درباره پویایی های قیمت پیچیده یک مدل بازار مالی ساده تک بعدی ناپیوسته به همراه معامله گران تعاملی نامتجانس

We develop a financial market model with heterogeneous interacting agents: market makers adjust prices with respect to excess demand, chartists believe in the persistence of bull and bear markets and fundamentalists bet on mean reversion. Moreover, speculators trade asymmetrically in over- and undervalued markets and while some of them determine the size of their orders via linear trading rules others always trade the same amount of assets. The dynamics of our model is driven by a one-dimensional discontinuous map. Despite the simplicity of our model, analytical, graphical and numerical analysis reveals a surprisingly rich set of interesting dynamical behaviors.

Spectacular financial market bubbles have repeatedly been observed in the past, often followed by equally stunning crashes. In some cases, these events even had an impact on the real economy, triggering deeper recessions, for instance. Moreover, the volatility in financial markets may be regarded as excessively high in the sense that prices fluctuate more strongly than warranted by the underlying fundamentals. Also extreme price changes, which make up a large part of financial market risk, occur quite frequently. Detailed empirical accounts on these intriguing phenomena are provided by Sornette, 2003, Shiller, 2005 and Shiller, 2008 and Lux (2010). Obviously, it is important to understand what drives the dynamics of financial markets. Bouchaud et al. (2009) present significant empirical evidence showing that asset prices mainly adjust with respect to the markets’ order imbalances which, of course, originate from the transactions of its market participants. Fortunately, we at least have some empirical evidence on how agents determine their speculative orders. As can be seen from empirical studies involving questionnaires (summarized by Menkhoff and Taylor, 2007), market participants rely on both technical and fundamental trading rules to determine the course of the market. Technical analysis is a trading method that seeks to identify trading signals from past price movements (Murphy, 1999). As a result, technicians - also called chartists - may have a destabilizing effect on the dynamics of financial markets. Fundamental analysis presumes that prices will mean-revert toward fundamental values (Graham and Dodd, 1951), generally inducing some kind of market stability. Similar insights are obtained in laboratory experiments in which human subjects trade in a controlled financial market environment (Smith et al., 1988 and Hommes et al., 2005). But how exactly do markets with a diverse ecology of interacting technical and fundamental traders function? Models with heterogeneous agents take exactly this issue into account. For recent surveys of this burgeoning field of research, see Chiarella et al., 2009, Hommes and Wagener, 2009 and Lux, 2010 and Westerhoff (2009), among others. While some stochastic versions of these models aim at matching the stylized facts of financial markets – several interesting contributions are presented in LeBaron, 2006 and Lux, 2009 and Chen et al. (2010)1 – other studies focus on deterministic setups to improve our basic knowledge of what drives prices in financial markets. Let us briefly outline a few of these frameworks in order to appreciate the insights made in this exciting field of research and to clarify the extent to which our model differs from previous works in this field. • One interesting finding is due to Day and Huang (1990), who show that endogenous price dynamics may be triggered by nonlinear trading rules. In their model, chartists apply a linear trading rule, and their orders destabilize the market close to the fundamental value. The trading behavior of fundamentalists is nonlinear. The more the price deviates from the fundamental value, the more aggressive they become. Eventually, orders placed by fundamentalists exceed orders placed by chartists, and prices are pushed back towards fundamental values. However, close to the fundamental value, chartists again dominate the market and the process repeats itself, albeit in an intricate, unpredictable way. Related models featuring nonlinear technical trading rules have been elaborated by Chiarella, 1992 and Chiarella et al., 2002, and others. • Another interesting insight is that when agents switch between technical and fundamental analysis, a similar dynamic behavior can emerge. Let us suppose the market is dominated by destabilizing chartists. In this case, it is likely that prices disconnect from fundamentals. However, when fundamental analysis becomes more popular, a period of price stability, together with a convergence towards fundamental values, may set in. Brock and Hommes (1998) develop a model in which agents switch between trading rules with respect to their past performance and thus display some kind of learning behavior. In Kirman (1991), agents have social interactions that may lead to swings of opinion. In Lux (1998), traders compare the performance of trading rules but are also subject to herding behavior. • A third natural mechanism of endogenous dynamics is based on market interactions. Let us assume a situation in which technical traders can switch between several financial markets. A market may temporarily become unstable if it attracts numerous chartists from other markets. However, when chartists leave the market again – e.g. when other markets appear to be more profitable – a period of convergence sets in. Models along these lines have been proposed by Westerhoff, 2004 and Chiarella et al., 2005 and Westerhoff and Dieci (2006). The contribution our paper makes is as follows: we develop a novel financial market model with five different types of agents. Technical traders believe in the persistence of bull and bear markets. For instance, these traders optimistically buy assets in a bull market. In contrast, fundamental traders expect prices to return towards fundamental values. In a situation where the market is overvalued (i.e. in a bull market), fundamentalists submit selling orders. Although these two building blocks are standard in the literature, we generalize them in our paper. First, speculators react asymmetrically in bull and bear markets. Here is an example: fundamentalists may trade more (less) aggressively when an asset is overvalued by 10 percent than when it is undervalued by 10 percent.2 Second, some speculators determine the size of their orders using linear trading rules. However, other speculators simply keep the size of their orders constant (always trading the same amount of assets) and only determine the direction of trade with their pertinent trading philosophy. Hence, there are two types of technical and two types of fundamental traders. Finally, a market maker, the fifth type of agent, adjusts prices with respect to excess demand in the usual way. Interestingly, our simple setup constitutes a one-dimensional discontinuous dynamical system which is sufficient to generate a very rich set of dynamical phenomena, including, for instance, irregular fluctuations between bull and bear market regimes, as observed in real markets and first modeled by Day and Huang (1990), yet in a different model environment. This does not, however, imply that the established and sophisticated mechanisms mentioned above do not play an important role in explaining the dynamics of financial markets. It does, however, demonstrate that at least part of the dynamics of financial markets may be due to rather simple deterministic mechanisms. In addition, our paper shows the relevance of discontinuous maps to the analysis of financial market dynamics, a rather new field of applied mathematics that has not yet yielded many results. Nevertheless, note that there are already several interesting economic models that feature piecewise-smooth or discontinuous maps, for example, the pioneering works by Day, 1982, Day, 1994, Day and Shafer, 1987 and Day and Pianigiani, 1991, which have also been used recently in Metcaf (2008) and Böhm and Kaas (2000). It is also worth mentioning the works by Hommes, 1991, Hommes, 1995 and Hommes and Nusse, 1991 and Hommes et al. (1995). Discontinuous models are furthermore discussed in Puu and Sushko, 2002, Tramontana et al., 2009 and Tramontana et al., 2010 and Bischi et al. (2010). The bifurcations occurring in a piecewise-smooth system may be quite different from those occurring in a smooth one. In fact, in the case of a piecewise-linear system (as in our model) the existing bifurcations are either border-collision 3 or contact bifurcations, 4 as the local bifurcations associated with the eigenvalues are always degenerate. The dynamic effects of such bifurcations may differ depending on the nature of the invariant sets and the global properties of the map. The term border-collision bifurcation was used for the first time by Nusse and Yorke, 1992 and Nusse and Yorke, 1995, and is now widely used in this context (i.e. for piecewise smooth maps). These bifurcations have been studied in recent years, mainly due to their relevant applications in engineering. The one-dimensional piecewise linear case, continuous and discontinuous, was considered by Banerjee et al., 2000, Jain and Banerjee, 2003, Avrutin and Schanz, 2006, Avrutin and Schanz, 2008 and Avrutin et al., 2006 and Di Bernardo et al. (2008). However, this simple subject (bifurcations occurring in one-dimensional piecewise-linear discontinuous maps) has still not been studied completely. In this paper we will be faced with some new cases that, to our knowledge, have not yet been considered in the existing literature. This case (called Case IV), which will be described in the last subsection of the paper, deals with the dynamics in the case of negatives slopes in the components and increasing jump of opposite signs (i.e. from a negative to a positive value), for which we can give a complete characterization. In fact, the simple (linear) components in the description of the model allow for a full analytical study on the possible dynamics. Moreover, particular cases that are often neglected in the literature may sometimes become relevant and deserve particular attention. Here we have considered and completely described one such case: the case with slopes both equal to +1+1 in the components (called Case II). Obviously, these mathematical results go beyond the economic model we propose in this paper, i.e. they are useful in general to characterize the dynamics of discontinuous maps. In this paper we analyze the deterministic skeletons of more elaborate stochastic versions of our approach. Our deterministic model can already explain some stylized facts of financial markets such as bubbles and crashes and excess volatility. However, a better matching of the stylized facts would require the inclusion of some kind of exogenous noise. For instance, one could add dynamic noise to the traders’ demand functions or randomize the traders’ reaction coefficients (see Westerhoff and Franke, 2009 for an example). Of course, such a model calibration would be most welcome since it helps us to identify relevant parameter regions. To keep the paper concise, we stick to deterministic models and leave stochastic extensions of our model for future work. The remainder of our paper is organized as follows. In Section 2, we present our model, derive the dynamical system governing the evolution of prices, and single out a few interesting economic scenarios for our model. In Section 3, we explore these scenarios classified into four cases. The results are given analytically in a number of theorems, while graphical and numerical tools are used to illustrate the different cases and show a number of simulations. The last section concludes the paper and offers interesting extensions and directions for future work.

What drives the dynamics of financial markets? Since prices adjust with respect to demand and supply, a number of interesting models have been proposed in the recent past, which explicitly studied how agents determine their speculative investment positions. These papers have a strong empirical foundation since their main building blocks are supported by questionnaire and laboratory evidence according to which speculators rely on both technical and fundamental analysis to predict the directions of future market movements. We contribute to this research field by developing a novel financial market model with four different types of technical and fundamental traders. In particular, we take into account that traders may react asymmetrically to bull and bear market situations and that they may either formulate their orders on the basis of linear trading rules or simply prefer to trade fixed amounts of assets. From an economic point of view, our main results may be summarized as follows: • In (special) Case I we assume only the existence of type 1 chartists and type 1 fundamentalists. As a result, the price dynamics either converges towards the fundamental value or it explodes. • (Special) Case II highlights the particular role of type 2 traders. We have here an example of an extremely simple financial market model with heterogeneous traders which is nevertheless able to generate periodic or quasi-periodic price dynamics, at least for a subset of the parameter space, and thus reveals one potential engine of excess volatility. • In Case III type 1 and type 2 chartists and type 1 and type 2 fundamentalists are active. However, type 1 chartists dominate type 1 fundamentalists and type 2 fundamentalists dominate type 2 chartists. Compared to (special) Case II, bounded price dynamics now is always chaotic. • Finally, in Case IV we explore parameter constellations in which type 1 fundamentalists (strongly) dominate type 1 chartist and type 2 chartist dominate type 2 fundamentalists. Also this setup has the potential to generate chaotic price dynamics (but, for instance, also coexisting locally stable fixed points). Interestingly, we now observe intricate bull and bear market dynamics: asset prices may circle in the bull market for some time but then the market crashes and is characterized by temporary bear market fluctuations. Such erratic bull and bear market dynamics was first studied by Day and Huang (1990), yet within a different model environment. We would like to stress that our setup, i.e. the assumed set of trading rules, is a quite natural extension of what has been explored so far. In this sense, we add a new view to the literature which will hopefully help us to penetrate the complicated dynamics of financial markets even further. Let us finally point out a few avenues for future research. First, it would be interesting to explore a model in which the trading rules do not only take current prices into account. For instance, a technical trading rule in which orders depend on the most recent observed price trend can be assumed. Then the model would result in a two-dimensional discontinuous dynamical system. Second, the focus could be placed on more general trading rules which determine orders on the basis of a linear function but where the absolute size of the orders is also bounded. Finally, one may try to calibrate such models so that they are able to mimic certain stylized facts of financial markets. One way may be to add random shocks to the model, and we refer interest readers to a related paper by Westerhoff and Franke (2009). Another way may be to add (many) more rules to the market, as in Farmer and Joshi (2002).