A new structural stochastic volatility model of asset pricing and its stylized facts
AA new structural stochastic volatility model of asset pricingand its stylized facts
Radu T. Pruna, Maria Polukarov, Nicholas R. Jennings
School of Electronics & Computer Science, University of Southampton, UK
Abstract
Building on a prominent agent-based model, we present a new structural stochastic volatility asset pricingmodel of fundamentalists vs. chartists where the prices are determined based on excess demand. Specifically,this allows for modelling stochastic interactions between agents, based on a herding process corrected bya price misalignment, and incorporating strong noise components in the agents’ demand. The model’sparameters are estimated using the method of simulated moments, where the moments reflect the basicproperties of the daily returns of a stock market index. In addition, for the first time we apply a (parametric)bootstrap method in a setting where the switching between strategies is modelled using a discrete choiceapproach. As we demonstrate, the resulting dynamics replicate a rich set of the stylized facts of the dailyfinancial data including: heavy tails, volatility clustering, long memory in absolute returns, as well as theabsence of autocorrelation in raw returns, volatility-volume correlations, aggregate Gaussianity, concaveprice impact and extreme price events.
Keywords:
Structural stochastic volatility, Method of simulated moments, Discrete choice approach,Herding, Stylized facts
1. Introduction
A financial market is a broad term describing a marketplace where buyers and sellers participate in the tradeof various financial assets at prices that reflect supply and demand. One of the most influential ideas inmodern financial markets is the Efficient Market Hypothesis (EMH) proposed by Fama (1970). It postulatesthat prices fully reflect all available information, meaning that prices should adjust instantly and correctlyto reflect new information. Consequently, all currently available relevant information regarding a particularasset is already incorporated in the market price. Only new information can change the price and it wouldbe an immediate reaction of the market triggered by its arrival.As a consequence of the random information arrival process, a market with such information efficiencyleads to random sequences of price changes that cannot be predicted. Mathematically, it is equivalent tosaying that prices follow martingales, where knowledge of past events never helps predict future values.It means that no profits can be generated from information-based trading since such earnings must havealready been captured. According to Lucas Jr (1978), prices fully reflect all available information and followmartingales when all investors have rational expectations.Nevertheless, many theoretical and empirical implications of the EMH have been tested over the years.A range of studies raised several questions regarding the basic doctrines of the efficient markets model,especially on its view for asset price dynamics. The attacks on both theoretical and empirical sides ofefficient market ideas developed and continued during the 1980s and 1990s with the work of Grossmanand Stiglitz (1980), Kindleberger (1978) Brock et al. (1992), Mehra and Prescott (1988) and Hansen andSingleton (1983), among others.Most of the findings and debates regarding market efficiency and rationality are still unresolved, suggest-ing that markets may have non-trivial internal dynamics (Hommes, 2006). Full rationality requires all agentsto have knowledge about the beliefs of all the other participants in the market, thus making it impossible
Preprint submitted to Elsevier May 2, 2016 a r X i v : . [ q -f i n . E C ] A p r n a heterogeneous framework. On the other hand, heterogeneity obviously complicates the market models,making their analytical tractability almost impossible. A computational approach is therefore better suitedfor investigating the heterogeneous world, offering a strong motivation for developing agent-based models offinancial markets (Chen et al., 2012).Generally speaking, agent-based models are systems in which a number of heterogeneous agents interactwith each other and their environment. In particular, agent-based simulations dealing with financial marketsare known as Agent-Based Computational Finance Models (Tesfatsion, 2002). Specifically, the financialmarkets can be viewed as examples of evolutionary systems populated by agents with different tradingstrategies. Recently, we have witnessed a transition in economics and finance from the rational agentapproach to agent-based markets populated by heterogeneous agents with bounded rationality (Hommes,2006; Lux, 2008; Chen et al., 2012). This comes as an alternative to the rational expectations frameworkwhere all agents have fully, unbounded rationality.In this context, the term bounded rationality, first introduced by Simon (1955), suggests that individualsare not capable of the optimization levels required by full rationality. Instead, because of their limitations,market participants make choices that are satisfactory, but not necessary optimal. This means that agentsusually follow strategies that have performed well in the recent past, according to different measures ofaccumulated wealth or profits. Interestingly, even the simplest trading strategies may survive the competitionbecause of the co-evolution of prices and beliefs in the heterogeneous world.One of the rapidly growing research directions in the field is to combine the agent-based modellingwith econometrics and numerical simulations (Chen et al., 2012). Nowadays, researchers do not only tryto illustrate the basic mechanisms, but also quantitatively recreate the statistical properties of financialmarkets. Specifically, heterogeneous Agent Based Models (ABM) that use simple trading strategies havesuccessfully generated a rich set of key properties that match the dynamics of asset prices in real financialmarkets. The ABM can provide useful insight on the behaviour of individual agents and also on the effectsthat emerge from their interaction, making the financial markets a very appealing application for agent-basedmodelling. This has led to the development of various models that aim to understand the links betweenempirical regularities of the markets and the complexities of the entire economic system (for recent surveysof this research area see e.g. Chen et al. (2012); Chiarella et al. (2008); Hommes (2006); LeBaron (2006);Westerhoff (2010)). However, while some models have been able to provide possible explanations for variousproperties of financial markets, no single one has produced or explained all the important empirical featuresof trading data, including volume, duration, price and especially asset returns (Lux, 2008; Tesfatsion, 2002).To this end, independent studies have revealed a set of statistical properties common across differentfinancial instruments, markets and time periods. Due to their robustness across various financial markets,they are called stylized facts in the econometrics literature (Cont, 2001). Specifically, these stylized factstarget a wide range of aspects in the market, from price series and returns, to trading volume and volatility.The heterogeneous agent models attempt to explain these statistical properties of financial time seriesendogenously, by considering the interaction of market participants (Lux, 1995, 1998; Brock and Hommes,1998; LeBaron et al., 1999). The most important stylized facts refer to the heavy tails in the distributionof asset returns, the persistence and clustering in return volatility, the characteristics of trading volumeand the cross correlations between returns, volumes and volatility (Cont, 2001). Models which are able toproduce outcomes close to the empirical stylized facts are very useful for studying the effects of regulations,trading protocols or clearing mechanisms. For this reason, the financial markets have become one of themost active research areas in agent-based modelling.Following the classification of Chen et al. (2012), there are, by and large, two main design paradigms offinancial agents. The first one is referred to as the N-type design, and considers a relatively small numberof agents with simple interactions, where all the agents’ types and rules are given at the beginning of thedesign. On the other hand, in the autonomous-agent design complex interactions may lead to a large numberof agents. Unlike the N-type design, in the autonomous-agent the agents are no longer divided into a fixednumber of different types, their behaviour being customised via artificial intelligence algorithms. However,we are ultimately interested in constructing a model that mimics the most important properties of financialdata and which is analytically tractable at the same time. Due to the high complexity of real markets it ishard to find the origins of critical events. Since we encounter a wide variety of trading strategies, motives2nd irrationality in the real world, a simple N-type design model is preferred. Focusing on how agentsbehave, we can observe the minimum conditions required to replicate the stylized facts in terms of bothheterogeneity and complexity of the rules. Specifically, in this work we will focus on a two-type design, aparticular case of the N-type design. Highly motivated and backed by empirical work, we will show howthis type of financial model successfully meets our goals. In the same time, we will keep a high degree oftransparency and clarity at the system level, making it easier for analysing the results.In more detail, the main objective of Agent-Based Computational Finance Modelling is to proposean alternative to the apparent randomness of financial markets, trying to explain the most importantproperties of financial data. In particular, we are interested in simple structures that can reproduce theempirical findings to a high degree and which are quantitatively close to the real ones. Models that relyon simple trading strategies have proven themselves very efficient in generating important dynamics of realfinancial markets. Specifically, the fundamentalist vs. chartist models (to be defined) that incorporate all theimportant mechanisms of real financial markets are capable of recreating their key properties and are simpleenough for analysis and computation. This strongly motivates our choice of focusing on fundamentalists vs.chartists model and we will show how effective it is in matching a rich set of stylized facts.In these models, the first type of agents, fundamentalists , act as a stabilizing force in the market.Fundamentalists base their trading strategies and expectations of future prices on economic and marketfundamentals such as earnings, growth and dividends. They believe in the existence of a fundamentalprice and invest relatively to its value. Therefore, a fundamentalist will most likely buy an asset that isundervalued; that is, one whose price is below the believed fundamental value, and sell an asset that isovervalued, that is, one whose price is above the fundamental value. In their opinion, the price series willeventually move or converge towards a long term equilibrium or fundamental value.In contrast, the other type of agents, chartists or technical analysts, forecast the future prices entirelyby modelling historical data. Surveys that brought to the attention of academics the use of technicalanalysis were conducted by Frankel and Froot (1987, 1990a,b), Allen and Taylor (1990); Taylor and Allen(1992), and more recently by Gehrig and Menkhoff (2004) and Menkhoff and Taylor (2007). Technicalanalysts do not take into consideration the market fundamentals and base their decisions entirely on observedhistorical patterns in past prices. They clearly believe that price developments display recurring patterns.Therefore, while the fundamentalists take into consideration only the fundamental price, the chartists basetheir decisions and trading strategies on historical prices.Following some of the well-known approaches in the literature, we consider the Structural StochasticVolatility model (FW) introduced by Franke and Westerhoff (2012). It is one of the most successful modelsin capturing the empirically observed traders’ behaviour (Barde et al., 2015). However, the price seriesgenerated by the FW model violates one of the core properties of financial time series – its non-stationarity.We overcome this problem by extending the original model and changing the motion of the fundamental valueover time. As a result, we drastically reduce the non-stationarity of the price series generated. Furthermore,a (parametric) bootstrap method is for the first time used to estimate the model’s parameters in this setting,thus overcoming the joint-point problem associated with other methods. Finally, we evaluate the model andshow it is able to match a rich set of stylized facts of real financial markets. Specifically, we make thefollowing contributions: • Checking for stationarity in the price series generated by the FW model, we observe that the unit roottest is rejected in more than one in every four simulations (see Section 2.3 for more details). This isequivalent to having a price distribution with a mean and variance that do not change over time. Insuch a setting, we usually observe a mean-reverting behaviour with the price fluctuating around thefundamental value and which can easily be exploited. This strongly contradicts the well-known factthat in financial markets the price series are non-stationary. A non-stationary process has mean andvariance that change over time and, as a rule, is unpredictable and cannot be forecasted.The failure of simulations to produce the non-stationary prices observed in real financial markets isdue to the unrealistic assumption of a constant fundamental value. Therefore, we build on the originalmodel by making a novel change in the motion of the fundamental value over time. Accordingly, weassume the fundamental value obeys a random walk. Specifically, we set the fundamental price to3ollow a Geometric Brownian Motion (GBM), which is the most widely used model of describing stockprice behaviour (Hull, 2006). As a consequence, the price series generated by the modified model(FW+) rejects the unit root test for stationarity in less than 5% of the simulations. • In the FW+ model, the agents interact stochastically, their switching between strategies being set bya discrete choice approach, influenced by a series of factors such as herding and price misalignment. Inthis setting, we estimate the model’s parameters numerical values using a powerful method of simulatedmoments introduced by Franke and Westerhoff (2014). The method, based on a (parametric) bootstrapprocedure used to evaluate the model’s goodness-to-fit, is different from the other bootstraps or MonteCarlo experiments discussed in the literature and overcomes the joint-point problem (see Section 3 formore details). This is the first time it is applied to a model where the interaction between agents isbased on a discrete choice approach. • One of the main objectives of agent-based financial modelling is to recreate the most important prop-erties of financial data. An in depth analysis of the time series generated by our FW+ model isundertaken, providing a wide range of tests and arguments that reinforce the presence of a rich set ofstylized facts including: lack of predictability and non-stationarity, absence of autocorrelation in rawreturns, fat tails, volatility clustering and long memory in absolute returns, volume-volatility relations,aggregate gaussianity, price impact and extreme price events. To date, this is the only model that isreported to match such a rich set of the stylized facts of real financial markets. Therefore, we illustrateone of the essential aims of the agent-based financial modelling. Namely, that many of the stylizedfacts arise and can be explained just from the interaction of market participants.The remainder of the paper is organised as follows. In Section 2 we formally define the fundamentalistvs. chartist asset pricing model and its key mechanisms, discussing its dynamic properties. In Section 3we estimate and validate the model’s parameters. Next, in Section 4, we present our results. We give athorough analysis of the price series generated by the agent-based model, integrating a wide range of testsand evidence that demonstrate the presence of a rich set of stylized facts. Section 5 concludes.
2. Model formulation
In this section, we extend the fundamentalists vs. chartists asset pricing model by Franke and Westerhoff(2012) (FW) to capture new properties usually observed in real financial markets. In Sections 2.1 and 2.2we formally define the asset pricing model, with its pricing dynamics and agents’ interactions. Next, inSection 2.3, we describe our contribution regarding the movement of the fundamental value, finishing witha discussion on the dynamics of the model in Section 2.4.
The asset price changes are determined by excess demand, as in the original FW model, following someof the most prominent examples in the literature (Beja and Goldman, 1980; Farmer and Joshi, 2002).Here, by excess demand we mean the precise positive or negative orders per trading period. The specificdemand of each trader type is kept as simple as possible, in the form of the demand per average trader.For fundamentalists, the demand is inversely related to the difference between the current price and thefundamental value. That is, at time t their core demand D ft is proportional to the gap ( p ft − p t ), where p t is the log price of the asset at time t , while the p ft is the fundamental log value at time t . Similarly, the coredemand of the chartists’ group, D ct , is proportional to the price changes they have just observed, ( p t − p t − ),where p t and p t − are the log prices at time t and t −
1, respectively.The wide variety of within-group specifications are captured by noise terms added to each of the coredemands. These terms encapsulate the within-group heterogeneity and scale with the current size of thegroup. Specifically, the noise is represented by two normally distributed random variables (cid:15) ft and (cid:15) ct for fun-damentalists and chartists, respectively. In other words, one can think of the noise variables as a convenientway of capturing the heterogeneity of markets populated by hundreds or thousands of different agents. Each4f the two noise terms are sampled at every iteration and added to the deterministic part of the demand,leading to the total demand per agent within the corresponding group.Thus, combining the deterministic and stochastic elements, we get the net demand of each group for theasset in period t as follows: D ft = φ ( p ft − p t ) + (cid:15) ft (cid:15) ft ∼ N (0 , σ f ) φ > , (cid:15) f > , (1) D ct = χ ( p t − p t − ) + (cid:15) ct (cid:15) ct ∼ N (0 , σ c ) χ ≥ , (cid:15) c ≥ , (2)where φ and χ are constants denoting the aggressiveness of traders’ demand and σ ft and σ ct are noisevariances.The agents are allowed to switch strategies at each iteration, so their market fractions fluctuate over time.For simplicity, we fix the agents’ population size at 2 N . Let n ft and n ct be the number of fundamentalistsand chartists in the market at time t , respectively. We define the majority index of the fundamentalists as: x t = ( n ft − n ct ) / N (3)Therefore, x t ∈ [ − , x t = − x = +1) corresponds to a market where all traders arechartists (fundamentalists). Moreover, the market fractions can be expressed in terms of the majority indexas: n ft / N = (1 + x t ) / , n ct / N = (1 − x t ) / . (4)The (scaled) total demand D t , given by the equation: D t = n ft D ft + n ct D ct = (1 + x t ) D ft / − x t ) D ct / µ > t + 1 results from equations 1-5 as: p t +1 = p t + µ x t ) φ ( p ft − p t ) + (1 − x t ) χ ( p t − p t − ) + (cid:15) t ] , (6) (cid:15) t ∼ N (0 , σ t ) , σ t = [(1 + x t ) σ f + (1 − x t ) σ c ] / . (7)Equation 7 is derived as the sum of the two normal distributions from Equations 1 and 2, multiplied bythe market fractions (1 ± x t ) /
2. The result is a new normal distribution with mean zero and variance equalto the sum of the two single variances. The combined variance σ t depends on the variations of the marketfractions of the fundamentalists and chartists. This random time-varying variance is a key feature of themodel, being termed the structural stochastic volatility (SSV) of returns (defined as the log differences inprices) in Franke and Westerhoff (2009). To complete the model, it remains to set up the motions of the market fractions n ft and n ct . They arepredetermined in each period and change only from one period to the next one. Following the discretechoice approach (DCA) introduced by Brock and Hommes (1997), the two market shares n st +1 ( s = f, c )can be determined using the multinomial logit model. In the basic setting, some payoff indices u ct and u ft are considered, usually derived from past gains of the two groups. The market fractions can be expressedas n st +1 = exp ( βu st ) /exp ( βu ft ) + exp ( βu ct ), where β is the intensity of choice. Dividing by exp ( βu ft ), themarket fraction of fundamentalists is given by n ft +1 = 1 / { exp [ − β ( u ft − u ct )] } .5ote that the difference in any utility variables, ( u ft − u ct ), can be viewed as a measure of relative attrac-tiveness of fundamentalist trading. Hence, we may change the notation, making use of the attractivenesslevel , a t , defined as the difference ( u ft − u ct ). The discrete choice approach is then given by: n ft +1 = 11 + exp ( − βa t ) , n ct +1 = 1 − n ft +1 . (8)We normalize all the demand terms in the price rule 6 by using a market impact µ = 0 .
01. Furthermore,we fix the intensity of choice β = 1. Of course, setting these values is just a matter of scaling the marketimpact on prices and the relative attractiveness level a t of fundamentalism, respectively.Note that the market fractions are directly influenced by the attractiveness level. An increase in theindex a t leads to an increase in the market share of the fundamentalists. For this reason, it is extremelyimportant to define the exact mechanism of the attractiveness level and all of its components.Following the FW model setting, we consider three principles that may influence the way in which theagents choose one of the two strategies. The first principle is a herding mechanism, meaning that the moreagents are in a group the more attractive that group becomes. This idea has been long used in the literature(see, for example, Kirman (1993) and Lux (1995)) and can be easily represented by a term proportional tothe most recent difference in the market fractions ( n ft − n ct ).The second principle is based on a certain predisposition towards one of the two trading strategies. Thiscan be directly captured by a constant α , which is positive (negative) if the agents have a priori preferencefor fundamentalism (chartism). Finally, the third component encapsulates the idea of price misalignment.This is empirically backed by Menkhoff et al. (2009), who observes that when the price is further away fromits fundamental value professionals tend more and more to anticipate its mean reversion toward equilibrium.In a setting where the current price tends to move far away from the fundamental value, chartism becomesriskier and the fundamentalism is more attractive. Hence, it is convenient to make a t rise in proportion tothe squared deviations of the price p t from the fundamental value p ft .Combining the three components of the attractiveness level, we have: a t = α + α n ( n ft − n ct ) + α p ( p t − p ft ) , (9)where α is the predisposition parameter, α n > α p > σ t . Secondly, the changesof the market fractions n ft and n ct are described by Equations 8 and 9, encapsulating a herding mechanismcorrected by a strong price misalignment. The whole system has a recursive structure that is easily forwarditerated. Financial time series are characterised by the lack of predictability, mathematically known as the martingaleproperty. It is a fundamental ingredient of theoretical models stating that knowledge of past events neverhelps predicting the future movements. The lack of predictability of price series is well known and docu-mented, with a well-established body of literature offering plausible generic explanations of this stylized fact(Lux, 2008). It is explained by traditional finance as a consequence of the informational efficiency, that is,all currently available relevant information is already embedded in the market price. Hence, the price canbe changed only by the arrival of new information.A similar, very common property of financial time series data is its non-stationarity. In particular, anon-stationary process is a stochastic process whose joint probability distribution changes when shifted intime. Consequently, its mean and variance change over time. Usually, stock prices are defined as examples ofrandom walks, a non-stationary process. It is commonly assumed that non-stationary data is unpredictableand cannot be forecasted.In more detail, the non-stationarity of financial data has been discussed and studied for a long time and,similarly to the martingale property, arises from the theory of efficient markets (Pagan, 1996). A property6 igure 1: Augmented Dickey-Fuller stationarity test results.Table 1: Model parameters φ χ σ f σ c α α n α p µ p σ p p ft = 0. In 10000 simulations, the unit root testwas rejected 2716 times with the p-value being less than the critical value at 10%. Moreover the test wasrejected 1557 times with the p-value less than the critical value at 5%. The p-values are obtained usingthe updated tables from MacKinnon (2010). Therefore, we can say that the price series generated by themodel using a constant fundamental value are stationary approximately 27% of the time, which stronglycontradicts the behaviour of real financial time series. The failure to pass the unit root test is mainly dueto the unrealistic assumption of a constant fundamental value. As a solution, we extended the setting byallowing the fundamental value p ft to change over time. Specifically, we set p ft to follow a geometric Brownianmotion. This is the most widely used model of describing stock price behaviour (Hull, 2006) and is usuallyapplied in quantitative finance. Mathematically, the fundamental price is given by dp ft = µ p p ft dt + σ p p ft dW t , (10)where W t is a Weirner process, µ p is the percentage drift and σ p is the percentage volatility. The exactvalues of the drift and volatility will be estimated together with all the other model’s parameters in Section3. With this change in the model, we run the Augmented Dickey Fuller test once again over 10000 differentsimulations. Now, the test is rejected only 750 times with the p-value less than the critical value at 10% and378 times with the p-value less than the critical value at 5%. Thus, we can say that the price series generatedby the new model (FW+) are non-stationary in more than 93% of the simulations (with 95% confidenceinterval). A visual representation of the number of non-stationary price series generated in both models can7 a) Simulated log prices(b) Simulated majority index(c) Simulated returns(d) Empirical returnsFigure 2: Simulated price series ( p t ), fundamental price series ( p ft ), majority index ( x t ) and returns ( r t ) over 6867 time periods,together with empirical S&P 500 returns.
8e seen in Figure 1. We plot the percentage of the non-stationary price series generated as we increase thenumber of simulations. The red and blue lines represent the initial and our modified model, respectively.The hard (dotted) lines are the computed ADF results at 10% (5%) critical values. We can observe howthe number of non-stationary price series is drastically reduced, from 27% with the initial model (at 10%critical size) to 4% in our modified model (at 5% critical size).Therefore, we can say that the new model produces more realistic price series, since in real financialmarkets the prices follow martingales where knowledge of past events does not help predict the mean of thefuture changes. Unlike the original (FW) model, our improved (FW+) setting better reflects some of thefundamental properties of real financial time series – their non-stationarity and unpredictability.
We now turn to explore the qualitative measures that demonstrate the existence of the most importantstylized facts of financial markets. An in-depth analysis of the price, returns, volatility and volume seriesgenerated by the FW+ model will be performed in Section 4. The numerical parameters used for all thesimulations are given in Table 1.In Figure 2 we present a simple run of the model over 6867 days, covering the same time span as theempirical data represented by Standard and Poor (S&P) stock market index from January 1980 to mid-March 2007. The top panel illustrates the (log) price series p t generated by the model and the fundamentalvalues p ft . We can observe the irregular swings in the prices, with a considerable amplitude, similar to thebehaviour of real financial series, reflected by the empirical returns in Figure 2(c). The second panel showsthe corresponding composition of the traders, or the fluctuations in the market fraction. The majority index x t is moving continuously, changing from periods of fundamentalists domination (positive values) to periodsof chartists domination (negative values). However, it shows that most of the time the market is dominatedby fundamentalists ( x t > . x t ≈ − . σ c > σ f , by comparingFigures 2 (b) and (c) we observe that the level of returns during a chartism domination exceeds its level in afundamentalism regime. Therefore, it appears that normal sequences of returns are interrupted by outburstsof increased volatility, when the majority of agents are chartists.
3. Estimation of the model’s parameters
In this section we discuss the formal numerical estimation of the model’s parameters, using one of the mostprominent methods from the literature, the Method of Simulated Moments (MSM). For an overview of themethod of simulated moments and its applications to agent-based models see the work of Winker et al.(2001); Gilli and Winker (2003); Amilon (2008); Franke (2009). This technique is based on an objectivefunction that is optimised across the set of FW+ model’s parameters ( φ, χ, σ f , σ c , α , α n , α p , µ p , σ p ). Themethod refers to a set of statistics, also known as moments, arising from the simulations. The basic idea isthat these moments should be close to their empirical counterparts, with the distance between them beingcaptured by the objective function. An agent-based model is not supposed to mimic the exact economy orfinancial markets, but rather it should explain some of the most important stylized facts of financial markets,9 lgorithm 1 Bootstrap estimation of the covariance matrixSet I = { , , . . . , T } for b = 1 : B do Draw T random numbers with replacement from I Construct bootstrap sample I b = { t b , t b , . . . , t bT } Calculate vector of moments m b = ( m b , . . . , m b ) from I b end for ¯ m = B (cid:80) b m b ˆΣ = B (cid:80) b ( m b − ¯ m )( m b − ¯ m ) (cid:48) W = ˆΣ − making this estimation method suitable for our purposes. In this paper, we calibrate the FW+ model usingthe same dataset of T=6866 daily observations of the Standard and Poor (S&P 500) stock market indexfrom January 1980 to mid-March 2007, as in the FW model. The estimation was conducted such that thefour most discussed statistical properties of empirical financial data are matched. These include absenceof autocorrelation in raw returns, heavy tails, volatility clustering and long memory. The returns, or (log)price changes, are expressed as percentage points, such as: r t = 100( p t − p t − ) . (11)Given this, the volatility of returns is defined as their absolute value v t = | r t | .In order to conduct the quantitative analysis, the four stylized facts were measured using a number ofsummary statistics, or moments. The first moment considered is the first-order autocorrelation coefficientof raw returns. It needs to be close to zero, so that it agrees with the empirical findings on this matter(Kendall and Hill, 1953; Fama, 1965; Bouchaud and Potters, 2003; Chakraborti et al., 2011; Cont et al.,2014). This will limit the chartists’ price extrapolations of the most recent price changes. Furthermore, ifthe model generates coefficients that are insignificant, all the autocorrelations at longer time lags will vanishtoo.Furthermore, the remaining three moments are dealing with the volatility of returns. First of all, themodel should suitably scale the overall volatility, thus limiting the general noise brought by the two variances σ f and σ c . The mean value of the absolute returns is considered. Next, the heavy tail is measured by theHill tail index of the absolute returns. The tail is specified as the upper 5% in order to eliminate bias andconsider a more accurate tail index.The long memory effects are captured by the autocorrelation function (ACF) of the absolute returns upto a lag of 100 days. In particular, the autocorrelation decays as we increase the lag, without becominginsignificant. The entire profile has to be matched and is sufficiently well represented by six different lagcoefficients ( τ = 1 , , , , , τ = 1 , , , , ,
100 for the absolutereturns), summarized in a (column) vector m = ( m , ..., m ) (cid:48) (the prime denotes transposition). Applyingthe method of simulated moments, they have to come as close as possible to their equivalent empiricalmoments, m emp , calculated on the daily S&P 500 stock market index.The distance between the two vectors m and m emp is defined as a quadratic function with a suitableweighting matrix W ∈ R x (defined shortly). Hence, the distance is given by: J = J ( m, m emp ; W ) = ( m − m emp ) (cid:48) W ( m − m emp ) . (12)The weight matrix W accounts for the moments’ sampling variability, its determination being crucialfor the model’s parameters. The idea is that the higher the sampling variability of a given moment i ,the larger the differences between m i and m empi that can still be considered insignificant. This behaviourcan be achieved by correspondingly small diagonal elements w ii . Moreover, the matrix W should support10 lgorithm 2 Parameter estimation for a = 1 : 1000 do ˆ θ a = minimise(funJ, method=Nelder-Mead) end for ˆ θ = ˆ θ ˜ a , where ˜ a such that ˆ J ˜ a = median of { ˆ J a } a =1 and ˆ J a = J [ m a (ˆ θ ; S ) , m empT ]def funJ( θ ):Simulate model using vector of parameter θ Get simulated moments m sim = ( m , . . . , m )Get J = J ( m e mp, m empT ; W ) = ( m sim − m empT ) (cid:48) W ( m sim − m empT )Return J possible correlations between single moments. One obvious choice for W is the inverse of an estimatedvariance-covariance matrix ˆΣ of the moments (Franke, 2009), W = ˆΣ − . (13)The covariances in ˆΣ are estimated by a bootstrap procedure used to construct additional samples fromthe empirical observations. In the literature, this is often carried out by a block bootstrap (Winker et al.,2007; Franke and Westerhoff, 2011, 2012). However, the original long-range dependence in the return series isinterrupted every time two non-adjacent blocks are pasted together. Hence, the independence of randomlyselected blocks cannot reproduce the dependence structure of the original sample. This is known as thejoint-point problem (Andrews, 2004). Since our estimation is concerned with summary statistics, we canovercome the joint-point problem by avoiding the block bootstrap.Correspondingly, in order to estimate the variance-covariance matrix ˆΣ, we use a new bootstrap methodfirst described by Franke and Westerhoff (2014). In their work, the authors apply this method to theFW model with the interaction between agents being modelled by the transition probability approach (Lux,1995). Therefore, we are for the first time applying this new bootstrap framework to a model where the agentsinteract stochastically according to the discrete choice approach Brock and Hommes (1997). Departing fromthe traditional block bootstrap, we sample single days and, associated with each of them, the history of thepast few lags required to calculate the lagged autocorrelations. In more detail, the procedure is presentedin Algorithm 1. We construct the set of time indices, I = { , , . . . , T } , and for each bootstrap sample b ,we can sample directly from it. Accordingly, a bootstrap sample is constructed by T random draws withreplacement from I . Repeating this B times, we have b = 1 , . . . , B index sets, I b = { t b , t b , . . . , t BT } , (14)from witch the bootstrapped moments are obtained.For a good representation, the bootstrap method is repeated 5000 times, obtaining a distribution for eachof the moments. Formally, let m b = ( m b , ..., m b ) (cid:48) be the resulting vector of moments, and ¯ m = (1 /B ) (cid:80) b m b be their mean values. Then, estimate of the moment covariance matrix ˆΣ becomes,ˆΣ = 1 B B (cid:88) b =1 ( m b − ¯ m )( m b − ¯ m ) (cid:48) . (15)Going back to the estimation problem, we are interested in the set of parameters that minimise thedistance function J from Eq. 12. In order to reduce the variability in the stochastic simulations, the timehorizon is chosen to be longer than the empirical sample period T , commonly defined as S = 10 · T . Repeatedsimulations over S periods (or days) are carried out, in search of the set of parameters that minimise theassociated loss. To this end, let θ be the vector of parameters and m = m ( θ ; S ) denote the moments towhich a vector θ gives rise. 11 igure 3: Distribution of objective function J. Furthermore, the comparability of different trials of θ is determined by a random seed a = 1 , , . . . , let ussay. Moreover, let m a ( θ, S ) denote the moment vector obtained by simulating the model with a parametervector θ over S periods on the basis of a random seed a . The parameter estimates on a random seed a ,denoted ˆ θ a , are the solution of the following minimisation problem,ˆ θ a = argmin θ J [ m a ( θ ; S ) , m empT ] , (16)where m empT is the moment vector for the empirical S&P 500 data .Although one may think that a simulation over S = 68660 days provides a large sample to base themoments on, the variability arising from such different samples is still considerable. Hence, it seems mostappropriate to carry out a great number of estimations ( a = 1000) and choose the one with the medianloss. The parameter set ˆ θ giving this associated loss will be our representative estimation (see Algorithm2). Specifically, using Equation 16 we have,ˆ θ = ˆ θ ˜ a , where ˜ a is such that ˆ J ˜ a is the median of { ˆ J a } a =1 , and ˆ J a = J [ m a (ˆ θ a ; S ) , m empT ] , a = 1 , . . . , θ resulting from the estimation has already been reported in Table 1. Thecorresponding minimized loss is 5 . J = J [ m ˜ a (ˆ θ ; S ) , m empT ] = 5 .
416 (18)A visual representation of the distribution of the objective function J can be observed in Figure 3. Inthe left panel, we plot the function as small changes of the last two parameters of the model, the drift andvolatility of the fundamental price. We observe how it doesn’t depart from small values, staying in theneighbourhood of the median we chose as optimal value. In the right panel, we plot the actual distributionof J .To sum up, the estimation of the model’s parameters is based on the minimisation of Equation 16, wherethe objective function J is defined by Equations 12, 13, 14 and the set of nine moments described earlier.The model was validated on the S%P 500 data, leading to the set of parameters presented in Table 1. Theseare the values of parameters we used to generate all the results discussed in the following section. For the actual minimisation problem we use the Nelder-Mead simplex search algorithm (Nelder and Mead, 1965). igure 4: Autocorrelation function of returns. The red lines (blue dots) indicate empirical (simulated) returns. The two upper(lower) lines represent the autocorrelation of volatility (raw returns) at lags τ = 1 , ...,
4. Results
In this section, we explore the statistical properties generated by our model. Specifically, we provide bothquantitative and qualitative measures that demonstrate the existence of the most important stylized facts offinancial markets. An in-depth analysis of the price, returns, volatility and volume series generated by thenew model will be performed. We show that the model is able to match a rich set of properties includingmartingales, absence of autocorrelations in raw returns, heavy tails, volatility clustering and long memoryin absolute returns, volume-volatility relations, aggregate Gaussianity, a concave price impact and extremeprice events.
The lack of predictability of price series is one of the most distinctive characteristics of financial time series.It has puzzled both researchers and investors over the years, making it one of the most discussed propertiesof financial data. A similar, very common property of financial data is its non-stationarity. Stock priceshave probability distributions whose mean and variance change over time. Accordingly, they are defined asexamples of random walks, a non-stationary process. This has motivated us to change the behaviour of thefundamental price to a geometric Brownian motion, one of our main contributions, such that the price seriesgenerated by the model are non-stationary (see Sec. 2.3).A further, well-known property of financial data states that price movements do not exhibit any significantautocorrelation (Cont, 2001). The autocorrelation function defined as: C ( τ ) = corr ( r ( t, ∆ t ) , r ( t + τ, ∆ t )) , (19)where corr denotes the sample correlation, rapidly decays and even for small interval time is close to zero.Many studies have found a rapid decline of ACF after the first lag, therefore confirming the absence of(linear) autocorrelation in returns at all horizons and making it a well accepted stylized fact (Working, 1934;Kendall and Hill, 1953; Fama, 1965; Bouchaud and Potters, 2003; Chakraborti et al., 2011).The absence of autocorrelation in returns can easily be demonstrated using the autocorrelation function.This basic stylized fact is demonstrated in Figure 4, where the ACF of raw returns is computed at lags from1 to 100. We compare the ACF of both simulated and empirical returns showing that it becomes close to13 a) Raw returns distribution. The red line represents a normaldistribution superimposed on the return distribution. (b) Volatility distribution. The red line represents an exponen-tial distribution superimposed on the volatility distribution.Figure 5: Distribution of (a) raw returns and (b) absolute returns. zero after the first lag. This behaviour is in line with the empirical findings, reinforcing the idea that pricechanges are not correlated. Another challenging topic in the econometrics literature is the distribution of returns. Consider the distri-bution on raw returns, plotted in Figure 5(a). We can immediately observe that the distribution displaysstrong deviations from Gaussianity. Returns of stock prices, like returns of many other financial assets,are bell shaped similar to the normal distribution, but contain more mass in the peak and the tail thanthe Gaussian distribution (Fama, 1965; Mandelbrot et al., 1963). Such distributions have excess positivekurtosis, being called leptokurtic. Specifically, the simulated return series has an excess kurtosis of 2.49 anda small negative skewness of -0.0055.The positive excess kurtosis implies a peakiness of the distribution bigger than normal and a slowasymptotic decay of the probability distribution function. This non-normal decay is coined as heavy (orfat) tail (Jansen and De Vries, 1991; Lux, 1996; Gopikrishnan et al., 1998; Jondeau and Rockinger, 2003;Bouchaud et al., 2008). Heavy tails are defined as tails of the distribution that have a higher density thanwhat is predicted under normality assumptions (LeBaron and Samanta, 2005). For example, a distributionwith exponential decay (as in the normal) is considered thin tailed, while a power decay of the densityfunction is considered a fat tail distribution.Next, we consider the distribution of absolute returns, or volatility. As we can see from Figure 5(b) thedistribution of absolute returns has a bigger decay than the exponential distribution superimposed on it,indicating a heavy tail. The power law behaviour in the distribution of absolute returns can be approximatedby fitting a distribution of the form, p ( x ) ∝ x − α , (20)where α is the power law exponent. Following the work of Alstott et al. (2014) and the key recommendationsfrom Clauset et al. (2009), we try to fit a power law distribution to the distribution of absolute returns.Accordingly, we find a power law exponent α = 4 . α emp =4 . The absence of autocorrelation discussed in Section 4.1 does not rule out the possibility of nonlinear depen-dencies of returns. It is well known that absence of serial correlation does not imply independence (Pagan,1996). Even simple visual representations of return series (see Figure 5(b)) reveal heteroscedasticity as aviolation of the assumption of independently and identically distributed returns. That is, volatility mea-sured as squared or absolute returns is not constant in time. Fluctuations of volatility were first noted byMandelbrot et al. (1963) in daily returns of cotton prices. The author reported that periods of high volatilityalternate with periods of low volatility. Further early examples of heteroscedasticity were found in Fielitz(1971); Wichern et al. (1976); Hsu (1977), to name a few.Furthermore, nonlinear representations of returns, such as absolute, squared or various powers of returns,exhibit a much higher positive autocorrelation that persists over time. The powers of absolute returns canbe seen as measures of volatility, indicating a high degree of predictability of volatility. This phenomenonis stable across different financial instruments and time periods, being a quantitative signature of the wellknown volatility clustering. That is, large price variations are more likely to be followed by large pricevariations. This behaviour has been first noted by Mandelbrot et al. (1963), while Cont (2005) provides anextensive study on volatility clustering in financial markets.We can observe this behaviour just by inspecting the returns generated by the model (Figure 2(c)).Periods of relatively small returns are interrupted by abrupt increases in returns. Moreover, these periodstend to be clustered together. Specifically, small price changes are followed by small ones and large pricechanges by large ones.A common way of confirming the presence of volatility clustering is by considering its autocorrelationfunction. Even though there are different ways of measuring volatility, the most commonly used ones arethe absolute returns, C a ( τ ) = corr ( | r ( t + τ, ∆ t ) | , | r ( t, ∆ t ) | ) . (21)Plotting out the autocorrelation of volatility measured as absolute returns in Figure 4, we observe apositive autocorrelation that persists over time, doubled by its slow decay. This is a clear presence ofvolatility clustering.A property closely related to the volatility clustering effect is the decay of the autocorrelation function.The long memory effect specifically addresses this decay. Mandelbrot (1971) was the first one to suggestthis stylized fact and observed it in many empirical studies (Mandelbrot and Taqqu, 1979). Long rangedependencies have been found across different markets and periods (Liu et al., 1997, 1999; Cont, 2005;Chakraborti et al., 2011; Cont et al., 1997). Usually, if the decay is slow, similar to a hyperbolic function,we can say that the corresponding process exhibits long memory. A possible explanation of this stylized factis that investors with different time scales interact in the market, which typically results in a mixture of long-short relaxation times (the delay to secondary reactions unfold), following the impact of exterior events orinformation. Thus, different relaxation times combine, leading to a hyperbolic decay in the autocorrelationfunction.One way of observing the decay in the autocorrelation function is by fitting a power law of the form, C a ( τ ) ∼ Aτ β , (22)with empirical studies finding the coefficient β ≤ . α ≈ .
472 (Figure 4).15 a) Absolute returns vs. Volume (b) Squared returns vs. VolumeFigure 6: Volume-Volatility relations. Cross correlation between volatility measured as (a) absolute returns and (b) squaredreturns and volume.
Another widely used method of testing the long memory effect is by using the Hurst exponent (Hurst,1951), which falls in the range 1 / < H < The relationship between volatility and volume traded is important for understanding how informationis transmitted and embedded in markets. It has been noticed and documented across different financialinstruments at different time scales. The volume-volatility relation was first observed by Osborne (1959),with the positive correlation between them being first detected by Ying (1966). For an early survey seeKarpoff (1987). In addition, the causalities of volume on price changes were studied in Chuang et al. (2009),concluding that it exhibits a V-shape relation so that the diffusion of volatility distribution increases withvolume.This stylized fact is at the heart of determining the underlying mechanisms of crashes and rapid changesin the stock prices. Furthermore, an intensive analysis on volume can help investors identify periods inwhich informational shocks occur, providing valuable information about future price changes. In Figure 6we can observe a significantly positive cross correlation between volatility and volume. In this setting, wedefine the volume at time t as the total absolute demand at t of both fundamentalists and chartists. It isimportant to mention that all measures of volatility are positively correlated with traded volume.The significantly positive cross correlation means that a small (large) trading volume is accompanied bya small (large) change in volatility. This behaviour is reflected in the way future prices are determined bythe market maker, demands (and therefore volume) directly influencing volatility changes. It is importantto note that the dependence between volume and volatility remains significant as we increase the time lag.From the investor’s point of view, this seems to be the only dependency that can be exploited. As we haveseen so far, the returns are not correlated (see Section 4.1) and even though the volatility displays positivecorrelation, it is less significant and slowly decays as we increase the lag, a signature of volatility clustering(see Section 4.3).A further easily observable property of returns is the aggregate Gaussianity (Cont, 2001). As we haveseen, the distribution of returns has excessive kurtosis and heavy tails, with a higher peak than the normaldistribution and power law tails. However, this behaviour changes as we increase the time scale over whichthe returns are calculated. More specifically, the distribution of returns changes its shape, becoming morelike the normal distribution. This stylized fact can easily be observed in different markets and time periods.16 igure 7: Aggregate Gaussianity. Distribution of returns calculated at different time scales.Table 2: Values of excess kurtosis and skewness as we increase the time lags at which returns are computed. time lag 1 10 25 50 100kurtosis 2.5116 2.2022 1.6289 1.0661 0.4954skewness -0.0049 -0.0128 -0.0324 -0.0409 -0.0648In more detail, the aggregate Gaussianity of the simulated returns generated by our model is clearly visiblein Figure 7. We plot the distribution of returns calculated for different time lags τ = 1 , , , , So far, we have discussed the volume-volatility and return-volatility correlations. The price impact tacklesthe correlations between the volume and the price changes. More specifically, we are interested in the impactof the volume traded on the prices. There are a few different theories regarding this relation, from linear tonon-linear dependencies (Kyle, 1985; T´oth et al., 2011; Moro et al., 2009; Gopikrishnan et al., 2000; Gabaixet al., 2003). However, the impact of a single order has been widely found to be a concave function ofvolume.Using our simulated data, we plot the changes in prices as a function of volume. As before, we refer tothe volume in time period t as the absolute value of the total agents’ demand in that specific time period. InFigure 8(a), we plot the price changes as a function of volume, clearly obtaining a concave function of priceimpact. We were expecting this kind of behaviour from the model definition. This is because the marketmaker computes the next price as a function of the agents’ demand multiplied by their market fraction.Clearly, if the absolute demands increase, the next price will also increase, but not linearly.Moreover, in Figure 8(b), we show the volume distribution. Although the exact distribution of volumesis hard to find, we notice that its tail can be described by a power law with α ≈ .
5. This is similar to theempirical findings regarding the distribution of the number of trades (Gabaix et al., 2003).17 a) Price impact (b) Volume distribution(c) Price series: crash from t = 5163 to t = 5175. (d) Majority index: crash from t = 5163 to t = 5175.Figure 8: Price impact function, the volume distribution and an extreme price event. Finally, we discuss one of the most recently observed phenomena in real financial markets, the so calledextreme price events. Following the definition of Johnson et al. (2013), a spike (crash) is an occurrence ofan asset price ticking up (down) at least ten times before ticking down (up) with a price change exceeding0.8%. On average, we observe 13 spikes or crashes in each simulation (6866 time periods). In Figure 8 (c)and (d) we present an example of a crash in prices occurring between periods t = 5163 and t = 5175. Weobserve a drop of 17 .
19% from the initial price at time t = 5163 in 8 (c). There is no surprise that the crashoccurs in a period of switch from fundamentalism (majority index > .
5) to chartism domination (majorityindex < − .
5. Conclusions
Heterogeneous agent-based models that rely on simple trading strategies have proven themselves very efficientin generating important dynamics of real financial markets. Indeed, the fundamentalists vs. chartists modelshave been shown to successfully capture empirically observed traders’ behaviour. In this paper, we described,evaluated and extended one of the most recent and successful in capturing real-life market dynamics. First,we presented its key mechanisms and their particular roles in the model structure. Second, motivated bythe violation of a central property of real financial price series, their non-stationarity, we proposed a novelchange in the system. In particular, considering the high percentage of stationary price series generated bythe model, we change the way the fundamental value is computed and make it follow a geometric Brownianmotion. This new model was then shown to overcome the initial violation.18he main objective of agent-based financial modelling is to propose an alternative to the apparent ran-domness of financial markets, trying to explain the most important properties of financial data. Specifically,we are interested in simple structures that can reproduce the empirical findings to a high degree and whichare quantitatively close to the real ones. To this end, we offer both a quantitatively and a qualitativelyanalysis of the simulated asset price series, its returns, volume traded and volatility. By providing a widerange of tests and arguments, we demonstrate the presence of a rich set of stylized facts including absenceof autocorrelations, heavy tails, volatility clustering, long memory, volume volatility relations, aggregateGaussianity, price impact and extreme price events. By doing so, our model is the first to match such a richset of the stylized facts.The precise numerical values of the model’s parameters were obtained following a formal econometricestimation, known as the method of simulated moments. However, we depart from some of the classicalexamples in the literature where a block bootstrap is used and follow the recent work of Franke and West-erhoff (2014), thus overcoming the well-known joint-problem associated with older methods. In so doing,this is the first time this more trustworthy alternative estimation has been applied on a model where theinteraction between agents is based on a discrete choice approach.To date, we have discussed a model where the market participants, i.e. fundamentalists and chartists,change their strategies according to a herding mechanism corrected by a price misalignment. However, weare also interested in exploring other empirically backed factors that force the traders to change their beliefsregarding future price movements. In particular, motivated by the growing literature of behavioural finance,more components can be incorporated in the attractiveness level and explore how they alter the interactionsbetween the agents. It would be interesting to investigate how different factors such as gossip, politicalrumours, or waves of optimism and pessimism can be added in the switching mechanism, making the agentsbehave in a more realistic way and determine how they change strategies.
Acknowledgements
This work was supported by the EPSRC ORCHID project (EP/1011587/1). The authors acknowledgethe use of the IRIDIS High Performance Computing Facility in the completion of this work.
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