The validity of heterogeneous agent models in assessing stock market indices : a study of mean-reversion and volatility clustering

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Master’s Thesis

The Validity of Heterogeneous Agent

Models in Assessing Stock Market Indices

A Study of Mean-Reversion and Volatility Clustering

Mark A. Nothdurft

Student number: 10826289 Date of final version: July 1, 2015 Master’s programme: Econometrics

Specialisation: Mathematical Economics Supervisor: Prof. C.H. Hommes

Second reader: M.C.W. van de Leur, PhD

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Statement of Originality

This document is written by Mark A. Nothdurft who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is 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 supervision of completion of the work, not for the contents.

Acknowledgements

I would like to thank Michiel van de Leur. His constant support, guidance and expertise in the field of mathematical economics have made it possible for me to undertake and complete this thesis. Michael provided constructive criticism at all stages and encouraged me to extend my ideas and to aim high with the study and the thesis content. Importantly, working with Michiel was always enjoyable as his approach is friendly and collaborative.

I would also like to thank Professor Cars Hommes for his expertise and time. He suggested a number of improvements and possible directions for the thesis, all of which were invaluable.

The experience, expertise and contribution from Michiel and Cars assisted me to deliver a more comprehensive final product.

I have met many people during my time in Amsterdam and I have made many wonderful friends and colleagues. I thank them for making my stay in the city memorable and for making me feel welcome while away from my home in Australia. I hope that we cross paths again in the future, professionally and personally.

I would like to acknowledge my parents, John and Simone and brother, Rhett, for their unwavering love and assistance throughout this important chapter of my life. While I was studying and preparing this thesis, the regular contact provided by them as well as my close Australian friends was an asset that I will always remember and cherish.

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Introduction

The framework of asset pricing models is founded on the Efficient Market Hypothesis (EMH). It is the cornerstone of standard economic and financial theory and widely regarded as a consis-tent hypothesis (Jensen, 1978). It assumes that stock market prices incorporate and reflect all relevant information. Under this hypothesis, stock prices are presumed to maintain their fun-damental value at all times which means agents are unable to earn profits by purchasing stocks when they are undervalued or selling stocks when they are overvalued. Purchasing riskier assets is the only way to earn higher returns under the EMH because it implies there is no opportu-nity for agents to outperform the market. Theoretically, optimal stock selection, market timing and speculative trading provide no incentive to traders and should be non-existent. However, statistical analysis of historical data and the sheer volume of trading on stock markets prove that the EMH relies on an unrealistic assumption.

Additionally, the EMH has been heavily criticised in response to the recent “Dot-com bub-ble”. Rapid technology development was the catalyst for extraordinary growth in internet stocks in the late 1990s. An index monitoring internet stock reportedly grew over 1000% in the space of two years. The bubble was characterised by: highly inflated market prices, speculative frenzies, and an eventual crash in prices. This event has been compared to the earliest recorded price bubble; the Dutch “Tulip Mania”. Tulips were a highly coveted asset in seventeenth century Holland and triggered expansive speculative trade and remarkable price growth (Mackay, 1841). In both instances, changes in fundamental market prices did not provoke such expansive dy-namics. This evidence proposes the question; “What is the cause of excess asset market returns if it is not attributed to changes in fundamentals”?

Modern theory proposes that the flaws in the efficient market assumption boil down to the existence of excess volatility in financial markets. Excess volatility refers to the situation where the variance in stock prices or interest rates exceeds the upper bounds implied by the variance of dividends and other fundamental measures (Shiller, 1988). It is important to consider that the variance of stock prices often exceeds the implied upper bounds by very large margins. For example, when historical stock prices are analysed and compared to price fundamentals, it is immediately noticeable that the fundamentals significantly underestimate the true variability

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of real-world markets (West, 1988). This evidence directly contradicts the EMH. Moreover, the existence of multiple equilibria in economic models, mean-reversion and volatility clustering raises further doubt about usefulness of the Efficient Market Hypothesis in explaining real world asset price volatility.

Results indicate that agents’ beliefs can have a substantial effect on stock prices and market variability. Specifically, excess volatility is likely to occur in a market where traders adopt heterogeneous beliefs about future prices. Naturally, when agents act on different price forecasts, the volume of trading increases as traders speculate against the beliefs of other traders. This opens up the possibility of earning larger profits without purchasing riskier assets and can lead to significant price movements. In light of this, price changes are now, to an extent, attributed to how agents form their expectations about future prices (Merton, 1980). Social optimism, fads, biases, loss aversion and trend chasing (Thaler, 1994) are a few examples of this contention.

Many asset pricing models that are based on the rational expectations hypothesis (REH) assume market expectations are summarised by a single representative agent because any irra-tional traders will be swiftly driven out of the market by the rairra-tional traders (Friedman, 1953). These models do not allow for price bubbles or deviations from fundamental values. Moreover, this assumption has been challenged in recent years with the introduction of heterogeneous agent models. Naturally, bounded rationality and heterogeneous agent models go hand-in-hand because real-world markets contain traders who form different price expectations. Asset pricing models assume expectations are based on simple, limited and variable information sets. This also fits seamlessly with the concept of herding behaviour as agents may look to maximise their profits by learning from agents with alternative beliefs. Moreover, reinforcement learning and heuristics are considered potential means for forming price expectations in financial mar-kets. They are also a prominent feature of heterogeneous agent models and the tremendous volume of stock trading must imply agents use markets to speculate against the beliefs of other agents (Frankel & Froot, 1990). The above provides theoretical and empirical evidence that dismisses the classical efficient market and rational expectations hypotheses. In contrast, the heterogeneous agent model appears more suited to explain the existence of excess volatility.

Market psychology became the focus of economic models in the 1950’s when Herbert Simon argued that perfect rationality required unrealistic assumptions about an agent’s ability to process information. Bounded rationality is now a dominant area of study where models for asset prices depend on the expectations of every agent in the market. In these models, since the accurate law of motion is unknown to traders, agents form beliefs in response to forecasting rules and historical trends. They implement adaptive learning techniques (Sargent, 1993) and update their beliefs in response to the relative profitability of their prediction. This is typically measured by a predetermined fitness function. In this sense, forecasts are (boundedly) rational because the majority of agents will simplify their decisions by basing their expectations on the predictor that has the highest past performance (Brock & Hommes, 1998).

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infor-mation from every possible source. They often attempt to learn from the price forecasts and performance of others. The herding phenomenon arises in an environment where agents change price forecasts based on the measure that is performing best (Lux, 1995). Herding behaviour, or the contagion effect, corresponds to individuals in a group acting in the same way without centralised direction. It is apparent in every-day decision making, stock markets and animal behaviour. In markets, herding can be a result of growing investor sentiment, irrationality and trend chasing and is a common cause of prices deviating from fundamental values. Specifi-cally, these deviations are in the form of bubbles and crashes, particularly following “large scale herding”.

Another noteworthy characteristic of markets is the tendency for stock prices to display skewness, fat tails and mean-reversion. Mean-reversion refers to the expectation that prices will be closer to their fundamental value in the following period. In other words, it is where the asset prices appear to remain around their fundamental value instead of diverging from them. Thomas Lux implemented a noise trader/infection model to study the creation of cyclical dynamics that are centred around fundamental values. This model was used to study why asset markets exhibit mean-reversion. Volatility clustering is another peculiar market characteristic. This phenomenon is where asset markets sometimes display minimal returns during some periods, but display expansive returns at other times. To this point, the number of theoretical models that are able to produce volatility clustering is scarce. Only simple and stylized models are able to exhibit qualitative volatility clustering (Gaunersdorfer, Hommes, & Wagener, 2008).

This paper applies the asset pricing model developed by Brock and Hommes (1998) to investigate the complicated and irregular price dynamics in financial markets. The model in-corporates heterogeneous agents who form price expectations based on a number of forecasting rules. The success of their forecast is determined by a function of realised profits and this performance measure is then applied to a discrete choice evolutionary process where agents can systematically change their beliefs within the available set. This framework is designed to analyse how market expectations affect the variability of price dynamics.

This thesis first looks to establish whether or not excess volatility and chaotic price dynam-ics are apparent in a simple two-type model consisting of fundamentalists and chartists. By restricting traders to two forecasting rules, the model analyses whether fundamentalists drive out chartists or chartists drive out fundamentalists in the market. Alternatively, whether agents with mixed forecasting rules can coexist with stable price dynamics. Another possible outcome is where the market displays herding behaviour. This phenomenon is explored extensively and the creation of chaotic dynamics through rational forecasting is analysed. It is the aim of this paper to use the asset pricing model with heterogeneous agents to replicate these dynamics and determine the validity of these models.

The model is then extended in an attempt to replicate market price dynamics with a greater number of trader beliefs. Although analytical tractability decreases significantly as more fore-casting rules are added, a four-type model is explored in order to add complexity and investigate

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how traders adopting conflicting, yet logical, expectations can cause excess volatility in the mar-ket. Under this complex environment, price dynamics are monitored to discover if the four-type model is able to accurately model mean-reversion and volatility clustering. Moreover, consid-eration is given to a-synchronous updating and heterogeneity in risk aversion and conditional variance expectations although the thesis first looks to develop a simple model without these advanced components.

The remainder of the thesis is organised as follows. Chapter 2 provides an overview of existing research and explains the theoretical framework for the asset pricing model. Chapter 3 explores a two-type example by analysing the price dynamics of the market with fundamentalists and chartists. The market dynamics are investigated using a four-type model in Chapter 4. This chapter provides a number of extensions to the model implemented in the previous chapter. It includes empirical estimation and outlines a number of important results. Finally, Chapter 5 concludes.

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Chapter 2

Asset Pricing Model with

Heterogeneous Beliefs

2.1 Theoretical Framework

Consider an asset pricing model where agents may choose between investing in a risk-free asset and a risky asset. The model looks to develop a dynamical price mechanism by implementing wealth maximisation, market clearing, heterogeneous beliefs and evolutionary selection. In order to establish a closed form solution for an agent’s wealth, let R = 1 + r > 0 denote the gross rate of return on the perfectly elastically supplied risk-free asset and pt denote the price

(ex-dividend) of the risky asset at time t. Assume yt is a stochastic dividend process for the

risky asset and zh,t is the number of shares of the risky asset demanded by an agent with belief

h. Hence, an individual’s dynamic wealth, Wh,t, can be defined as:

Wh,t+1= R (Wt− ptzh,t) + (pt+1+ yt+1) zh,t

= RWh,t+ (pt+1+ yt+1− Rpt) zh,t. (2.1)

Notice zh,t is a function of the asset price. This price is determined to ensure the market

allows all traders to fulfil their optimal demand for shares. For simplicity, consider the case where all agents adopt a myopic mean-variance maximisation strategy. Using this method, all agents aim to maximise their expected wealth only in the subsequent period while placing an upper bound on the level of risk. In this case, an agent’s optimal demand for shares will depend on their conditional expectations of future wealth, their risk aversion and their forecast of conditional variance of their return. Therefore, let Eh,t[·] and Vh,t[·] denote the beliefs of

agent h about the conditional expectation and conditional variance at time t. Thus, where a > 0 represents an endogenous risk aversion parameter, the following problem solves for an agent’s optimal demand for shares.

max zh,t n Eh,t[Wh,t+1] − a 2Vh,t[Wh,t+1] o

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= max zh,t n Eh,t[RWt+ (pt+1+ yt+1− Rpt) zh,t] − a 2Vh,t[RWt+ (pt+1+ yt+1− Rpt) zh,t] o = max zh,t n RWt+ zh,tEh,t[pt+1+ yt+1− Rpt] − a 2Vh,t[pt+1+ yt+1− Rpt] z 2 h,t o (2.2) Therefore, the utility maximisation problem can be solved by taking the first order conditions with respect to zh,t and setting Vh,t[pt+1+ yt+1− Rpt] = σ2. Section 2.3 provides a specific

explanation for the form of the conditional variance. Consequently, the first order conditions of (2.2) imply: zh,t = Eh,t[pt+1+ yt+1− Rpt] aVh,t[pt+1+ yt+1− Rpt] = Eh,t[pt+1+ yt+1− Rpt] aσ2 . (2.3)

2.2 Market Clearing Condition

In order to establish a suitable pricing equation, the model must not allow for an asset surplus or shortage in the market. Additionally, any displacement should be corrected swiftly. In simple terms, the model assumes the asset market clears at all times. This is a reasonable restriction given the market mechanisms at work, particularly for the large number of high liquidity assets that exist in markets. Following neoclassical economic theory and assuming perfect competition, the market clearing condition is achieved when the aggregate demand for the asset is equal to the aggregate supply. In this case, a cleared market represents a situation where the total quantity of goods sought after by consumers exactly equals the amount produced by suppliers with no residual supply or demand.

Define nh,t to correspond to the fraction of agents of type h and zs as the supply of outside

shares. Additionally, suppose there are H different types of agents in the market. Therefore, using (2.3), the market clearing equation reduces to:

H X h=1 nh,tzh,t = zs H X h=1 nh,t Eh,t[pt+1+ yt+1− Rpt] aσ2 = z s H X h=1 (nh,tEh,t[pt+1+ yt+1] − nh,tRpt) = aσ2zs. (2.4)

The model considers a special case of the market clearing equation. Walrasian market clearing is defined by a constant, zero-supply of outside shares. Thus, assuming zs = 0, equation (2.4) can be rearranged to solve for the equilibrium dynamic pricing equation.

pt= 1 R H X h=1 nh,tEh,t[pt+1+ yt+1] + pt−1t (2.5)

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Note that the independent and identically distributed white noise term, t∼ N(0, ρ) is included

to account for minor shocks to the pricing equation. This error term allows for shocks to the demand or supply of stocks because it may reflect the price adjustment required to maintain equilibrium. Moreover, the error term is scaled by the current asset price because it is reasonable to assume that when prices are higher, shocks to the system should have a greater magnitude compared to when prices are low. In a way, the error terms are based on a percentage of the current price of the asset which ensures they have an equivalent effect on future price movements for any value of pt.

2.3 Model Assumptions

Evidently, any market will be comprised of a number of boundedly rational agents each imple-menting a combination of heuristics and forecasting rules to make investment decisions. There are effectively an infinite number of ways that an agent can form price expectations. Therefore, it is important to clarify a number of key assumptions about the model and the beliefs of an agent of type h at this point. Each assumption is designed to simplify and restrict the form of the model. This ensures the effect of an agents’ expectations on price dynamics can be isolated and analysed.

Assumption I: Homogeneous Conditional Variance Expectations

Equation (2.3) utilises the model’s first major assumption. It states that agents maintain identical and constant beliefs about the conditional variance of returns. This assumption is based on the hypothesis that agents tend to form a widespread census in regards to the variability of stock markets. In light of this, a homogeneous conditional variance measure is included to isolate the effect of an agent’s price expectations. An algebraic form for agents’ estimation of market volatility is summarised below.

Vh,t[pt+1+ yt+1− Rpt] = Vt[pt+1+ yt+1− Rpt] = σ2 (2.6)

Imposing this assumption has a number of benefits. Firstly, the pricing equation in (2.5) is significantly more simple under this assumption and secondly, this assumption allows steady state and stability analysis to be conducted on the model. This helps understand the local dynamics of the heterogeneous agent model. Chapter 4 generalises this assumption. It considers a heterogeneous conditional variance measure in an attempt to increase the complexity of the model and map stock market indices in a more accurate manner.

Assumption II: Dividends are a Martingale

yt is defined as an independent and identically distributed stochastic dividend process. The

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walk. This means that dividend can be expressed as:

yt= yt−1+ yt−1εrwt εrwt ∼ N(0, ρrw).

Additionally, all agents maintain identical and accurate beliefs about all future dividend payouts. Therefore every agent holds the same expected dividend payout for every period.

Eh,t−1[yt] = Et−1[yt] = E [yt] = yt−1 (2.7)

Although assuming dividends follow a random walk and expectations are homogeneous and known to the market may be a limiting assumption, it allows the model to take advantage of the deviations form of the model. This form summarises the difference between the asset price and its fundamental value. It is important to note that this assumption only applies to the expectations. The assumption’s only function at this point is to provide a pricing equation for the model in deviations form. This will be especially convenient when the model’s stability conditions are assessed. Section 2.4 provides greater detail about the fundamental solution and the model in deviations form.

2.4 Model in Deviations Form

Chapter 3 explores the existence of steady states and their stability conditions. Algebraically, it is convenient to analyse these properties using the model in deviations form. The following components of the model ensure the pricing equation (2.5) can be transformed appropriately and the results remain valid.

Most importantly, the model assumes a fundamental solution, p∗_t, can be found. Accord-ing to Radner (1979), if the no-arbitrage condition is satisfied and the market is complete, a fundamental solution must exist. In all cases, the fundamental price must be unique. In order to establish a closed form solution for the fundamental price, the model takes advantage of Assumption II. With accurate and endogenous beliefs about future dividends, the fundamental price is be estimated by implementing a discounted cash flow valuation. Therefore, the funda-mental price in the model is equal to the present value of all future dividends. In light of this, p∗_t can be expressed as a function of yt, R and conditional expectations. The following equation

outlines the derivation of the fundamental price of an asset. Importantly, the fundamental price of the asset follows a random walk because of the way dividends are determined.

p∗_t = Eh,t[yt+1] R + Eh,t[yt+2] R2 + ... = ∞ X s=1 yt Rs = yt R − 1 = yt r (2.8)

Additionally, the model requires price expectations to be of a specific form. This is essential because agents are likely to change and adapt their expectations over time. Section 2.6 out-lines the adaptive belief system implemented by the model. Setting expectations to a certain

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form allows for time-varying solutions to be easily incorporated into the model. This assump-tion ensures heterogeneous beliefs on future prices are a funcassump-tion of past deviaassump-tions from the fundamental value.

Eh,t[pt+1] = p∗t + Eh,t[xt+1]

= p∗_t + fh,t(xt−1, ..., x1) (2.9)

In this case, xt= pt− p∗t denotes price deviations at time t and fh,t(xt−1, ..., x1) represents

a deterministic function that can differ across trader types. The shorthand notation fh,t will be

used throughout the paper.

Thus, the equilibrium pricing equation in (2.5) can be reformulated in deviations form. Appendix A.1 outlines the detailed derivation of this form of the model.

xt= 1 R H X h=1 nh,tEh,t[xt+1] + pt−1t = 1 R H X h=1 nh,tfh,t+ pt−1t (2.10)

2.5 Belief Types (Forecasting Rules)

Realistically, there is a large number of different forecasting rules that traders adopt in the market. If they are reduced to a deterministic function outlined in (2.9), the beliefs may range from very simple to extremely complex functions. With modern day technology, the majority of traders have access to a means of trading derived from complicated forecasting functions. This poses a problem in an artificial setting as the analytical tractability is severely diminished. Conveniently, it is shown that simple forecasting rules are able to accurately replicate real-world price dynamics (C. H. Hommes, 2013). This is useful as it allows the model to apply logical and theoretically sound forecasting rules.

Specifically, the model investigates price dynamics in the presence of two categories of agents: fundamentalists and chartists. A fundamentalist’s trading strategy is simple. They believe prices are determined by underlying fundamentals and expect prices to converge to the discounted cash flow valuation outlined in (2.8). Therefore, this means a fundamentalist will always buy when the current price is below the fundamental price of the asset and sell when the current price is above the fundamental price. They expect all future prices to return to their fundamental value and deviations to return to zero. Let v be a fundamentalist’s belief of the mean-reversion parameter. Therefore, in the context of the model and following the restrictions above, a fundamentalist’s deterministic function reduces to:

fh,t(xt−1, ..., xt−L) = vxt−1. (2.11)

Notice if υ = 0, a fundamentalist will believe deviations will be constantly equal to zero and if υ = 1, the forecasting rule corresponds to a situation where agents follow the EMH and

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believe that prices follow a random walk. Both values of υ are unrealistic assumptions. Thus, the model implements the restriction 0 < υ < 1.

On the other hand, chartists, also known as technical analysts, form their expectations in a response to past price movements. They believe future prices can be predicted using information on historical trends, patterns or cycles as well as the asset’s fundamental value. Although Chapter 4 considers a number of extensions, forecasting rules based on the price direction in the previous period will be the starting point for the analysis. Let g > 0 denote the trend sensitivity of agent h. Thus, a chartist’s deterministic function is defined by a linear forecast:

fh,t(xt−1, ..., xt−L) = xt−1+ g (xt−1− xt−2) . (2.12)

Here, the chartist forecasting rule represents a situation where agents extrapolate the most recent trend using the most recent deviation as a point of reference. However, there are many different forms of this rule that could be considered. For example, Chapter 4 utilises a more complicated chartist rule where agent’s predictions take into account longer trends and price movements over a greater number of periods.

A key component in the four-type model is the effect of contrarians, a special type of technical analyst. Like chartists, contrarians believe future prices can be predicted and base their price forecasts on previous market dynamics. However, as their name suggests, contrarians assume prices will move against the current trend. They will adopt a short position in the risky asset if price dynamics exhibit a positive trend because they believe prices will be lower in the future. Similarly, they will take up a long position if there is a negative trend. Let κ < 0 be a contrarians trend sensitivity. Therefore, a simple deterministic function outlining a contrarian’s expected price deviations can be summarised by:

fh,t(xt−1, ..., xt−L) = xt−1+ κ (xt−1− xt−2) . (2.13)

A more complex forecasting rule is implemented for contrarians in Chapter 4.

2.6 Evolutionary Selection

Agents in the market typically change their beliefs about future prices. This is especially the case if a trader’s strategy is relatively under-performing. It is natural to assume that an agent will always form their price forecast in a way that will maximise their expected utility. Therefore, if an agent is experiencing low profits using their strategy, they are likely to adopt a more profitable forecasting rule. Thus, it is important to consider the evolutionary dynamics of nh,t, the fraction of agents of type h. The model implements a discrete choice evolution

process developed by Brock and Hommes (1997) where agents take advantage of reinforcement learning. This process ensures more successful strategies will attract more followers. Moreover, the fractions of agents are updated according to a fitness function, Uh,t. This function, referred

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Let Ch ≥ 0 be the total costs for an agent of type h. For example, the costs could be the

outlay attached to attaining information about market patterns, cycles and trends that assists agents in forming their beliefs. Typically, fundamentalists incur a greater cost than technical analysts because it requires a greater amount of information to be able to evaluate the market’s fundamental values. We denote an agent’s profit as:

πh,t = (pt+ yt− Rpt−1) zh,t−1−

a 2σ

2_{− C}

h. (2.14)

A risk-adjustment term is implemented to cap the level of risk agents are exposed to. By substituting optimal demand, (2.3) into the profit equation, (2.14), realised profits can be defined as a function of deviations and agent beliefs:

πh,t = (xt− Rxt−1) Eh,t−1[xt− Rxt−1] aσ2 − a 2σ 2_{− C} h = (xt− Rxt−1) fh,t−1− Rxt−1 aσ2 − a 2σ 2_{− C} h. (2.15)

Therefore, consider the following autoregressive performance measure:

Uh,t = πh,t+ ηUh,t−1. (2.16)

Here, η denotes a memory strength parameter. Notice that η = 1 corresponds to infinite memory meaning Uh,t is determined by accumulated wealth while η = 0 corresponds to a one

period lag so that the fitness function is exactly equal to the most recent profit. Chapter 3 is focused on a zero-memory situation. However, Chapter 4 considers the more general case where 0 < η < 1. In this situation, the fitness function corresponds to an exponentially moving average of past realised profits.

Having defined realised profits and a performance measure, the fraction of agents of type h can now be specified. Let:

nh,t = exp (βUh,t−1) Zt−1 , Zt−1 = H X h=1 exp (βUh,t−1) . (2.17)

An integral component to the model is the intensity of choice, β ≥ 0, which measures how quickly agents switch to more successful strategies. Interestingly, if β is equal to zero, the fractions are evenly divided between the available strategies and if β approaches infinity, its neoclassical limit, all traders will select the strategy with the fitness function that performed best in the previous period. Additionally, the model is extended to include asynchronous updating of beliefs by introducing a persistence (inertia) parameter. This will be explained in greater detail in Chapter 4.

2.7 Generalised Model Specification

Combining Sections 2.1 to 2.6 yields the generalised model specification. Assessing the validity of models based on the following framework will be the focal point for the remainder of the thesis.

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xt= 1 R H X h=1 nh,tfh,t+ pt−1t nh,t = exp (βUh,t−1) PH h=1exp (βUh,t−1) Uh,t−1= (xt−1− Rxt−2) fh,t−2− Rxt−2 aσ2 + ηUh,t−2− a 2σ 2_{− C} h

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Chapter 3

Fundamentalists vs. Chartists

This chapter explores the price dynamics of a market with two types of agents, fundamentalists and chartists. Their interaction is intriguing because these agents will often have conflicting expectations of the future price of an asset. By implementing stability analysis and relevant MATLAB simulations, the fundamentalist and chartist model aims to provide insight into the characteristics of asset prices. The model also delves into the existence of excess volatility and irregular price dynamics. Specifically, consideration is given to mean-reversion, volatility clustering and herding, all of which are evident in historical market dynamics.

Furthermore, the simulated model is exposed to a number of statistical tests to provide a holistic representation of it’s characteristics. Additionally, identical tests are conducted on the daily log returns of the DAX30 index in order to assess the performance of the asset pricing model with heterogeneous agents. To ensure the tests are comparable, one period in the asset pricing model corresponds to one day. The aim of this chapter is to form a model that exhibits the same price dynamics as a stock index.

3.1 Model Specification

Firstly, denote fundamentalists to be agent type 1 and chartists to be agent type 2. Thus, the forecasting rules this model will implement can be expressed as:

f1,t = vxt−1, f2,t = xt−1+ g (xt−1− xt−2)

Therefore, by combining the generalised model specification outlined in Section 2.7 and these forecasting rules it is possible to express this dynamical model as a system of equations with only two state variables. In order to reduce the model to a system of this form it is important to consider the differences in the fractions of agents, mt= n1,t− n2,t. Studying the differences does

not reduce the information the model provides and is much more convenient than proceeding with the individual fractions. Note that Appendix A.2 gives a detailed derivation of the model specification. This difference can be simply expressed using the hyperbolic tangent function:

mt= tanh

β

2 [U1,t−1− U2,t−1]

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If the differences in the agent’s performance measures are then substituted in, mt can be

given a closed form solution. Furthermore, using the difference of fractions and the deterministic functions for fundamentalists and chartists, the price deviations at time t can be reformulated. Together, these two functions provide the complete model specification to be used for analysing the price dynamics in a market with fundamentalists and chartists.

mt= tanh β 2 (xt−1− Rxt−2) (v − g − 1)xt−1+ gxt−2 aσ2 − C1 xt= 1 R 1 + mt 2 vxt−1+ 1 − mt 2 (xt−1+ g (xt−1− xt−2)) + pt−1t (3.1)

In later sections, the model is transformed further because it will be convenient to work with a first order system of equations when analysing the stability conditions.

3.2 Dynamical Systems Theory

The long-term qualitative behaviour of the heterogeneous agent model is explored in this section. The system is analysed rigorously because it is important to assess whether the system contains any steady states. Moreover, determining if the simulated asset price has a tendency to converge to its equilibrium value or exhibit cyclical or chaotic dynamics provides a deeper understanding about the behaviour of asset prices. Throughout the steady state and stability analysis, the effect of the independent error term is ignored.

3.2.1 Steady States

The steady state of a dynamical system refers to members of the invariant set. The most general member of this set is a “fixed point”. If Fλ(·) defines the law of motion in a simple

one-dimensional setting, x∗ is a fixed (or equilibrium) point of the system if it satisfies: x∗= Fλ(x∗) .

Specifically, ignoring the effect of any random shocks, a system will never change if it begins at any of the fixed points. Alternatively, if the system converges to its fixed point, it will remain there. This concept can be directly translated into higher order systems. Therefore, denote x∗ and m∗ as the fixed points of price deviations and difference of fractions respectively. The steady states of the system are therefore solved by setting x∗ = xt = xt−1 = ... = xt−N and

m∗ = mt= mt−1= ... = mt−N. Therefore, the equation reduces to:

x∗= 1 R 1 + m∗ 2 vx ∗ +1 − m ∗ 2 x ∗ x∗= 0, or m∗ = 1 + v − 2R 1 − v . (3.2)

Given |mt| < 1, the restrictions on R and v mean the second solution is not feasible. For a

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can be characterised by the point: (x∗, m∗) = 0, tanh −βC1 2 . (3.3)

It is of interest to study the stability conditions of this steady state. In the following section, the point is substituted into the Jacobian matrix of the system to determine the local behaviour of price deviations.

3.2.2 Stability Analysis

As aforementioned, analysing the stability of the steady state promises a deeper understanding of the price dynamics. At this stage, the model provides little indication about the nature of the fixed point. The stability analysis in this section helps to understand the price dynamics in the local neighbourhood of the steady state and the conditions that are required for the model to converge to this fixed point.

In order to assess the stability of the steady state expressed in (3.3), the Jacobian ma-trix must be derived. To achieve this, the model outlined in (3.1) must be transformed to a four dimensional, first order system of the vector form (x1,t+1, x2,t+1, x3,t+1, x4,t+1) =

F (x1,t, x2,t, x3,t, x4,t). Therefore, the dynamical system can be written as:

x1,t+1= 1 R 1 + mt+1 2 vx1,t+ 1 − mt+1 2 (x1,t+ g (x1,t− x2,t)) x2,t+1= x1,t x3,t+1= x2,t x4,t+1= x3,t. (3.4)

Due to the intensity of the algebra, mt+1 is still included as a component of the system.

Conveniently, it has been derived in such a way that ensures it is a function of the elements of the system. In the case of fundamentalists and chartists, the difference of fractions can be expressed as: mt+1= tanh β 2 (x1,t− Rx2,t) (v − g − 1) x3,t+ gx4,t aσ2 − C1 .

The Jacobian is a matrix comprised of all first-order partial derivatives of the system. This matrix contains information about the local behaviour of price deviations. The generalised Jacobian matrix is not reported due to the sheer algebraic magnitude. However, the Jacobian matrix evaluated at the steady state will be essential for the stability analysis. It is reported in equation (3.5). J =        g 2R[g + v + 1 + (v − g − 1)m ∗_] g 2R(m ∗_{− 1)} ₀ ₀ 1 0 0 0 0 1 0 0 0 0 1 0        (3.5)

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The stability of the system depends on the eigenvalues of this Jacobian. The eigenvalues, λ are the solution to the characteristic equation |J −λI4| = 0. Hence, the characteristic polynomial

becomes: λ2 λ2− λ 2R(g + v + 1 + (v − g − 1)m ∗ ) − g 2Rm ∗ + 1 = 0. (3.6)

Solving the characteristic equation for λ and substituting in the parameters of the model yields the four eigenvalues of the system. Appendix A.4 provides a detailed derivation of the characteristic equations and gives an algebraic solution for the eigenvalues. The parameters used in the model are taken from Amilon (2008), where the efficient method of moments (EMM) estimator is calculated. This estimator is an extension of quasi-maximum likelihood estimation. Thus, substituting in the parameters v = 0.55, g = 1.11, C1= 0.5 and β = 1.99 yields:

λ1 = 0, λ2 = 0, λ3 ≈ 0.8757 − 0.705655i, λ4 ≈ 0.8757 + 0.705655i.

Since ||λ3,4|| ≈ 1.12463 > 1, these eigenvalues demonstrate that the steady state point

(x∗, m∗) constitutes an unstable focus of the system because the magnitude of the non-zero eigenvalues are greater than one. In general, this point is unstable.

Additionally, the bifurcations of the system are of interest. Bifurcations affect the local stability of dynamical systems because they represent qualitative changes of the dynamical behaviour. Codimension one bifurcations are the focus of this section. These bifurcations occur when a single parameter is varied. Although bifurcations of a higher order may provide interesting information, they are beyond the scope of this thesis. If the magnitude of the product of the two eigenvalues is equal to one and they cross the unit circle, the system experiences a Hopf bifurcation. This bifurcation is the cause of the creation of an unstable focus and chaotic price dynamics. Therefore, for this bifurcation to occur, we must have λ1λ2 = 1 and |λ1+ λ2| < 2.

− g 2Rm ∗ _{= 1,} − 1 2R(g + v + 1 + (v − g − 1)m ∗₎₎ < 2

If these conditions are satisfied, the model estimated in the following section should exhibit an absence of convergence and unpredictable, chaotic dynamics. Importantly, the value of m∗ must be negative. This implies that chartists must outnumber fundamentalists in the model for a Hopf bifurcation to generate chaotic dynamics. A saddle node bifurcation and a number of other bifurcations may also occur when an eigenvalue is equal to one. Alternatively, a period-doubling bifurcation occurs when an eigenvalue is equal to negative one. These bifurcations are not explored in detail because they are not able to produce the chaotic price dynamics evident in stock market indices.

3.3 Characteristics of Asset Prices

This section outlines the key aspects of real-world asset markets and compares them to the characteristics of the price deviations achieved through MATLAB simulations. The code used

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to simulate the features of the model is shown in Appendix C.1. Figure 3.1 displays the daily log-returns of the DAX30 index. The closing price of the German index from 01/11/1988 to 12/31/2008 is sourced from Yahoo Finance. This index is used as a reference point for the analysis because it is major European stock index with high liquidity. Additionally, the data is taken over a long enough period of time to reduce standard error and appropriately test the hypotheses that are outlined below.

Additionally, the following table outlines the parameter values derived from EMM estimation that are kept constant throughout the Chapter 3 estimations:

R a g v β C1 C2 η ρ×103 ρrw_×10₃

1.037 1.00 1.11 0.55 1.99 0.50 0.00 0.00 5.25 6.36

Table 3.1: Parameter values used in estimation of the heterogeneous agent model with funda-mentalists and chartists.

Figure 3.1: DAX30 Log Returns and MATLAB Price Deviation Simulations of the two-type model.

The DAX30 has a number of fascinating characteristics. Firstly, the Jaque-Berra Test in-dicates non-normality because histograms display excess kurtosis and skewness. Moreover the returns appear to maintain a constant mean which is close to zero and there are periods where large returns trigger subsequent large returns. The figure also shows 5000 replications of simu-lated log returns from the asset pricing model with fundamentalists and chartists. Hypothesis

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testing provides a definitive answer as to whether the heterogeneous agent model accurately maps the features of asset prices.

Figure B.1 provides another view of the simulations. It reports the price deviations from fundamental values as well as the actual price at time t. Notice the price deviations, xt do not

converge to the steady state value, x∗ = 0. Instead the system experiences deviations larger than can be explained. Thus, chaotic dynamics are present as predicted in the stability analysis in Section 3.2.2. The bubbles appear to be very short and overshoot the fundamental value of the asset when they crash. The following subsections explore the validity of the heterogeneous agent model with respect to certain characteristics. There are a number of summary tables throughout this section. It should be noted that within these tables, an asterisk denotes a coefficient that is significant at the 5% level and standard errors are reported in parentheses.

3.3.1 Mean-Reversion

Mean-reversion refers to the tendency of stock prices and interest rates to return to their fun-damental value. The amount of evidence on this phenomenon has rapidly expanded in recent years and there is significant theoretical and empirical reasoning behind the existence of mean reversion. Firstly, variance tests reject the hypothesis that stock prices follow a random walk and there exists autocorrelation in long-term returns. More specifically, asset returns exhibit negative serial correlation (Cecchetti, Lam, & Mark, 1988). In light of this, there is widespread consensus that asset prices revert to their fundamental level instead of diverging boundlessly (Wong & Lo, 2009).

Stationary processes and mean-reversion go hand in hand. A stationary process maintains a time-invariant mean and auto-covariance. The variance of the system must also be finite. Moreover a stationary process exhibits joint probability distributions that do not vary with time. Hence, if Xt is a stochastic process, it is strictly stationary if the joint distribution of

(Xt1, ...Xtk) is the same as the joint distribution of (Xt1+h, ...Xtk+h).

These processes display autocorrelation functions that slowly decay or have a cut-off point. Additionally, random shocks have a transient effect on the system. It is most important to note that a stationary process will fluctuate around its mean. Although mean-reversion is easy to identify visually, it is typical for a unit root test to be implemented in order to test if a process is stationary. Therefore, the Augmented Dickey-Fuller (ADF) Test - an adaptation of the unit root test - is used to determine if the real-world asset returns and the simulated log returns are stationary processes and thus exhibit mean-reversion. This test is preferred to the standard Dickey-Fuller test because of the potential for serial correlation problems. The ADF testing procedure involves fitting the following model to the simulated and actual log returns:

∆Rt= α1+ β1t + θRt−1+ δ1∆Rt−1+ ... + δN∆Rt−N+ εt.

In this model, ∆Rtcorresponds to the change in log returns of the DAX30 or the simulated

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to the log returns of the data, let β1 = 0. By doing this, the model will not include a time

trend. This is consistent with the characteristics of the time series for both the DAX30 log returns and simulated log returns.

Simulated Log Returns DAX30 Log Returns

θ : -0.492173* -1.019669* (0.019005) (0.014121) δ1: 0.241090* (0.019289) δ2: 0.144799* (0.018650) δ3: 0.059967 (0.017854) δ4: -0.140283* (0.017062)

Table 3.2: Estimation coefficients of an Augmented Dickey Fuller Test on simulated and DAX30 log returns with one regressor and four lags displayed.

The Schwarz Information Criterion (SIC) is used to determine the optimal number of lags to include. Low values of this criterion function represent a better model fit and more accurate test results. Table 3.2 summarises the first four lag coefficients of the regression and Figures B.2 and B.3 display the extensive test output that is produced by Eviews. This regression will allow the use of the ADF test. When conducting the test, θ is the parameter of interest. The null hypothesis, θ = 0 states that the dependent variable’s characteristic equation has a unit root. This test considers the alternate hypothesis that θ < 0 because returns are known to exhibit negative serial correlation. The intuition behind the test is that if the return series ∆Rt is not

stationary, the lagged return level, Rt−1 will provide no relevant information about predicting

the next price change.

Holding all else constant, the statistically significant negative values of θ indicate that when the model experiences an increase in returns, the rate of change of the following return is less than zero. This negative value implies the log returns are likely to exhibit mean-reversion. However, the ADF test statistic will provide a more accurate result.

Simulated Log Returns DAX30 Log Returns Test Statistic: -25.94992 -72.21004

Probability: 0.0000 0.0000

Table 3.3: Augmented Dickey-Fuller Test for unit roots.

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both the log returns simulated by MATLAB and the DAX30 log returns given the 1% critical value is equal to -3.431477. In light of this, it can be concluded that neither process has a unit root. This means that both processes are stationary and display mean-reversion. A Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test may also be implemented to assess this feature of asset prices. It should be noted that the output is consistent with the ADF test however it is not reported.

Although it is clear both the log returns on the DAX30 index and the simulated price are integrated processes and exhibit mean-reversion, the test statistics remain different. A number of extensions to the model are implemented in Chapter 4 in an attempt to replicate actual returns more accurately. For example, introducing a heterogeneous risk measure where agents become more risk averse when prices deviate further from the fundamental values may increase the level of mean-reversion in the simulated prices. First, attention is drawn to other notable characteristics of the current model.

3.3.2 Volatility Clustering

The Autoregressive Conditional Heteroskedasticity (ARCH) model by Engle (1982) has paved the way for the investigation of second moments that vary over time in financial markets. However, only recently has the effect of time-varying volatility been considered in asset pricing models and dynamic hedging strategies (Bollerslev, Chou, & Kroner, 1992). It is interesting to note that stock prices, interest rates and exchange rates display what is known as volatility clustering. In asset markets, it is an event where intervals of time experience an increased level of volatility and other intervals of time experience relatively low levels of volatility (Grannan & Swindle, 1994). Put simply “large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes” (Mandelbrot, 1963).

Econometricians and financial analysts continue to face the difficult challenge of understand-ing the source of volatility clusterunderstand-ing. Generic quantitative analysis indicates that although asset returns Rtare uncorrelated, the absolute value of returns, |Rt| and their squares display a

pos-itive, significant and slowly decaying autocorrelation function. That is, corr (|Rt|, |Rt+τ|) > 0

for any value of τ ranging from a matter of minutes to months. The existence of volatility clustering can be identified by implementing an ARCH Lagrange Multiplier Test. This tests for autocorrelation in the squared errors of a regression model. In order to conduct this test on the log returns of the model and the DAX30 log returns, an appropriate autoregressive model must be estimated. By analysing the autocorrelation and partial correlation functions of both time series, it can be seen that an AR(1) model is most suited to both data sets. Thus, price deviations and actual returns must first be estimated using an autoregressive model of order one.

Rt= c + φ1Rt−1+ ˆat

The real and simulated log returns are fitted using this simple model in order to generate a sample residual series ˆat. Next, another autoregressive model is used to estimate the squared

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residual series ˆa2_t. This output is used to determine whether the size of the errors contain autocorrelation. The sample errors are fitted to the following auxiliary model:

ˆ

a2_t = ˆc + ρ1ˆat−12 + ... + ρNaˆ2t−N+ εt.

Looking at the partial autocorrelation function for the squared residuals indicates that no lag is appropriate for the errors of the simulated log returns and nine lagged values are optimal for the DAX30 log returns. Thus, clustered volatility appears to be more likely to occur in the actual log returns because the optimal number of lags for the DAX30 is greater than the optimal number of lags for the simulated log returns. The complete regression outputs are expressed in Figures B.4 and B.5.

ρ1 : 0.022771 0.094413* (0.014144) (0.014151) ρ2 : 0.075471* (0.014187) ρ3 : 0.102052* (0.014201) ρ4 : 0.078157* (0.014245)

Table 3.4: Squared residual regression estimation coefficients.

Four lags of the regression output are summarised in the Table 3.4. Each of the coefficient values in the DAX30 log returns regression are far greater than the coefficient values of the simulated log returns. Furthermore, the squared errors of the DAX30 display significant auto-correlation for all lagged values of the model while the squared errors of the simulated returns do not share this trait. This suggests that the fundamentalists and chartists model fails to model the volatility of asset prices.

Using the residual regression output, the ARCH LM Test can be applied to evaluate the null hypotheses; ρ1 = 0 for the simulated log returns and ρ1 = ... = ρ9 = 0 for the DAX30

log returns. In other words, the null hypotheses state that there is no autocorrelation in the sample squared error terms and therefore no volatility clustering. If the standard R2-statistic is calculated, it can be shown that the LM statistic, nR2 asymptotically converges to a χ2_N distribution.

LM-statistic: 2.591526 550.9887

Probability: 0.1074 0.0000

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As Table 3.5 shows, the ARCH LM Test rejects the null hypothesis for the DAX30 log returns. This means the squared residual processes DAX30 returns exhibit significant autocor-relation. In light of this, the DAX30 stock index contains a high degree of volatility clustering. On the other hand, the test fails to reject the null hypothesis for the simulated log returns in the two-type model. This confirms the indication given by the regression output.

Therefore, the two-type heterogeneous agent model proves to be an insufficient representa-tion of financial markets because it is unable to incorporate volatility clustering and thus fails to correctly capture the variability of the market. This is a common issue faced by economists. However, there remains further elements of real-world asset markets to consider and include in the model. Chapter 4 integrates many new features including heterogeneous conditional variance expectations and asynchronous updating of beliefs. These additions to the model are included to change the nature of volatility caused by expectations and random shocks to the market. It is of interest to test if introducing a persistent effect on the fractions of agents creates volatility clustering in the model.

3.3.3 Herding and Agent Behaviour

To this point, a number of characteristics of asset prices have been explored such as mean-reversion and volatility clustering. However, there has been very little discussed about the cause of these irregular price dynamics. Of course, fundamental values are changing over time and that will have an effect, but it has also been proved that market psychology and heterogeneous expectations are other sources of the excess volatility that is evident in financial markets. Herd mentality is one of the most common causes of extraordinary events in financial markets and many perceive it to be a hard-wired attribute of human behaviour. This perhaps explains why it has an ever-present effect on asset prices.

Herding is caused by social conformity and the opinion that such a large group could not be wrong. Noise traders are especially susceptible to the pressure of the contagion effect. These traders are defined by having extremely limited market experience, information and expertise. They are often guilty of being susceptible because even if they are convinced that a certain action or belief is incorrect or irrational there is still a chance that they will follow the herd if they suspect there is information that they are unaware of. This market behaviour can often spark highly volatile price dynamics. As demonstrated in previous sections, the DAX30 log returns display significant mean-reversion and volatility clustering. Although the model developed with fundamentalists and chartists does not summarise all the characteristics of stock markets accurately, it still provides insight into the cause of irregular price dynamics. Conclusions can be drawn because the model exhibits herding and periodic switching of beliefs without centralised direction. An example is displayed in Figure 3.2. The figure shows a significantly shorter time span of an identical model to that of Figure 3.1, namely 100 periods. Moreover, the figure contains the fraction of agents who adopt a chartist belief, n2,t.

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Figure 3.2: Market price deviations from fundamental values and fractions of chartists in the two-type model.

causing deviations to maintain a value close to zero. However the fractions of agents rarely settle around a particular value. Instead, the number of traders who adopt a chartist forecasting rule changes dramatically. On the other hand, there are a number of exemptions to this statement. Perhaps the most noticeable feature of the model in such a short time frame is the price bubbles that develop at approximately period 50 and 75. In total, the price bubbles last approximately ten periods which is a relatively short time span when compared to bubbles in real-world asset markets. Furthermore, Figure B.2 in the appendix provides a view of price deviations and the fractions of chartists over a longer period of time.

It is clear that price bubbles occur relatively frequently when the heterogeneous agent model is estimated. This confirms that the irregular dynamics in the model are not caused by sensitiv-ity to initial conditions. Each bubble and period of high volatilsensitiv-ity has one thing in common; the fractions of chartists and fundamentalists rapidly switch. Perhaps traders are unable to predict future prices during a bubble or a crash and hence frantically switch between the forecasting rules available because the relative profitability of each belief type changes dramatically.

By turning attention back to Figure 3.2, it can be seen that the asset price is significantly different from its fundamental value when all of the traders in the market adopt a chartist forecasting rule. Once the model returns to a situation where agents continue to rapidly switch beliefs the price deviations return to their average value. In this case, the herding of all traders towards a particular forecasting rule is the catalyst for expansive price movements. When

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adopting the chartist belief, traders assume the deviations will follow the apparent trend. Thus, if there is a positive price deviation, and all traders predict the trend will be elongated, the trend will continue. This emphasises the contention that agent psychology can have a significant effect on market dynamics. This simulated example can be compared to the “Tulipmania” and “Dot-com Bubble”. The inflated asset prices during these periods must have been caused by the heightened optimism and belief of consumers at the time.

Another fascinating dynamic of the model occurs when the model experiences substantial deviations. It is rare for fundamentalists to drive out chartists in the model. However, chartists will drive out fundamentalists from the market as soon as the asset price deviates far enough away from the fundamental value. Thus, due to the results of the stability analysis, chaotic dynamics are more likely to occur because the number of chartists outweigh the number of fun-damentalists. Interestingly, chartists and fundamentalists predominantly coexist in the market which may suggest that neither strategy is more profitable than the other in the long-term.

3.4 Summary

This chapter presents an asset pricing model with two heterogeneous agents. Traders are able to select their forecasting rule between two conflicting belief types; fundamentalists and chartists. Ignoring random shocks, the steady state and stability analysis indicates that this model has one unstable fixed point when parameters are derived from efficient method of moments estimation. Moreover, having a greater number of chartists in the market is more likely to result in chaotic price dynamics.

These findings are confirmed by the simulations of the model. The price of the asset fails to converge to its fundamental value and displays unpredictable dynamics. Interestingly, as the level of deviations increase, irregularity in the series of log returns can be observed. Thus, this model shows that price bubbles are necessary to produce increased variability in the returns. However, the model fails to generate persistent and appropriate price deviations because bubbles tend to last for a short period of time. Moreover, cyclical dynamics tend to arise following the creation of a bubble.

The DAX30 stock index is used as a reference point to determine the validity of the funda-mentalist and chartist model in summarising the characteristics of stock prices. The existence of mean-reversion and volatility clustering are the focal point of the assessment and consideration is given to excess kurtosis and skewness. Hypothesis testing illustrates that the model is able to generate mean-reversion and non-normality of returns. However, in line with the literature in the field, volatility clustering proves to be an illusive trait of the model.

The fundamentalist and chartist model also provides a number of insights into the behaviour of agents when price bubbles are formed. Agents are likely to rapidly switch between forecast-ing rules because the profitability of each belief type is not constant. Additionally, although chartists drive out fundamentalists during bubbles, it appears that neither forecasting rule is more profitable in the long-term.

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The model is extended in Chapter 4 by introducing two additional forecasting rules for traders to adopt including a contrarian belief type. Under this extended model it is important to assess the relative profitability of the forecasting rules and analyse how agent behaviour causes excess volatility and irregular price dynamics.

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Chapter 4

Four-Type Model

Despite being more accurate than traditional models, the asset pricing model with two agent types in Chapter 3 imposes a number of conditions that may be unrealistic. Chapter 4 attempts to correct the flaws in the fundamentalist and chartist model evaluated in the previous chapter by introducing a greater number and variety of agents. Additionally, the model specification is altered in an attempt to replicate actual stock market dynamics which results in a more advanced model with increased volatility clustering and mean-reversion.

4.1 Model Specification

As aforementioned, this section outlines the specification of the asset pricing model with four heterogeneous agents. The deviations form of the model is no longer considered because the steady state and stability analysis loses its analytical tractability due to the intensity of the algebra. Moreover, given the results in outlined in Sections 3.2.1 and 3.2.2, further stability analysis would be unlikely to provide a deeper understanding of the model dynamics. Thus this chapter considers a new pricing equation similar to the one summarised in (2.5). In light of this, an agent’s deterministic function summarising their forecasting belief must be reformulated as a conditional expectation of the future asset price. These expectations are determined by a function of past realised prices and fundamental prices and outlined in (4.1)-(4.4).

Similar to the two-type model, the first agent in the four-type model is a fundamentalist who believes prices will revert towards their fundamental value and v represents the mean reversion factor. The fundamentalist’s forecasting rule is unchanged from the previous model.

E1,t[pt] = p∗t−1+ v pt−1− p∗t−1

(4.1) The second type of agents are trend followers. Unlike fundamentalists, they believe future prices can be predicted based on observed trends and patterns in past prices. Thus, trend followers have identical belief type to chartists in the two-type model. However, a new, more complicated forecasting rule is applied to the four-type model. This price expectation involves extrapolating past price trends into the future using a geometrically declining moving-average

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of past price changes. The effective time window is determined by α2 and λ represents the

degree of extrapolation where 0 < α2 < 1 and λ > 0.

E2,t[pt] = pt−1+ λ T

X

i=0

α2(1 − α2)i(pt−1−i− pt−2−i) (4.2)

Contrarians are the third type of agent in the model. As their name suggests, contrar-ians believe prices will move contrary to the observed trend. Thus, their forecasting belief maintains a very similar structure to that of a trend follower. Another geometrically declining moving-average forecasting rule is applied to the observed trend. Similar to trend followers, the restriction 0 < α3 < 1 is imposed. However, given trend followers extrapolate the trend using a

positive factor and contrarians extrapolate using a negative number, the restriction κ < 0 must be enforced. This ensures the contrarians move against the observed trend.

E3,t[pt] = pt−1+ κ T

X

i=0

α3(1 − α3)i(pt−1−i− pt−2−i) (4.3)

The final type of agent is denoted as anchor and adjustment traders. They exploit a simple example of adaptive learning where their price forecast is primarily centred around a moving average of past prices, pav_t and adjusted according to the most recent price trend. The moving average of prices is restricted to an average over a limited number of periods to ensure the average price reflects relevant price information. The limit is set to fourteen periods in the simulation. As aforementioned, one period corresponds to one day. The implementation of this type of forecasting rule has logical reasoning. Market psychology analysis indicates that an average trader is likely to extrapolate a recent price trend from a reference point which can be referred to as an anchor point (C. Hommes, 2011). In this case, the anchor is the average of the historical mean of the price and the price reported in the previous period.

E4,t[pt] =

pav_t−1+ pt−1

2 + (pt−1− pt−2) (4.4)

Combining these price forecasts with the pricing equation determines the model’s law of motion. However, before the system is simulated a number of new features are added. Addi-tionally, the existing framework is edited to make the model a more realistic representation of an actual stock market index.

4.1.1 Heterogeneous Risk Aversion

A time-varying, heterogeneous risk aversion parameter, ah,t is introduced to act as a stabilising

force that has an increasing effect the further the price deviates from its fundamental value (Amilon, 2008). This heterogeneous risk aversion parameter ensures a fundamentalist’s risk aversion reduces as the price deviation increases. Thus, when prices are significantly misaligned from their fundamental value, a fundamentalist will become less risk adverse than when the price and the fundamental value are comparable. This is a reasonable assumption because as the price extends further away from its fundamental value, a fundamentalist believes it is

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increasingly likely to revert back. Thus, they are willing to risk more given the increase in potential for gains. A closed form expression for the heterogeneous risk aversion parameter is:

a1,t=

a

1 + φ|pt−1− p∗t−1|

(4.5) ah,t= a, h = 2, 3, 4.

The parameter φ represents a fundamentalist’s sensitivity to the price deviation. It is clear that if the sensitivity parameter is set to zero, the model reverts back to a case of homogeneous risk aversion where all agents adopt identical and constant risk aversion.

4.1.2 Heterogeneous Conditional Variance Expectation

Equation (2.3) assumes agents form identical and constant conditional variance forecasts. How-ever, this may be an unrealistic assumption of the model. This chapter generalises this as-sumption to allow for heterogeneous volatility measures. It follows the work of De Grauwe and Grimaldi (2006) where agents evaluate risk according to equation (4.6). The expected condi-tional variance is defined by a weighted average of the squared (one-period forward) forecasting errors made by the agents. Therefore, σ2_h,t can be expressed as a function of expected prices and actual prices:

Vh,t[pt+1+ yt+1− Rpt] = σ2h,t= T

X

i=0

ψ (1 − ψ)i(Eh,t−1−i[pt−i] − pt−i)2. (4.6)

This equation implements the restriction 0 < ψ < 1 such that ψ (1 − ψ)i are geometrically declining weights. It is important to consider that this variance does not represent the actual variability of market returns. It is in fact an agent’s prediction of the volatility at time t. When the model is estimated in Chapter 3, the expected variance is held identical and constant. This is backed up by the assumption that there is a general consensus by traders in the market about it’s variability. This assumption is now lifted. The level of volatility clustering should naturally increase given the volatility expectations are a weighted sum of past forecasting errors.

As outlined in Chapter 2, introducing heterogeneous forecasts for the volatility of the asset price drastically increases the intensity of the algebra in the model. Equation (4.7) shows the updated rule for the law of motion in the simulated market. It is clear that any form of algebraic stability analysis would not provide enough value to be worth the time required.

pt= 1 R H X h=1 nh,t σ2 h,t Eh,t[pt+1+ yt+1] ! 1 PH h=1 nh,t σ2 h,t + pt−1t (4.7)

4.1.3 Simplified Profit Measure

Chapter 3 considers a utility measure based on realised profits in the previous period which implies the profitability of a forecasting rule is affected by the short-term level of demand. In reality, it may be difficult for agents to accurately determine their optimal level of demand, zh,t.

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In light of this, an adjusted profit measure is proposed whereby an agent’s profit is increased if they are able to predict the direction of the asset price movement. In other words, if agents forecast positive excess returns and maintain a long position in the asset, or predict negative excess returns and hold a short position in the stock their profits will be positive. However, if they forecast the sign of the return incorrectly, their profit measure will be negative. The updated profit measure to be used in Chapter 4 can be summarised as:

πh,t = (pt+ yt− Rpt−1) s (zh,t−1) −

ah,t

2 σ

2

h,t− Ch. (4.8)

Here, s (zh,t−1) corresponds to the signum function where s (zh,t−1) = 1 if zh,t−1 > 0,

s (zh,t−1) = 0 if zh,t−1 = 0 and s (zh,t−1) = −1 if zh,t−1 < 0. By implementing this simplified

profit measure, the performance measure of each forecasting rule depends on the level of profit per unit of the risky asset demanded. It is hoped that the model will exhibit the features of asset prices in a more accurate manner because the quantity of shares demanded will not have an effect on the profitability of an agent’s forecast. Perhaps this will correct the flaws in the previous simulations.

4.1.4 Asynchronous Updating

The discrete choice model for adaptive learning and evolutionary selection is extended in this chapter to allow for asynchronous updating. This is achieved by introducing a degree of inertia to the fractions of agents in the form of an autoregressive term. Put simply, a portion of agents δ adopts the identical forecasting rule as they did in the previous period. Whereas (1 − δ) of agents are free to adopt the forecasting rule using the discrete choice model where the belief with highest performance measure will attract a greater number of traders. The following rule summarises nh,t, the dynamics of the fractions of agents:

nh,t = δnh,t−1+ (1 − δ)

exp (βUh,t−1)

PH

h=1exp (βUh,t−1)

. (4.9)

Notice, when δ = 0, the dynamics reduce to the discrete choice model from Chapter 3 and δ = 1 corresponds to a situation where agents are unable to update their beliefs because they are forced to maintain their initial forecasting strategy. This chapter considers the more general case of 0 < δ < 1 in order to model the fact that not all agents update their belief type at the same time. Moreover, many agents may not adapt their forecasting rule in every period.

4.2 Simulations

This section outlines the characteristics of the simulated asset pricing model with four heteroge-neous agents. Importantly, the extensions to the model summarised in Section 4.1 are included in the simulations. The validity of the four-type model is compared with the fundamentalist and chartist model in order to ascertain the performance of the more advanced model. Additionally, the DAX30 is used as a reference point to determine whether the extended heterogeneous agent

The validity of heterogeneous agent models in assessing stock market indices : a study of mean-reversion and volatility clustering

Master’s Thesis