Contagion risk and spillover effects in asset prices

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Contagion Risk and Spillover Effects in Asset

Prices

Author: Steyn Verhoeven Supervisor: Prof. dr. Jan Tuinstra 2nd Reader: Prof. dr. Cees G. H. Diks

Master’s Thesis in Financial Econometrics

Faculty of Economics and Business, University of Amsterdam

September 27, 2014

Abstract

This thesis investigates the dynamics of financial contagion and the impact of trading restrictions on financial market stability. Asset prices are modelled in a well-known asset pricing model with heterogeneous beliefs. In particular, this the-sis expands the model of Brock and Hommes (1998) to include multiple risky assets and time-varying variance. The expectations of agents about correlation are shown to serve as a self-fulfilling prophecy. In particular, financial contagion and spillover effects can be observed when agents believe assets are correlated. The results on trading restrictions suggest that the stabilizing effects of restrictions on demand are limited. Furthermore, trading restrictions may have the adverse effect of enforcing volatility spillovers to other markets. The results of this thesis show that policy makers should be reluctant when imposing demand constraints on the market.

Keywords: Multi-asset pricing model, Heterogeneous agents, Adaptive beliefs, Trading restrictions, Contagion risk, Volatility spillovers

∗

Acknowledgements: I would like to thank Jan Tuinstra for his valuable comments, guidance and useful discussions during the writing of this thesis. Further I would like to thank Mikhail Anufriev for his interest in the subject and helpful suggestions during the first phase of the research.

∗

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1 Introduction

The global financial crisis that began in 2007 resparked the debate on whether inter-national regulatory policies such as the Basel Accords have a positive effect on the stability of financial markets, or in fact contribute to its instability during a crisis. Economic studies have been performed to investigate if the Basel II Accord, initially published in June 2004, contributed to containing the financial turmoil during the 2007-2008 crisis. Cannata and Quagliariello (2009) argue that although not to blame for the financial crisis, the Basel II Accord failed to address underlying market dynamics, such as procyclicality of capital requirements. Jorion (2009) and McAleer et al. (2012) show that prevailing risk models, even when flawlessly implemented according to regulatory requirements, have been proven to fail under severe financial stress.

What causes financial turmoil to persist despite extensive regulatory control? An analysis of the 2007-2008 crisis by Mishkin (2010) reveals that a seemingly small dis-ruption of the financial system led to wide economic distress, in particular due to the great interconnectivity within the financial system. In other words, an outside shock (in this case, the subprime mortgage crisis) in the first phase of the crisis was arguably amplified by the dynamics within the system during the second phase of the crisis. From a risk perspective, it will be useful to distinguish these two phases as being the result of two different type of risks. The first is exogenous risk, which refers to the risk from shocks that are generated outside the system. The second is endogenous risk, which refers to the risk from shocks that are generated and amplified within the system. In hindsight, it may appear that regulators have previously underestimated endogenous risk in financial markets.

Described by Danielsson and Shin (2002), endogenous risk arises whenever (i) indi-viduals react to their environment and (ii) the individual actions affect the environment. The financial market is a prime example of a system where endogenous risk arises. In fact, Zigrand (2010) and Danielsson et al. (2011) argue that different financial crises, although originating from different events, develop in much the same way through feed-back effects that are built into the financial system itself. Regulators primarily focus on exogenous risk: that is, they try to ensure that financial institutions are able to withstand shocks from outside the system that may affect their capital or liquidity. In this sense, they view each financial institution on its own, largely neglecting the market-wide dynamics. However, arguably it is precisely these market-market-wide dynamics that pose the greatest danger during financial crises. In fact, Danielsson and Shin (2002) and Danielsson et al. (2012) demonstrate that by neglecting these market-wide dynamics, risk regulation might have the perverse effect of increasing financial instability.

An example of a risk that is built into the system of financial markets is the risk of financial contagion. Financial contagion is generally defined as a significant increase in comovements of prices and quantities across markets, that cannot be explained by

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fundamentals underlying the asset prices1. An important distinction is made between interdependence and contagion, where the first refers to transmission channels that are real and stable, such as export-import relations, and the latter refers to the spreading of instability through speculation, panic or herd behaviour. A stylized fact of financial contagion is the spillover of volatility from one market to another during periods of fincial turmoil. These spillover effects are used extensively in empirical studies as a tool to detect and measure financial contagion (see e.g. Bae et al., 2003 and Kleimeier, 2008).

Financial contagion is widely regarded by policymakers as one of the key problems during the global financial crisis. Markets that are prone to contagion have a signif-icantly higher probability of suffering a crisis (OECD, 2012). To enforce stability on financial markets and thereby mitigate contagion risk, various policies have been pro-posed for regulating asset markets. A prime example are restrictions on short-selling, which are aimed at reducing speculative trading. However, the effectiveness of such trading restrictions remains the subject of debate. In their paper on short-selling con-straints, Anufriev and Tuinstra (2013) show that restrictions on short-selling may in fact increase volatility on markets and enforce mispricing. In a similar setting, In ’t Veld (2013) shows that constraining investment positions have potentially adverse effects on market stability. This thesis expands on the research by Aufriev and Tuinstra and In ’t Veld by analysing the effect of trading restrictions spillover effects. In particular, it is investigated whether or not trading restrictions have a positive effect on the stability of prices and are able to mitigate the risk of financial contagion. To investigate the effect of trading restrictions in a closed setting, a dynamic system is introduced in this thesis to model the behaviour of agents on financial markets.

In behavioural economics, models have been proposed that attribute price volatility to boundedly rational behaviour of market participants. So-called agent based models have been shown to replicate endogenous price fluctuations observed in financial mar-kets (Cont and Wagalath, 2014). Agent based models relax the typical Efficient Market Hypothesis (EMH) constraint that all market participants should be rational, and in-stead are based on the assumption that agents are boundedly rational. In particular, Brock and Hommes (1998) have proposed an asset pricing model with heterogeneous agents that produces endogenous price fluctuations through an adaptive belief system. The heterogeneous agents model by Brock and Hommes forms the basis of the model proposed in this thesis.

Brock and Hommes introduced a framework with two agent types, fundamental-ists and chartfundamental-ists. They allowed agents to adjust their beliefs through a switching mechanism that is based on past performance. Similar models have been proposed

1_{See Pericoli and Sbarcia (2012) for an extensive discussion of the definitions of contagion currently}

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by Gaunersdorfer (2000), Chiarella and He (2003), Brock et al. (2006), Chiarella et al. (2007) and Gaunersdorfer et al. (2008). To be able to investigate the occurrence of financial contagion and spillover effects, the framework of Brock and Hommes will be ex-tended to incorporate multiple risky assets that are connected through beliefs of agents about correlation between assets. Similar multi-asset frameworks have been proposed by Westerhoff (2004), Chiarella et al. (2007) and Chiarella et al. (2013). This thesis deviates from the aforementioned research by introducing a framework in which agents may adopt different strategies on different markets. For example, an agent may adopt a chartist belief on one market, but a fundamentalist belief on another. This is a key adaptation of the model as it ensures that comovements of asset prices are not caused by a direct link between agent fractions on both markets, but instead are enforced by beliefs about correlation.

The results in this thesis show that spillover effects can be observed when agents believe that assets are correlated, even when the underlying dividend processes are independently distributed. As such, beliefs of agents about correlation between assets serve as a self-fulfilling prophecy. The result on trading restrictions in this thesis are ambiguous. In contrast to Anufriev and Tuinstra (2013), in this thesis, simulations of the model with restrictions on short positions show a decrease in the mispricing of the restricted asset. However, restrictions on short positions also lead to an increase in volatility spillovers. Restrictions on long positions are shown to have a positive effect on the stability of non-fundamental steady states in the model. Furthermore, restrictions on long positions lead to smaller volatility on the unrestricted market. However, similar to In ’t Veld (2013), overall the stabilizing effects of trading restrictions are shown to be limited.

This thesis is structured as follows. Section 2 describes the heterogeneous agents model with and without trading restrictions, respectively. In Sections 3.1, the dynamics of the deterministic model without restrictions is analysed. In Section 3.2, stochastic dividends are introduced and the occurrence of spillover effects and financial contagion are investigated for the model without restrictions. In Section 3.3, the impact of trading restrictions on market stability and contagion risk is analysed. Finally, Section 4 gives concluding remarks. The proofs of propositions are included in the appendix.

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2 The Model

2.1 Heterogeneous Agents Model

The model that is used in this thesis is an extension of the asset pricing model with het-erogeneous beliefs introduced by Brock and Hommes (1998). I consider a generalization of the model of Brock and Hommes by assuming there is one riskless asset and N risky assets, where N is arbitrary. For this purpose, the model will be structured in vector and matrix notation, with the N -dimensional vectors of prices and dividends given by

pt= (p1,t, . . . , pN,t)0,

dt= (d1,t, . . . , dN,t)0.

Following Brock and Hommes, I assume agents have heterogeneous expectations of prices but homogeneous beliefs about variance. However, in contrast to Brock and Hommes, I include endogenous or time-varying variance in the model. In the next section, I will describe the model under rational expectations. Subsequently, the model under heterogeneous expectations will be described.

2.1.1 Rational expectations

As above, let pt be an N -dimensional vector of (ex-dividend) prices per share at time

t, and let dt be an N -dimensional vector of stochastic dividends of the risky assets. In

addition, there is a riskless asset, which is assumed to be perfectly elastically supplied, yielding a constant return r and gross return R = 1 + r. At time t, agents compose their portfolio with shares of the risky assets, such that the evolution of wealth of each agent can be described as

Wh,t+1= RWt+ z0ht[pt+1+ dt+1− Rpt] ,

where zht = (zh,1t. . . zh,N t)0 is an N -dimensional excess demand vector of the number

of shares of each asset purchased by an agent with strategy h at time t. It is assumed that agents have a CARA-utility function with constant absolute risk aversion ah, with

is expressed by

uh(W ) = − exp(−ahW ).

The optimal portfolio of shares at time t is determined by maximizing the expected CARA-utility of wealth at time t + 1,

max

zht

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It can be shown that maximizing CARA-utility is equivalent to optimizing the following myopic mean-variance problem (Ang, 2014):

max

zht

Eht(Wh,t+1) −

ah

2 V arht(Wh,t+1). (2.1) By our definition of wealth, this is equivalent to solving

max zht z0_ht[Eht(pt+1+ dt+1) − Rpt] − ah 2 z 0 htV arht(pt+1+ dt+1)zht.

In this model, I introduce what is called endogenous or time-varying variance. In their paper, Brock and Hommes (1998) assume perceived variance of prices is not only identical for all trader types but also constant over time. However, similar to Gauners-dorfer (2000) I argue that when updating beliefs about conditional means, it makes sense to assume that agents also update their beliefs about conditional variance. For simpli-fication purposes, I will stick to the homogeneous nature of beliefs about conditional variance. Nelson (1992) shows that it is much easier to estimate conditional variance than conditional means, which provides justification for assuming homogeneous beliefs about conditional variance.

Let Ωt := V arht(pt+1+ dt+1) represent the homogeneous beliefs of agents about

the variance-covariance matrix at time t. Then the individual excess demand at time t equals

zht=

1 ah

Ω−1_t [Eht(pt+1+ dt+1) − Rpt] . (2.2)

Now it is assumed that there is a Walrasian auctioneer such that an equilibrium exists between supply and demand of shares. Let s be an N -dimensional vector of supply of outside shares and let nhtbe the fraction of agents using strategy h at time t. Then the

Walrasian market clearing equation can be written as

H X h=1 1 ah Ω−1_t [Eht(pt+1+ dt+1) − Rpt] nht= s.

In the remaining part of this thesis, it is assumed that risk aversion is equal across traders, such that ah = a, ∀h. The dividends dt are assumed to follow a martingale

process defined by

dt+1= dt+ σζζt+1, (2.3)

where ζ_t+1 is an independently and identically distributed standard normal random variable with E(ζ_t) = 0 and V ar(ζ_t) = I. Furthermore, agents are assumed to have homogeneous and correct beliefs about the dividends. Under these assumptions, the

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Walrasian equilibrium equation can be solved for the prices at time t: pt= 1 R "_H X h=1 Et(pt+1)nht+ dt− aΩts # . (2.4)

In the context of a dynamic system of pricing equations, it is important to be clear about the timing of the expectations formations. At the beginning of time t, the dividends dt

are realized and agents form their expectations about the payoff pt+1+ dt+1 in period

t + 1 based on realized prices up until period t − 1 and the dividends dt. As such, a

positive feedback structure exists in the model that causes realized prices to be higher (lower) when predictions of future prices are higher (lower).

Under rational expectations, there would be only one type of trader such that Eht(pt+1) = Et[pt+1]. Furthermore, we assume in this case that common beliefs about

the variance-covariance of prices and dividends are constant such that Ωt= Ω0. Then

the equilibrium equation (2.4) reduces to pt=

1

R[Et(pt+1) + dt− aΩ0s] .

Assuming that the transversality condition limt→∞Et(p∗t)/Rt = 0 holds, there is only

one solution that solves for p∗_t ≡ pt= Et(pt+1). This solution is the fundamental price

vector that would prevail in a perfectly rational world, p∗_t = dt− aΩ0s

R − 1 . (2.5)

In case of zero outside supply, equation (2.5) simplifies to the discounted sum of expected future dividends.

2.1.2 Heterogeneous expectations

In this section I will consider agents who are boundedly rational and have heterogeneous expectations about asset prices and returns. The predictions of agents depend on their view and knowledge of the market. Chiarella et al. (2013) propose a system where an agent type consists of a single prediction strategy that is applied to all markets. Similar to Brock and Hommes (1998), they introduce the following agent types: fundamentalists and trend-followers, or chartists. Fundamentalists have perfect knowledge of the fun-damental price and believe that prices will eventually move towards their funfun-damental value. Chartists on the other hand, do not require knowledge of the fundamental price, but instead rely solely on the extrapolation of positive or negative trends. In this the-sis, I propose an alternative system where agents may choose to apply a fundamentalist strategy on one market and a chartist strategy on another. In other words, agents are allowed to adopt a different strategy for each market.

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two risky assets. Let each asset be analysed by two experts, with one expert being a fundamentalist and one expert being a chartist. At the beginning of every period, an agent decides for each asset which expert to follow, based on past performance. This way, an agent may decide to follow the fundemantalist expert on one market but a chartist expert on another market. For analytical tractability, in the remaining part of this thesis the case of N = 2 risky assets is considered, which ensures that the number of possible combination of strategies remains limited. The following assumption is made about the beliefs of agents about future prices:

Assumption 2.1. The beliefs of agents about future prices are assumed to be of the form Eht(pt+1) := fht= f1h(p1,t−1. . . p1,t−L) f2h(p2,t−1. . . p2,t−L) ! ,

where fihis an autonomous function that describes the beliefs of agent h about the price

of asset i in period t + 1, based on past prices. Note that fih only depends on asset i,

i.e. the price of the other asset is not used as a predictor.

Under Assumption 2.1, the price equation (2.4) can be written as

pt= 1 R " _H X h=1 fhtnht+ dt− aΩts # . (2.6)

The system described above leads to four agent types being distinguished in this model. Type 1 can be described as true fundamentalists, following a fundamentalist strategy on both markets. Agents of type 2 apply a fundamentalist strategy on the first market, but are chartists on the second market. Vice-versa, type 3 agents can be considered chartists on the first market but fundamentalists on the second market. Finally, agents of type 4 apply a chartist strategy on both markets. The aforementioned types are characterised by the following prediction vectors:

f1t= p∗t + α(pt−1− p∗t), f2t= p∗t + α1(p1,t−1− p∗1,t) γ2(p2,t−1− p∗2,t) ! , f3t= p∗t + γ1(p1,t−1− p∗1,t) α2(p2,t−1− p∗2,t) ! , f4t= p∗t + γ(pt−1− p∗t).

Here γ = diag(γ1, γ2), where γj > 1, represents the extrapolation, with high (low)

values of γj corresponding to strong (weak) extrapolation by trend followers, and α =

diag(α1, α2) with αj ∈ [0, 1] represents the speed at which fundamentalists expect prices

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believe prices will immediately revert back to their fundamental value, whereas αj = 1

represents the case where fundamentalists are naive traders. The definitions of αj and

γj imply that expected reversion to the fundamental price and trend extrapolation may

vary across markets, but not across agents following the same strategy on the same market.

2.1.3 Time-varying variance

As mentioned in Section 2.1.1, in this model agents are assumed to have homogeneous beliefs about variance. In particular, the following assumption is made concerning the expectations of agents about the variance-covariance matrix.

Assumption 2.2. The beliefs of agents about conditional variances of prices and divi-dends are assumed to be of the form

V arht(pt+1+ dt+1) = Ωt:= (1 − λ)Ω0+ λVt−1.

where Vtis an estimate of the N ×N variance-covariance matrix of prices and dividends

and λ ∈ [0, 1] is a parameter that represents the sensitivity of the beliefs to this variance-covariance estimate.

Assumption 2.2 implies that beliefs about the variance-covariance matrix are a weighted average of a fixed variance-covariance matrix Ω0 and observed volatility in the market.

The advantage of Assumption 2.2 is that it allows for time-varying variance, but at the same time ensures that beliefs about variance cannot become equal to zero. In particular, λ = 0 represents the case of constant beliefs about variance-covariance of prices and dividends. The matrix Ω0 is defined as follows:

Ω0:=

σ₁2 ρ12σ1σ2

ρ12σ1σ2 σ22

! .

As the estimate Vt of the variance-covariance matrix, I propose an

exponentially-weighted moving average (EWMA). This estimate is widely used in risk management practices and has the advantage over equally weighted moving average windows because it is less sensitive to extreme jumps in returns2. In particular, I assume that Vt follows

the following EWMA process:

ut= δut−1+ (1 − δ)pt, (2.7a)

Vt= δVt−1+ (1 − δ)(pt− ut−1)(pt− ut−1)0, (2.7b)

where δ ∈ (0, 1) is a weighting parameter and ut defines the exponentially weighted

moving averages of prices.

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2.1.4 Evolutionary learning

A key aspect of the model with heterogeneous agents is the ability of agents to learn from realizations of the market and update their beliefs accordingly. This is called evo-lutionary learning and requires that agents have knowledge of the performance of each strategy in the previous period. With respect to performance, they are several possible approaches. Brock and Hommes (1998) base performance on realized profits, whereas Gaunersdorfer et al. (2008) use risk adjusted realized profits to compute performance. The latter is considered here, as this provides consistency with the mean-variance max-imization problem each agent faces. At the beginning of period t, the realized risk adjusted profits for each strategy are publicly available and given by

πht= z0h,t−1[pt+ dt− Rpt−1] −

a 2z

0

h,t−1Ωt−1zh,t−1. (2.8)

In addition to the risk adjusted profits defined above, a strategy cost Ch ≥ 0 is

intro-duced for each strategy h, leading to the performance measure Uht= πht− Ch.

In particular, applying a fundamentalist strategy is more costly than applying a chartist strategy, as it is assumed that the former requires knowledge about the fundamental price, which can only be retrieved at an extra cost. The chartist strategy on the other hand, only relies on prices that are directly observable in the market. More explicitly, in this thesis it is assumed that applying a fundamentalist strategy on market i indures an additional cost ci > 0, such that C1 = c1 + c2, C2 = c1, C3 = c2 and C4 = 0.

In other words, applying a fundamentalist strategy on both markets is more expensive than applying a fundamentalist strategy on one market, which in turn is more expensive than applying a chartist strategy on both markets.

At the beginning of each period, the fraction for each strategy h will be given by the discrete choice probability

nht= exp (βUh,t−1) /Nt, Nt≡ H

X

h=1

exp (βUh,t−1) , (2.9)

where β represents the intensity of choice, which is a measure of the willingness of agents to switch between strategies. Once again, it is important to be clear about the timing of decisions. The fractions of strategies are determined at the beginning of each period t, and can therefore only depend on prices up until period t. In other words, the fractions in each period depend on the performance of the previous period.

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2.1.5 Complete dynamic model

Based on the definitions given above, the optimal demand of each agent type h is given by

zht =

1 ah

[(1 − λ)Ω0+ λVt−1]−1[fht+ dt− Rpt] , h = 1, 2, 3, 4.

Then the general dynamic model in (2.4) reduces to the nonlinear dynamical system                                        pt = _R1 h P4 h=1fhtnht+ dt− a[(1 − λ)Ω0+ λVt−1]s i , p∗_t = _R−11 [dt− aΩ0s] , ut = δut−1+ (1 − δ)pt, Vt = δVt−1+ (1 − δ)(pt− ut−1)(pt− ut−1)0,

n1t = exp (βU1,t−1) /Nt, Nt≡P4h=1exp (βUh,t−1) ,

n2t = exp (βU2,t−1) /Nt, n3t = exp (βU3,t−1) /Nt, dt = dt−1+ σζζt, (2.10) where n4t= 1 − n1t− n2t− n3t, Uht= z0h,t−1[pt+ dt− Rpt−1] − a 2z 0 h,t−1[(1 − λ)Ω0+ λVt−2]zh,t−1− Ch, zht= 1 ah [(1 − λ)Ω0+ λVt−1]−1[fht+ dt− Rpt] , f1t= p∗t+ α(pt−1− p∗t), f2t= p∗t+ α1(p1,t−1− p∗1,t) γ2(p2,t−1− p∗2,t) ! , f3t= p∗t+ γ1(p1,t−1− p∗1,t) α2(p2,t−1− p∗2,t) ! , f4t= p∗t+ γ(pt−1− p∗t).

In summary, an adaptively heterogeneous beliefs model is proposed to simulate asset prices in a closed setting. In the next section, an adaptation of the model is proposed that includes restrictions on demand.

2.2 The Model with Restrictions

In the previous section, an asset pricing model with heterogeneous agents was proposed. In particular, the model in Section 2.1 assumes that agents are able to take any position in the risky assets, without restrictions. In this section, a model with trading restrictions is considered. More specifically, for the model in this section it is assumed that there

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is a maximum amount that can be invested in the first risky asset, in either a long or short position. The demand for the second asset remains unrestricted.

The general trading restrictions as proposed in this section provide a way to in-vestigate to what extent restrictions on financial markets positively influence market stability, if at all. In the first part of this section, the new individual and aggregate demand curves are discussed. In the second part of this section, pricing equations in a Walrasian equilibrium are computed.

2.2.1 Aggregate demand curve

For the individual demand of agents, it is assumed that there is a lower bound z₁min≤ 0 and lower bound zmax

1 ≥ 0 that limits the position that an agent can take in the

first risky asset. More specifically, agents are assumed to have the following individual demand function with respect to the first asset:

˜ z1ht =         

z₁min, when z1ht< z1min

z1ht, when zmin1 < z1ht< z1max

z₁max, when zmax₁ < z1ht,

(2.11)

where z1htis the invididual demand for the first asset that would prevail without trading

restrictions, i.e. the first element of the demand in equation (2.2). For computational purposes, it is assumed from here on that there is no excess supply of shares, i.e. s = 0. Furthermore, let Ωt= σ_1t2 ρ12tσ1tσ2t ρ12tσ1tσ2t σ22t ! . Then the first element of equation (2.2) can be written as

z1ht =

1 a det(Ωt)

σ_2t2 [f1ht+ d1t− Rp1t] − ρ212tσ1tσ2t[f2ht+ d2t− Rp2t] . (2.12)

Note that when either the lower or upper bound constraint is binding for the demand of the first risky asset, agents optimize the problem in (2.1) for fixed ˜z1ht, which results

in the following expression for the demand of the second risky asset: ˜

z2ht(˜z1ht) =

1

aσ_2t2 (f2ht+ d2t− Rp2t− aρ12tσ1tσ2tz˜1ht) . (2.13) Let A ⊂ {1, . . . , 4} be a subset of strategy types such that in equilibrium, z_1ht∗ < z₁minfor each h ∈ A. Furthermore, let B ⊆ {1, . . . , 4} be a subset such that zmin

1 < z1ht∗ < z1max

for each h ∈ B and finally let C ⊂ {1, . . . , 4} be a subset such that z₁max < z_1ht∗ for each h ∈ C. Note that A, B and C may be empty sets, but not all at once since A ∪ B ∪ C = {1, 2, 3, 4} should hold. Using these definitions, the Walrasian equilibrium

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on both markets is given by X h∈A zmin₁ ˜ z2ht(z1min) ! nht+ X h∈B zhtnht+ X h∈C z₁max ˜ z2ht(z1max) ! nht= 0 (2.14)

2.2.2 Prices under constraints

To find the equilibrium price vector under constraints, the Walrasian equilibrium equa-tion in (2.14) needs to be solved. Figure 2.1 shows a sketch of the equilibrium when both an upper and lower bound is imposed on the individual demand of agents. It can be seen that the aggregate demand shows kink points where the individual demand of an agent becomes contrained. Furthermore, Figure 2.1 reveals that two cases need to be considered. In the first case, B 6= ∅, i.e. there are agent types for whom neither of the constraints are binding. In the other case, B = ∅, i.e. all agent types are constrainted by one the restrictions. Figure 2.1 illustrates the latter case.

Proposition 2.1. Assume B 6= ∅. Furthermore, let nA:=Ph∈Anht, nB :=Ph∈Bnht

and nC :=Ph∈Cnht. Then the equilibrium equation (2.14) implies that the equilibrium

prices are given by:

p1t = 1 RnB " X h∈B

f1htnht+ nBd1t+ a(1 − ρ212t)σ1t2(nAz1min+ nCz1max)

−ρ_12tσ1tσ2t−1 (nA+ nC) X h∈B f2htnht− nB X h /∈B f2htnht ! # , (2.15) and p2t= 1 R " ₄ X h=1 f2htnht+ d2t # . (2.16)

Proof. See Appendix A.

Notice that the price of the second assets is equivalent to the price without trading restrictions, assuming s = 0. In other words, imposing trading restrictions on the first market does not directly impact the price on the second market. Furthermore, note that when the restrictions or not binding for any strategy type, i.e. nA= nC = 0 and

nB= 1, then equation (2.15) reduces to the price without restrictions.

Now let B = ∅ for a set of prices P . Then the aggregate demand curve on the first market is flat for all p1t ∈ P and equal to P_h∈Az1min+

P

h∈Czmax1 . Note that

in particular, when B = ∅ in equilibrium, this requires the aggregate demand curve to be flat and equal to s = 0. Therefore, when B = ∅ in equilibrium, this implies that there is no unique solution to the equilibrium equation in (2.14). Instead, there is a set of prices P₁∗ where each p1t ∈ P1∗ solves the equilibrium equation in (2.14).

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Figure 2.1: Aggregate demand and equilibrium price on the first market, for individual restricted demand functions and aggregate supply s = 0.

As an example, Figure 2.1 illustrates a case where aggregate demand equals aggregate supply for any p1t ∈ [4, 5]. To solve this problem of nonuniqueness, in these cases

p1t := arg minx∈P1∗|x − p

∗

1t| is chosen. In other words, in this case the price of the

first asset is defined as the solution to the equilibrium equation that is closest to the fundamental price.

To determine which prices solve the equilibrium equation in case B = ∅ in equi-librium, once again Figure 2.1 is used as an example. First, note that the aggregate demand function is monotonically decreasing in p1t. Once again, let P1∗ be a set of

prices such that for each p1t ∈ P1∗ the equilibrium equation in (2.14) is solved. Since

the aggregate demand function is monotonically decreasing in p1t, the set P1∗ can be

represented by the interval P₁∗ = [P_{1,lef t}∗ , P_1,right∗ ]. From Figure 2.1, it can be seen that P_{1,lef t}∗ equals the largest kink point in the individual demand of all agents in A, whereas P_1,right∗ equals the smallest kink points in the individual demand of all agents in C. Let (2.12) be the unrestricted demand of agent h for the first asset, then the restricted demand of agent h shows a kink at z1ht(p1t) = zmax1 and z1ht(p1t) = zmin1 , which occurs

at prices pkink(1)_1ht = 1 R f1ht+ d1t− a(1 − ρ 2 12)σ1t2zmax1 − ρ212tσ1tσ2t[f2ht+ d2t− Rp2t] and pkink(2)_1ht = 1 R f1ht+ d1t− a(1 − ρ 2 12)σ21tz1min− ρ212tσ1tσ2t[f2ht+ d2t− Rp2t] ,

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respectively. Using the above definitions for the kink points in agents’ demand functions, the set of solutions to the equilibrium equations can be represented by the interval

P₁∗= max h∈A p kink(2) 1ht , min_h∈Cp kink(1) 1ht .

To choose the price p1t∈ P1∗ that is closest to the fundamental price p∗1t, three cases

can be considered: p1t=         

maxh∈Apkink(2)_1ht , if p∗1t< maxh∈Apkink(2)_1ht ,

p∗_1t, if p∗_1t∈ P₁∗,

minh∈Cpkink(1)_1ht , if p∗1t> minh∈Cpkink(1)_1ht ,

Now all that remains is to determine for which strategy types the constraints are binding in equilibrium. This is done by following an algorithm that can be summarized by the following steps. First, the strategy types are ordered by demand for the first asset in ascending order. To order the demand functions, the demand functions can be evaluated at an arbitrary price since the demand is linear in p1t. Next, it is determined

which types are in set A. To do this, it is assumed that all types are in A, and subsequently a temporary equilibrium ˆpt is computed. If, for the first type in the

ordered list, z1ht(ˆp1t, ˆp2t) > z1min, this type is added to B and prices are computed

again. This process is repeated for subsequent ordered types until a type is found for which the constraint is binding or until there are no types left in A.

To determine which types are in set C, a similar process is followed. First, all types that were assigned to set B are now assumed to be in set C. Next, temporary prices are computed and in reversed order, it is determined for each subsequent type in C whether the constraint z1ht(ˆp1t, ˆp2t) ≤ z1max is binding. If the constraint is not binding, the type

is assigned to set B and new prices are computed. After these iterations, definite sets A, B and C are obtained and equilibrium prices can be computed.

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2.2.3 Complete dynamic model

With the exception of individual demand and asset prices, the model in this section remains the same as the model described in Section 2.1. The dynamic model can therefore be described by the following state vectors:

                                       pt = (p1t, p2t)0, p∗_t = _R−11 dt, ut = δut−1+ (1 − δ)pt, Vt = δVt−1+ (1 − δ)(pt− ut−1)(pt− ut−1)0, n1t = exp β ˜U1,t−1 /Nt, Nt≡P4h=1exp β ˜Uh,t−1 , n2t = exp β ˜U2,t−1 /Nt, n3t = exp β ˜U3,t−1 /Nt, dt = dt−1+ σζζt, (2.17) where p1t =                              if B 6= ∅ : _Rn1 B " P

h∈Bf1htnht+ nBd1t+ a(1 − ρ212t)σ1t2(nAz1min+ nCz1max)

−ρ_12tσ1tσ2t−1 (nA+ nC)Ph∈Bf2htnht− nBPh /∈Bf2htnht # , if B = ∅ :         

maxh∈Apkink(2)_1ht , when p∗1t< maxh∈Apkink(2)_1ht

p∗_1t, when p∗_1t∈ P₁∗

minh∈Cpkink(1)_1ht , when p∗1t> minh∈Cpkink(1)_1ht

, p2t = 1 R " ₄ X h=1 f2htnht+ d2t # , ˜ zht =         

(z₁min, ˜z2ht(z1min))0, when z1ht < zmin1

(z1ht, z2ht)0, when z1min < z1ht < z1max

(z₁max, ˜z2ht(z1max))0, when z1max< z1ht,

˜ z2ht(˜z1ht) = 1 aσ2_2t(f2ht+ d2t− Rp1t− aρ12tσ1tσ2tz˜1ht) , πht = ˜z0h,t−1[pt+ dt− Rpt−1] − a 2˜z 0 h,t−1[(1 − λ)Ω0+ λVt−2]˜zh,t−1, Uht = πht− Ch, h = 1 . . . 4,

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with A, B and C being determined by the algorithm described in the previous section, pkink(1)_1ht = 1 R f1ht+ d1t− a(1 − ρ 2 12)σ21tz1max− ρ212tσ1tσ2t[f2ht+ d2t− Rp2t] pkink(2)_1ht = 1 R f1ht+ d1t− a(1 − ρ 2 12)σ21tz1min− ρ212tσ1tσ2t[f2ht+ d2t− Rp2t] ,

and the remaining definitions identical to Section 2.1.5.

In summary, an adaptively heterogeneous beliefs model is proposed that allows for both an upper and lower bound on individual demand. In the next two sections, the model without restrictions is analysed for the deterministic and stochastic case, respec-tively. After that, the dynamics of the model are analysed when trading restrictions are imposed.

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3 Results and Analysis

3.1 Dynamics of the Deterministic Model

In this section, the dynamics of the model without trading restrictions are analysed when dividends are assumed to be constant. Section 3.2 expands on this analysis by introducing stochastic dividends. In Section 3.3, the impact of trading restrictions will be discussed. First, I will show that the model has a unique steady state, after which the local stability of the steady state is analysed. Next, I analyse qualitative changes or bifurcations of the model and analyse the model dynamics for fixed parameters. 3.1.1 Model stability

Assume that dividends and therefore fundamental prices are constant, such that dt= ¯d

and p∗_t = p∗. Then the system of (2.10) reduces to the deterministic system                              pt = _R1 h P4 h=1fhtnht+ ¯d − a[(1 − λ)Ω0+ λVt−1]s i , ut = δut−1+ (1 − δ)pt, Vt = δVt−1+ (1 − δ)(pt− ut−1)(pt− ut−1)0, n1t = exp β ˜U1,t−1 /Nt, Nt≡P4h=1exp β ˜Uh,t−1 , n2t = exp β ˜U2,t−1 /Nt, n3t = exp β ˜U3,t−1 /Nt, (3.1)

Analysing this so-called deterministic skeleton of the model is aimed at gaining initial insight in the effect of parameters on the dynamics of the model. For the case of two risky assets, the system in (3.1) is a 17-dimensional dynamical system3. The system has a unique steady state at (pt, ut, Vt, n1t, n2t, n3t) = (p∗, p∗, 0, n∗1, n∗2, n∗3), where the

fundamental price is given by

p∗= 1

R − 1(¯d − aΩ0s),

and the fractions are given by n∗_h= exp(−βCh)/P_hexp(−βCh) (∀h).

Proposition 3.2. For the system (3.1), we have that

(i) If R ≥ γj for all j ∈ {1, 2}, then the fundamental steady state (p∗, p∗, 0, n∗1, n∗2, n∗3)

of the system is always locally asymptotically stable.

(ii) If R < γj for some j ∈ J0 ⊆ {1, 2}, then the steady state is locally asymptotically

3

The state vectors pt, ut, Vt, n1t, n2tand n3tcan be expressed in terms of pt−1, pt−2, pt−3, ut−1,

Vt−1, Vt−2, n1,t−1, n2,t−1 and n3,t−1, which have N , N , N , N , N (N + 1)/2, N (N + 1)/2, 1, 1 and 1

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stable if, for all j ∈ J0,

Gj(β) < (R − αj)/(γj − αj),

where Gj(β) represents the fraction of agents that follow the chartist strategy on

market j at the fundamental steady state. In particular, G1(β) := n∗3t+ n∗4t= exp(−βC3) + exp(−βC4) P4 h=1exp(−βCh) , G2(β) := n∗2t+ n ∗ 4t= exp(−βC2) + exp(−βC4) P4 h=1exp(−βCh) .

A pitchfork bifurcation occurs at β = β∗ where Gj(β∗) = (R − αj)/(γj− αj) for

j ∈ J0.

Proof. See Appendix A.

Proposition 3.2 implies that the local stability of the fundamental steady state depends on the intensity of choice β and the parameters α and γ, but is not directly affected by the variance-covariance matrix Ω0. Note that the functions Gj(β) are monotonically

increasing in β since C4< C2, C3 < C1, so if R < γj for j ∈ {1, 2}, then the system will

be stable for low values of β and become unstable for high values of β. More specifically, violation of the condition Gj(β) < (R − αj)/(γj− αj) for some j ∈ {1, 2} will result in

instability of the fundamental steady state price of asset j.

Furthermore, Proposition 3.2 shows that, when R < γj for j ∈ {1, 2}, an increase in

α or γ has a destabilizing effect on the fundamental steady state. This is an intuitive result, since an increase in α means fundamentalists are less inclined to push prices towards the fundamental steady state and instead act more like traders with naive expectations, whereas an increase in γ means chartists are more inclined to drive prices away from the fundamental steady state.

3.1.2 Stability regions

Consider Proposition 3.2. When Gj(β∗) = (R−αj)/(γj−αj), a bifurcation or qualitative

change of the dynamics of the system occurs. More specifically, a pitchfork bifurcation occurs due to one of the eigenvalues of the characteristic polynomial being equal to χ = 1 at β = β∗, which becomes clear from the proof of Proposition 3.2 (see Appendix A). A pitchfork bifurcation implies that the fundamental steady state becomes locally unstable and two additional locally stable non-fundamental steady states are created.

To better understand what happens for β ≥ β∗, consider the following example of two risky assets where

Ω0= σ2 1 ρ12σ1σ2 ρ12σ1σ2 σ22 ! (3.2)

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(a) Price of asset one (b) Price of asset two

(c) Largest Lyapunov exponent (asset one) (d) Largest Lyapunov exponent (asset two)

Figure 3.1: Bifurcation diagrams of the first (a) and second (b) risky asset with respect to β and the corresponding largest positive Lyapunov coefficient exponents (c and d).

with σ1 = 0.6, σ2= 0.4 and ρ12= 0.5 and furthermore ¯d = (0.1, 0.1)0, α = diag(0.1, 0.1),

γ = diag(1.6, 1.2), a = 1, r = 0.025 and s = (0.1, 0.1)0, For the strategy costs I set C1 = 2, C2 = C3 = 1 and C4 = 0, and finally λ = 0.5 and δ = 0.98. These parameter

values imply that there exists a fundamental value p∗ = (p∗₁, p∗₂) which becomes unsta-ble as the intensity of choice β increases since R < γj for j ∈ {1, 2}. More specifically,

from Proposition 3.2 it follows that a bifurcation of the system occurs at β₁∗ ≈ 0.4754 and β₂∗ ≈ 1.6650. Figure 3.1 shows bifurcation diagrams of the model with respect to β. From Figure 3.1 it can be seen that in fact for β₁∗ < β < β₂∗, the fundamental steady state price of the first asset is unstable but the fundamental steady state price of the second asset remains stable. This result is a consequence of the different values for γ1 and γ2, which imply that chartists extrapolate more on the first market than

on the second market. Note that the bifurcation diagrams in Figure 3.1 in fact show two attractors on each market, one above and one below the fundamental price. The bifurcation diagrams are generated by simulating the model for N = 1000 periods for each β, after an initial start-up period. The areas in the diagram represent the price

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 3.2: Time series plots of pt and p∗t (a-c), and fractions of

fun-damentalists and chartists on the first (d-f ) and second market (g-i) for β = 1.0 (a, d and g) β = 1.6 (b, e and h) and β = 2.2 (c, f and h).

values that have been obtained from the simulations.

Figure 3.1 also reveals information about further bifurcations of the dynamical sys-tem. At β₁∗∗≈ 1.3883, a secondary Hopf bifurcation occurs where both non-fundamental steady states of the first asset price lose stability and branch into limit cycles. This bi-furcation is of particular interest as it results in quasi-periodic price fluctuations of the first asset. For β₁∗∗< β < β₂∗, the fundamental steady state of the second asset remains stable even as the price of the first asset starts to fluctuate. The price dynamics for these different qualitative cases of β are illustrated in Figure 3.2.

Figure 3.2 shows the fluctuations of asset prices and the fractions of fundamentalists and chartists on each market, given by (1−Gj(β)) and Gj(β), respectively, for market j.

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The figure reveals that even though the fractions n1,t, n2,t, n3,t and n4,t are unstable for

β = 1.6, the fraction of chartists that are active on the second market, given by n3t+ n4t

in fact remains constant. In other words, in this particular case, the correlation between the assets does not cause instability to spill over from one market to another. However, with respect to possible spillover effects, the following note can be made about Figure 3.1. Unlike the primary bifurcation of the first asset, the primary bifurcation of the second asset is not followed by a stable region of two non-fundamental steady states. Instead, the dynamics become chaotic almost immediately after the primary bifurcation, as clearly seen in the plot of the largest Lyapunov exponent. This phenomenon will be analysed further in the next section.

Figure 3.3 shows the stability regions of the first and second asset, subject to the parameters β, α1, γ1, costs of fundamentalists on the first market, and the correlation

coefficient ρ12. The stability regions are based on the largest Lyapunov coefficient

exponent (LCE), where the dynamics are considered to be unstable for all cases where LCE > 0. The figure also shows the boundaries of the stability regions, where a Hopf bifurcation occurs. It becomes clear from Figure 3.3 that the stability of the system after the first bifurcation depends on the values of the parameters β, α, γ, Ch, ρ12and

λ. In the next section, the sensitivity of the system to changes in these parameters is discussed in more detail.

3.1.3 Sensitivity to parameter changes

The previous sections give initial insight in the stability of the fundamental steady state and its dependency on the intensity of choice, β. In this section, the stability of the dynamical system in (3.1) after the primary bifurcation is discussed, in particular the effect of the model parameters α, γ, Ch, ρ12 and λ on the dynamics of asset prices. To

investigate these parameters, the same initial state vectors and parameters are used as defined in section 3.1.2.

Mean reversion and trend extrapolation. From Proposition 3.2 it can be derived that an increase in α leads to an earlier occurence of the first bifurcation of asset prices. As discussed in section 3.1.1, this is an intuitive result since the stabilizing force of the fundamentalists is smaller for high values of α. Similarly, from Proposition 3.2 it follows that an increase in γ leads to an earlier occurence of the first bifurcation of asset prices. The intuition behind this result is that for higher values of γ, the chartists have predictions of prices that deviate farther from the fundamental prices, and thereby their destabilizing effect on prices is stronger.

To investigate the marginal effect of αj and γj for j ∈ {1, 2} after the primary

bifur-cation of prices, time series plots of the asset prices are analysed. Numerical simulations reveal that an increase in αj or γj leads to larger price oscillations on market j, while

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(a) Stability of asset one subject to α1 (b) Stability of asset two subject to α1

(c) Stability of asset one subject to γ1 (d) Stability of asset two subject to γ1

(e) Stability of asset one subject to c1 (f) Stability of asset two subject to c1

(g) Stability of asset one subject to ρ12 (h) Stability of asset two subject to ρ12

Figure 3.3: Stability regions for the first (a, c, e and g) and second asset (b, d, f and h) with β on the horizontal axis and α1 (a-b), γ1 (c-d), c1 (e-f )

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(a) (b) (c)

(d) (e) (f)

Figure 3.4: Time series plots of pt and p∗t for the case of ρ12 = 0 (a-c)

and ρ12 = 0.5 (d-f ), with c2 = 1 (a and d), c2 = 1.5 (b and e) and c2 = 2

(c and f ). Furthermore, β = 3 and λ = 0 in panels a-f .

assets offer further insight in the marginal effect of αj and γj on dyamics. The top

panels of Figure 3.3 show the stability regions for the first and second asset subject to α1 and γ1, with ρ12 = 0.5. It can be seen that when assets are considered to be

correlated, the mean reversion and trend extrapolation parameters of the first market affect the stability regions of both markets. In particular, an increase in α1 or γ1 leads

to an earlier occurence of the Hopf bifurcation on both markets.

Strategy costs. To analyse the effect of strategy costs on price dynamics, recall the assumption that C1 = c1+ c2, C2 = c1, C3 = c2 and C4 = 0, where cj with j ∈ {1, 2}

is a parameter representing the marginal cost of applying a fundamentalist strategy on market j. If cj increases, it becomes relatively more attractive to apply a chartist

strategy on market j. In particular, Proposition 3.2 states that when cj increases,

ceteris paribus the fraction of chartists on both markets at the fundamental steady state increases as well. Equivalently, an increase of cj leads to an earlier occurence of

the primary bifurcations of asset prices.

To determine the marginal effect of an increase of cj on one market, two cases are

considered. The first case is the case of no correlation between assets, with ρ12 = 0

and λ = 0. In other words, the first case represents the case of fixed variance and zero covariance between assets. The top panels of Figure 3.4 show the prices of both assets

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 3.5: Bifurcation diagrams of the second asset price (a-c), pt and

p∗_t (d-f ), and fractions of fundamentalists and chartists on the second market (g-i) for ρ12= −0.5 (a, d and g), ρ12= 0 (b, e and h) and ρ12= 0.5

(c, f and i), with β = 3 for d-i.

for increasing costs of the fundamentalist strategy on the second market. The time series plots show that without any perceived correlation between assets, an increase of c2 leads to larger price fluctuations on the second market, but has no effect on the prices

on the first market.

The second case is the case of correlated assets, where in particular ρ12= 0.5. The

lower panels of Figure 3.4 show the prices of both assets in this case, for increasing costs of the fundamentalist strategy on the second market. It can be seen that when assets are correlated, an increase of c2 leads to larger price fluctuations on the first market as

well as the second market. In other words, when assets are correlated, a spillover effect can be observed in the system. The implications of this phenomenon are discussed in

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(a) Price of asset one (b) Price of asset two

Figure 3.6: Bifurcation diagrams of the first (a) and second (b) risky asset with respect to ρ12, with β = 3 and λ = 0.5.

the next paragraph.

Correlation between assets. To analyse the effect of correlation between assets on price dynamics, the bifurcation pattern of the second asset is considered. Figure 3.5 shows the bifurcation diagrams and time series plots of the second asset for ρ12 =

−0.5, ρ12 = 0 and ρ12 = 0.5. For ρ12 = 0, when assets are uncorrelated, the primary

bifurcation yields two stable non-fundamental steady states. However, when the assets are correlated, i.e. ρ12 6= 0, prices are unstable after the primary bifurcation. In the

latter case the correlation between assets causes the instability of the first market to spill over to the second market for β ≥ β₂∗, as can also be seen in the bottom panels of Figure 3.3. An explanation for this can be found in the performance measure. When ρ126= 0, fluctuations in the prediction error on the first market affect the performance

measure of agents who are fundamentalists on the second market differently than of those who are chartists on the second market.

In addition to a qualitative change in dynamics, numerical simulations also show a quantitative effect of correlation on price fluctuations. The bifurcation diagrams in Figure 3.6 with respect to ρ12 show that an increase (in absolute value) of ρ12 leads to

increases in the fluctuations of prices when the system is unstable. This is an intuitive result, since high perceived correlation between risky assets means diversification is less effective and instability on one market may spill over to other markets. This spillover effect is discussed more extensively in the next sections.

Beliefs about conditional variance. With respect to the effect on correlation on price dynamics, it needs to be noted that ρ12, or Ω0 in general, affects the price

dy-namics through the perceived variance-covariance matrix Ωt, which depends on the

parameter λ. Therefore, it is important to determine the role of the parameter λ in the aforementioned price fluctuations. Consider the example of positive correlation between

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(a) Price of asset one, ρ12= 0 (b) Price of asset two, ρ12= 0

(c) Price of asset one, ρ12= 0.5 (d) Price of asset two, ρ12= 0.5

Figure 3.7: Bifurcation diagrams of the first (a and c) and second (b and d) risky asset with respect to λ, where ρ12= 0 for a and b, ρ12= 0.5 for c

and d and β = 3.

the risky assets, with ρ12 = 0.5 and β = 3. The lower panels in Figure 3.7 show the

bifurcation diagrams of the risky assets with respect to λ for ρ12 = 0.5. The figures

reveal that an increase in the value of λ causes a decrease in the price fluctuations on both markets. To see why this is the case, time series of asset prices and perceived variance are simulated for λ = 0, λ = 0.5 and λ = 1. The time series plots in the bottom panels of Figure 3.8 show that when λ increases, the beliefs about correlation do not change much, but the perceived variance of both asset decreases significantly. Apparantly, based on an EWMA prediction rule, agents predict variance terms in Vt

that are significantly lower than the fixed variance terms in Ω0. As a result of smaller

perceived variance for both assets, the fluctuations in asset prices are also smaller. To determine the effect of λ on asset prices when the risky assets are assumed to be uncorrelated, the bifurcation diagrams with respect to λ are analysed for the case where ρ12= 0. The upper panels in Figure 3.7 show that in this case, the two non-fundamental

steady states of asset two are stable for any λ > 0. The spillover effect that is observed when ρ12 6= 0 therefore remains absent when ρ12 = 0, regardless of the value for λ.

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

Figure 3.8: Time series plots of ptand p∗t (a-c), fractions of

fundamental-ists and chartfundamental-ists on the first (d-f ) and second market (g-i) and conditional variance and correlations (j-l) for λ = 0 (a, d, g and j) λ = 0.5 (b, e, h and k) and λ = 1 (c, f , i and l), with ρ12= 0.5.

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However, due to smaller perceived variance on the second market, the non-fundamental steady states are closer to the fundemantal price for larger values of λ. With respect to the first asset, similar to the case of correlated assets, larger values of the parameter λ lead to smaller price fluctuations.

In summary, the parameters α, γ, ci and β have both a qualitative as well as a

quantitative effect on the stability of the system. In particular, higher values of the aforementioned parameters lead to more instability. The perceived correlation ρ12

de-termines to which extent the price on one market affects the other. Higher perceived correlation leads to more volatility on both markets. Finally, the parameter λ deter-mines to which extent perceived variance is taken into account. In general, a higher value of λ leads to lower volatility on both markets since the perception of agents about variance is low.

3.2 Dynamics of the Stochastic Model

In this section, the dynamics of the model in (2.10) are analysed, where the dividends now follow a martingale process. For the standard deviation of the dividend process, σζ = diag(0.002, 0.002) is chosen. For the other paramaters, the same initial values

are used as defined in Section 3.1, with the exception of the extrapolation parameter, which is now set at γ = diag(1.3, 1.2), for the convenience of having price oscillations that are similar in size. In te first part of this section, the presence of spillover effects is investigated. Thereafter, realized volatility and trading volume are discussed. Finally, underlying contagion risk factors are identified.

3.2.1 Spillover effects

In this section, the presence of spillover effects as described in Section 3.1.3 is investi-gated for the case of stochastic dividends. A general definition of a spillover effect is the presence of externalities of a process that affect those who are not directly involved. In this thesis, a more narrow definition is used when referring to spillover effects. More specifically, the following definition is considered.

Definition 3.1. A spillover effect in the context of the dynamical system in (2.10) is defined as the occurrence of one of the following two circumstances:

(i) The instability of prices in a correlated market that would otherwise be stable under the assumption of no correlation (i.e. ρ12= 0).

(ii) Increased volatility of prices in a correlated market compared to the volatility under the assumption of no correlation.

In other words, spillover effects are considered to represent the risk of financial conta-gion. With respect to the assumptions about the correlation coefficient, it is important

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(a) Prices for ρ12= 0 (b) Prices ρ12= 0.5

Figure 3.9: Time series plots of pt and p∗t for ρ12= 0 (a) and ρ12= 0.5

(b), with β = 3.

to be clear about the interpretation. Assumption 2.2 states that the beliefs of agents about the variance-covariance matrix of future prices may vary over time, but revolve around the fixed variance-covariance matrix Ω0. Note that this implies that although

assumptions are made about Ω0 and therefore ρ12, these assumptions concern the

be-liefs of agents about future variance and do not necessarily enforce correlation in the realization of prices other than through the price expectations of agents. In particular, recall that the dividend processes underlying the fundamental prices of the risky assets are i.i.d. and therefore uncorrelated. In other words, the results in this section are con-ditional on the beliefs of agents about correlation, but do not rely on the existence of actual correlation between prices. This is an important distinction as it implies that the results about spillover effects may be applicable to a wide range of asset classes.

To investigate the phenomena described in Definition 3.1, the time series plots of asset prices are analysed for the model with and without perceived correlation. Figure 3.9 shows the prices of both assets for ρ12 = 0 and ρ12= 0.5, respectively, with β = 3.

In the case without correlation, both prices fluctuate above their fundamental price, with the price of the second asset closely following the oscillations of the fundamental price. This case resembles the case of ρ12= 0 in Figure 3.5, where the deviation from

the fundamental price of asset two remains constant while the price of the first asset shows larger fluctuations. In the case of strongly correlated assets, the price of the second asset shows moer irregular deviations from the fundamental price, which can be seen in the right panel of Figure 3.9. The figure also shows that the peaks in the price of the second asset coincide with the peaks in the price the first asset. This is evidence of a spillover effect as defined in Definition 3.1, as the correlation between assets leads to increased volatility and simultaneous price bubbles on both markets.

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(a) Positions in asset one, ρ12= 0 (b) Positions in asset one, ρ12= 0.5

(c) Positions in asset two, ρ12= 0 (d) Positions in asset two, ρ12= 0.5

Figure 3.10: Positions in the first (a and b) and second asset (c and d) for ρ12= 0 (a and c) and ρ12= 0.5 (d and e), with β = 3.

To analyse the difference in dynamics between the two cases described above, the individual traders are analysed. Figure 3.10 shows the positions of each strategy type for ρ12= 0 and ρ12= 0.5. For the case where ρ12= 0, it can be seen that the positions

in the first asset of the first and second type coincide, idem for the third and fourth type. Similarly, the positions in the second asset of the first and third type coincide, idem for the second and fourth type. This implies that when assets are not correlated, the individual demands of agents on a market where they share the same belief are equal, regardless of their beliefs about other markets. The right panels of Figure 3.10 show that when ρ12 = 0.5, the individual demands of agents on a market where they share

the same belief differ depending on their beliefs about the other market. This implies that there is a spillover of demand to the correlated market, which results in the price dynamics observed in Figure 3.9.

Note that the peaks in demand coincide with the peaks in asset prices, which rein-forces the hypothesis of this thesis that trading restrictions may increase stability and mitigate spillover effects. In the next section, the trading volumes are analysed in more detail.

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3.2.2 Volatility and trading volume

In this section, the relation between price volatilities and trading volumes is investigated. Following Chiarella et al. (2013), the absolute price difference |pt− pt−1| is used as a

proxy for volatility on the markets. The trading volume is defined as the sum of absolute differences of positions of agents between period t and t − 1:

Zt=

X

h

min{nh,t−1, nht}|zht− zh,t−1| + ˜Zt, (3.3)

where the first part of the equation represents the volume traded by agents who did not switch strategy types and ˜Ztrepresents the volume traded by agents who switched

strategies between period t and t − 1. To compute the value of ˜Zt, it needs to be

determined between which strategy types migration has taken place, which is done by applying the following algorithm. Initially, let ˜Zt:= 0. The migration per strategy type

is computed as the difference of the fractions between period t and t − 1: ˜

nht:= nht− nh,t−1.

Note that ˜nht may be positive or negative, depending on whether agents migrated

towards or away from the strategy type. If ˜nht is positive for some h, an alternate

strategy type is located for which ˜njt < 0 holds. At this point, it is assumed that a

fraction of agents min{|˜nht|, |˜njt|} migrated from strategy j to strategy h4, yielding an

additional trading volume ˜

Zt:= ˜Zt+ min{|˜nht|, |˜njt|}|zht− zj,t−1|

The migration is accounted for in the fractions by setting

˜ nht:=    ˜ nht− min{|ñht|, |ñjt|}, if ñht > 0 ˜ nht+ min{|ñht|, |ñjt|}, if ñht < 0

This process is repeated for each strategy type until all migrations are accounted for, i.e. ˜nht = 0, ∀h. The resulting value of ˜Zt is substituted in equation (3.3) to yield the

absolute trading volume for period t.

To analyse the dynamics of the volatilities and trading volumes, time series plots of |∆pt| and Zt are analysed for the two risky assets. Figure 3.11 shows the volatilities

and trading volumes of both assets for ρ12 = 0 and ρ12 = 0.5. As observed in the

previous section, spikes in the trading volume are accompanied by increased volatility. Furthermore, volatility and trading volume are higher for the case of ρ12= 0.5 compared

4_{Note that this assumption can be interpreted as the lower bound estimation of actual trading}

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(a) First market for ρ12= 0 (b) First market for ρ12= 0.5

(c) Second market for ρ12= 0 (d) Second market for ρ12= 0.5

Figure 3.11: The trading volumes and volatilities of the first (a-b) and second asset (c-d) for ρ12 = 0 (a and c) and ρ12 = 0.5 (b and d), with

β = 3.

to the case of no correlation. In particular, for ρ12= 0 the trading volume on the second

market is close to zero, whereas for ρ12= 0.5 the trading volume on the second market

shows large spikes that coincide with increased trading volume on the second market. In fact, for ρ12 = 0.5, the trading volumes and volatilities as presented in Figure

3.11 appear to be strongly correlated, corresponding to the expectations of agents about correlation. To investigate the relationship between expectations about correlation and realized correlation, simulations are done for ρ12 ∈ [−1, 1]. The top panels of Figure

3.12 show the realized correlation between volatility and trading volume on a particular market, based on 100 simulations. It can be be seen that for ρ12 = 0, volatility on a

market is strongly correlated with trading volume on the same market. However, when |ρ12| increases, the realized correlation between trading volume and volatility of the

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(a) Correlation between volatility and trading volume for asset one

(b) Correlation between volatility and trading volume for asset two

(c) Volatility correlation (d) Trading volume correlation

Figure 3.12: The correlation between trading volumes and volatilities for the first (a) and second asset (b) and the correlations of price volatilities (c) and trading volumes (d) of the risky assets, where β = 5.

same market decreases. This is an intuitive result, since if |ρ12| increases, the alternate

market become increasingly important, therefore trading volume on a market plays a smaller role in the formation of prices on the same market.

From the bottom panels in Figure 3.12, it can be seen that an increase in |ρ12| results

in increased realised correlation, of both volatility and trading volume, between markets. The beliefs of agents about correlation, through the feedback effect of the model, thus serve as a self-fulfilling prophecy. This self-fulfilling prophecy poses a contagion risk under certain market conditions, resulting in the spillover effects illustrated in Section 3.2.1. The market conditions underlying this contagion risk are summarized in the next section.

3.2.3 Contagion risk factors

The results brought forward in the previous sections involve the instability of asset markets, which have been shown to spread through the realisation of volatility spillovers. The spreading of instability of a market to one more connected markets is called financial contagion. This section is aimed at listing the factors underlying the risk of financial

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 3.13: Time series plots of pt and p∗t (a-c) and positions in the

first (d-f ) and second asset (g-i) for β = 1 (a, d and g), β = 1.4 (b, e and h) and β = 1.8 (c, f and i) with ρ12= 0.8.

contagion. Based on the results of the analyses in the previous sections, the following risk drivers are highlighted: (i) instability of fundamental prices, (ii) perceived correlation between assets and (iii) trading volume on the market. The remaining part of this section elaborates on the influence of each risk driver on the stability of financial markets. Instability of fundamental prices. Section 3.2.1 describes the occurrence of volatil-ity spillovers when markets are correlated. In particular, the right panel of Figure 3.9 shows simultaneous price bubbles on correlated markets conditional on β = 3. For this value of the intensity of choice, the fundamental steady state price is unstable on both markets. Figure 3.13 shows the prices and positions of agents for increasing values of

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(a) (b) (c)

(d) (e)

Figure 3.14: Time series plots of pt and p∗t (a), fraction on the first (b)

and second market (c) and positions in the first (d) and second asset (e), with β = 5 and ρ12= −0.8.

β. For β = 1 (the left panels), the fundamental steady state price of the second asset is locally stable and no spillover effects can be observed. For β = 1.8 (the right panels), the fundamental steady state of both assets are locally unstable and the correlation between the assets causes financial contagion.

The simulations in Figure 3.13 show that instability of the fundamental steady state is a necessary condition and a driving factor of financial contagion. In fact, financial contagion only occurs when the fundamental prices on both markets are unstable. This is illustrated in the middle panels of Figure 3.13, where the fundamental price of the first asset is unstable for β = 1.4, but the fundamental price of the second asset is stable and no financial contagion arises. Furthermore, Section 3.1 concluded that higher values of β, α as well as γ lead to larger price oscillations. These results imply that increased activity on a market, either through a high intensity of choice, weak fundamentalist beliefs or strong trend extrapolation poses an increased risk of financial contagion.

In the context of financial turmoil, the above results pose an additional interpre-tation. During times of crisis, the intensity of agents to choose the best strategy may increase, as a sign of panic. In the dynamical system in (2.10), this is equivalent to an increase in β. The above results imply that this increased intensity of choice not only causes additional volatility on markets, but may also enforce financial contagion.

Contagion risk and spillover effects in asset prices

Contagion Risk and Spillover Effects in Asset

Prices

Contents

1

Introduction

2

The Model

3

Results and Analysis