analysis
This chapter describes and discusses the results of the theoretical analy-sis. I investigate what kind of bifurcations occur, when these bifurcations occur, and how the different parameters affect stability.
To be able to investigate the model it first needs to be defined as a dynamical system. To do this, I denote pt = f (pt−1, pt−2), zt= pt, and
xt= zt−1. The system can now be written as
" zt xt # = " f (zt−1, xt−1) zt−1 # (13)
To determine when bifurcations take place I need to evaluate the Jacobian in the steady state and look at the eigenvalues. A short summary of the theory about dynamical systems is given in section 3.2 .The Jacobian of
this system is given by J = "∂f (z t−1,xt−1) ∂zt−1 ∂f (zt−1,xt−1) ∂xt−1 1 0 # = "∂f (p t−1,pt−2) ∂pt−1 ∂f (pt−1,pt−2) ∂pt−2 1 0 # . (14)
To determine the eigenvalues of the Jacobian evaluated in the steady state it is necessary to determine the steady state price p. Plugging in pt= pt−1= pt−2= p in (9) gives the following equation
p2+ θrp − θy = 0, (15)
for which the only positive solution is the steady state price
p = −θr +pθ
2r2− 4θy
2 . (16)
Plugging this in into the Jacobian yields
J∗ = " θg(α−r−1)−rp − 1 − g g − θg(α−r−1)p 1 0 # . (17)
From this the characteristic equation can be derived:
λ2− θg(α − r − 1) − r p − 1 − g λ − g + θg(α − r − 1) p = 0. (18)
To see whether a saddle-node bifurcation can occur, I plug λ = 1 into the characteristic equation. This leads to
2 + θr
p = 0. (19)
Which is not possible because θ, r, p > 0. So no saddle-node bifurcations occur in this system.
To investigate if a period-doubling bifurcation can occur, I plug λ = −1 into the characteristic equation. This yields
α = r + 1 + p θ +
r
For such parameter combinations, a period-doubling bifurcation may oc-cur.
When the discriminant of the characteristic equation is negative this means that the Jacobian evaluated in the steady state has complex eigenvalues. If there are complex eigenvalues, a Hopf bifurcation can occur. To find for what values a Hopf bifurcation occurs, we need to solve − g − θg(α − r − 1) p = 1 (21) this yields α = 1 + r + (1 + g)p gθ . (22)
At such parameter combinations a Hopf bifurcation occurs, provided that the discriminant of the characteristic equation is smaller than zero
θg(α − r − 1) − r p − 1 − g 2 + 4 g − θg(α − r − 1) p < 0. (23)
Plugging in the solution for when a Hopf Bifurcation occurs (22) into the discriminant gives the following solution:
θ2r
2
p2 − 4 < 0 (24)
Which holds for most values of θ, r, and p. Except for when θ is very large, which is in my opinion not realistic.
Therefore, of the bifurcations that I have tried to find in this theo-retical analysis, both a period-doubling and Hopf bifurcation can occur. Figure 1 shows the corresponding values of α and g for which the period-doubling- or Hopf bifurcation occur.
Figure 1: Values of α and g for bifurcations
If the investors are contrarians, with g < 0, the system is sta-ble for α ∈ 1 + r + (1+g)pgθ , 1 + r + pθ + 2gr where the lower bound is the value for which a Hopf bifurcation occurs and the upper bound is the value for which a period doubling bifurcation occurs. Contrar-ily, when considering trend followers (g > 0) the system is stable for α ∈ 1 + r + pθ + 2gr, 1 + r + (1+g)pgθ . In this interval, the lower bound is the value of α for which a period doubling bifurcation occurs and the upper bound is the value for which a Hopf bifurcation occurs. Outside of these intervals the system is unstable.
Interestingly, the upper bound of the interval of α, such that the system converges to a steady state, for contrarians is always lower than
α = r + 1 + p
θ. (25)
While, the lower bound of the interval of α for trend followers is always larger than (25).
With the theoretical analysis I have found that both a period-doubling- and Hopf bifurcation occur in the model. When these bi-furcations occur depends on the values of α and g. In the next chapter I perform numerical simulations, and compare these to the theoretical results.
5
Simulations
The previous chapter discussed the theoretical analysis of the model. In this chapter I discuss the results of the simulations with Matlab. In this chapter I focus on the effect of g and α and fix the other parameters as θ = 1000, r = 0.05, and y = 20. The fundamental price is equal to p = 118.6141 for these values. For most of the simulations either the value of g or α is also fixed and I investigate how the other variable influences the stability of the system.
It is necessary to get an idea of what can happen to the price in the system. When the system does not reach a steady state for the combination of α and g the system tends to result in both very high and negative prices for the risky asset. Because it is not realistic that prices become negative I have introduced a lower bound of 0.001 for the simulations of the system. To get a good idea of the price dynamics, I have plotted the price for 60 periods, after the system has already gone through 1,000 periods, for fixed values of both g and α. For these simulations the system starts close to the steady state price p that is determined in Chapter 4. The resulting graphs are shown in Figure 2.
1000 1020 1040 1060 time 0.00 200.00 400.00 600.00 800.00 1,000.00 price g=-1, α=2 1000 1020 1040 1060 time 0.00 200.00 400.00 600.00 800.00 1,000.00 price g=1, alpha=2 1000 1020 1040 1060 time 0.00 0.50 1.00 1.50 2.00 price ×107 g=-1, alpha=1 1000 1020 1040 1060 time 118.61 118.61 118.61 118.61 118.61 price g=0.5, α=1.3
Figure 2: Price after 1,000 periods
The graphs above show some different price dynamics. It can be seen that when the system does not end up in a steady state it tends to reach the lower bound of zero, and oscillate between two or more prices. For g = 0.5 and α = 1.3 the system converges to the steady state2, which
corresponds to what I found in the theoretical analysis. For g = −1 and α = 2 the price explodes and moves from the lower bound to extremely high prices. While for the two other graphs, the prices also go up and down, but never get as extreme as for g = −1 and α = 2.
It is interesting to look into for which values the system converges to the steady state and where the changes happen. First I want to determine for what values of g the system is in a steady state when α is fixed. The system runs for 10,000 periods and plots the next 100 periods for the specific value of g, for g ∈ [−2, 2]. The system starts with prices that are close to the steady state price p. Again the system has a lower bound of 0.001 and values that are larger than 1,000 are shown in the
graph as 1,000. The resulting graphs are shown below.
Figure 3: Bifurcation Diagrams, varying g
From the graphs in Figure 3 it can be seen that for negative and very large values of α there is only a very small interval of values of g for which the system converges to the steady state. It turns out that the largest intervals of g are when α has a value quite around 1. So, the system reaches the steady state for a large interval of g when people have approximately the same beliefs for the price in period t as in period t + 1. Remarkably, the system only reaches the steady state for a small interval of g when α = 2, while this is a simple version of the model where investors have the same beliefs with respect to the price trend for both period t and t + 1.
I also need to look at how α influences the stability of the system for fixed values of g, and whether the starting value of the system is of influence. In Figure 4 multiple bifurcation diagrams of the system are shown. Again, the system started with values close to the steady state, has run for 10,000 periods and the next 100 periods are plotted for every
value of α. Also, the system has a lower bound of 0.001 and prices that are larger than 1,000 are shown as 1,000 in the bifurcation diagrams.
Figure 4: Bifurcation Diagrams, varying α
From these graphs it can be seen that for every value of g investi-gated here, there is a value of α for which the system converges to the steady state, except for g = 0. It can be seen that bifurcations take place for the combinations of g and α that I found in the theoretical analysis. Figure 5 shows more bifurcations diagrams for values of g closer to zero.
0.5 1 1.5 α 0.00 200.00 400.00 600.00 800.00 1,000.00 price g=-0.4 0.5 1 1.5 α 0.00 200.00 400.00 600.00 800.00 1,000.00 price g=-0.2 0.5 1 1.5 α 0.00 200.00 400.00 600.00 800.00 1,000.00 price g=0.2 0.5 1 1.5 α 0.00 200.00 400.00 600.00 800.00 1,000.00 price g=0.4
Figure 5: Bifurcation Diagrams, varying α
When g is closer to zero, the interval of values of α for which the system converges to the steady state is larger. In other words, when g is further from zero, the interval of α for which the system is stable grows smaller. So, the more extreme the beliefs of the investors in the system are, the more specific the value of α has to be for the system to be stable. Therefore, when the investors have extreme beliefs, the chance of chaotic price fluctuations occurring is a lot larger.
When g is negative the system converges to the steady state for smaller values of α compared to when g is positive. This is also what we would expect from the theoretical analysis and Figure 1. From this it can be concluded that when the investors are trend followers the system converges to a steady state when the investors believe that the trend will continue, but not as strong, in period t + 1. While, for contrarians the system converges to a steady state when they either believe that the trend continues weakly in period t+1 or when they believe that the trend will be slightly reversed in period t + 1. As explained in the theoretical
analysis, for contrarians the upper bound of the interval of α, for which the system converges to a steady state, is always lower than a specific value. While the lower bound of the interval for trend followers is always higher than this value. When using the parameters as they are defined in this section, this value is α=1.16863.
Although bifurcations do occur at the combinations of g and α that I expected from the theoretical analysis. The dynamics of the system outside of the values of α for which the system converges to the steady state are not what I expected. When a period-doubling bifurcation was expected, generally the price does not converge to two steady states, but is a lot more diverse. The explanation for this is that the system has a lower bound of 0.001, which is necessary because without this lower bound the price would become negative. The lower bound causes the dynamics of the system to change. On top of that I did not look at the negative steady state price in the theoretical analysis, this could also explain some of the unexpected dynamics that occur in the simulations. To illustrate that the lower bound influences the dynamics of the system, Figure 6 shows two bifurcation diagrams of the system when there is no lower bound. Values smaller than -1,000 are shown as -1,000.
0.5 1 1.5 α -1,000.00 -500.00 0.00 500.00 1,000.00 price g=-0.5 0.5 1 1.5 α -1,000.00 -500.00 0.00 500.00 1,000.00 price g=0.5
Figure 6: Bifurcation Diagrams, no lower bound
3This value is determined by plugging the parameter values for θ, r, and the
To conclude, the simulations show that the dynamical system in-deed converges to the steady state for the combinations of g and α found in the theoretical analysis. When investors have extreme beliefs, the chances of chaotic price fluctuations occurring are larger. And the values of α for which the system converges are smaller for contrarians than for trend followers. Finally, it is important to note that as a result of the lower bound of 0.001 the dynamics when the system does not converge to the steady state are different from what we would expect from the theoretical analysis.
6
Comparison to Brock and Hommes (1998)
To compare the results of the theoretical analysis and simulations found in the previous sections to Brock and Hommes (1998), I need to do a theoretical analysis of their model when assuming that all investors have the same beliefs. By looking at this model I can see what type of bifurcations occur and if these are similar to my findings.
In their model, Brock and Hommes (1998) determine the price in period t by pt= ¯ pe t+1+ y 1 + r , (26)
where ¯pet+1 is the average expectation of all investors for the price in the next period. The expectations of the price in the next period for an investor is given by
pet+1 = pt−1+ g(pt−1− pt−2). (27)
Because I assume that all investors have the same beliefs, this ex-pectation per investor is also the average exex-pectation of the price. Plug-ging this into the price formula (26) yields
pt=
pt−1+ g(pt−1− pt−2) + y
1 + r . (28)
pt−1= pt−2= p into (28). This gives:
p = y
r (29)
The Jacobian evaluated in the steady state is given by:
J∗ = "1+g 1+r −g 1+r 1 0 # . (30)
This leads to the following characteristic equation:
λ2− 1 + g 1 + r
λ + g
1 + r = 0 (31)
To find out if a saddle-node bifurcation can occur I plug λ = 1 into the characteristic equation. This yields
1 − 1
1 + r = 0, (32)
Which is not possible because r > 0. Consequently, no saddle-node bifurcations occur in this system.
To see whether a period-doubling bifurcation occurs I plug λ = −1 into the characteristic equation. Which results in
g = −1 − 0.5r. (33)
So, for this combination of the variable g and the parameter r a period-doubling bifurcation may occur.
To find the combination of the variables and parameters for which a Hopf bifurcation may occur we need to find out for what values there are complex eigenvalues and when the following equation holds
g = 1 + r. (34)
The eigenvalues are complex when the discriminant of the characteristic equation is negative. Plugging (34) into the discriminant leads to the
following condition
−3(1 + r) + 1
(1 + r)2 < 0, (35)
which always hold. So, for the combination g = 1 + r a Hopf bifurcation occurs in the model by Brock and Hommes (1998) with one belief type. From this theoretical analysis of the model by Brock and Hommes (1998) it follows that both a period-doubling- and Hopf bifurcation can occur. These are the same kinds of bifurcations as that I found in the theoretical analysis of my model. So, the dynamics of the system are equal with respect to the bifurcations that occur.
But, there are also differences between the dynamics of the two models. While the bifurcations in the model of Brock and Hommes (1998) only depend on g and r, the bifurcations in the model of my thesis depend on g, α, θ, y and r.
7
Conclusion
In this paper I investigated an asset pricing model where all investors have the same beliefs. The aim of the paper is to check the robustness of the results of Brock and Hommes (1998), by looking at how the variables g and α influence the stability of the model.
The theoretical analysis showed that the stability of the model de-pends on the combination of g and α. Saddle-node bifurcations do not occur in the model. Both period-doubling- and Hopf bifurcations do oc-cur in the model. In the model of Brock and Hommes (1998) for one belief type, these are also the bifurcations that occur. While they also found saddle-node bifurcations for their model with different belief types. Nevertheless, the dynamics that occur are equal for both models when only one belief type is considered.
The simulations show that the prices differ strongly for different values of g and α and that the system is only in a steady state for a intervals of certain combinations of g and α. When the beliefs of investors are more extreme, so when g is far from zero, these intervals get smaller.
These findings endorse the findings of Brock and Hommes (1998), they find that bifurcations occur in their model when either trend followers or contrarians have extreme beliefs and these bifurcations cause chaotic price fluctuations. I also found that the values of α for which the system converges are smaller for contrarians than for trend followers.
To conlude, my thesis confirms the robustness of some of the results of Brock and Hommes (1998) but also calls for new research with respect to their model. In this new research the model should be adapted in such a way that it incorporates both the expectations for price in this period and the expectations for the price in the next period. This is necessary because my thesis clearly shows that the system can only be stable for certain combinations of g and α. On top of that future research could be done into why the price of my model also goes to negative values and how this influences the dynamics of the system.
References
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