مدل تجربی رفتاری از نقدینگی و نوسانات

We develop a behavioral model for liquidity and volatility based on empirical regularities in trading order flow in the London Stock Exchange. This can be viewed as a very simple agent-based model in which all components of the model are validated against real data. Our empirical studies of order flow uncover several interesting regularities in the way trading orders are placed and cancelled. The resulting simple model of order flow is used to simulate price formation under a continuous double auction, and the statistical properties of the resulting simulated sequence of prices are compared to those of real data. The model is constructed using one stock (AZN) and tested on 24 other stocks. For low volatility, small tick size stocks (called Group I) the predictions are very good, but for stocks outside Group I they are not good. For Group I, the model predicts the correct magnitude and functional form of the distribution of the volatility and the bid-ask spread, without adjusting any parameters based on prices. This suggests that at least for Group I stocks, the volatility and heavy tails of prices are related to market microstructure effects, and supports the hypothesis that, at least on short time scales, the large fluctuations of absolute returns |r| are well described by a power law of the form P(|r|>R)∼R-αr, with a value of αr that varies from stock to stock.

1.1. Toward a more quantitative behavioral economics In the last two decades the field of behavioral finance has presented many examples where equilibrium rational choice models are not able to explain real economic behavior1 (Hirschleifer, 2001, Barberis and Thaler, 2003, Camerer et al., 2003, Thaler, 2005 and Schleifer, 2000). There are many efforts underway to build a foundation for economics directly based on psychological evidence, but this imposes a difficult hurdle for building quantitative theories. The human brain is a complex and subtle instrument, and in a general setting the distance from psychology to prices is large. In this study we take advantage of the fact that electronic markets provide a superb laboratory for studying patterns in human behavior. Market participants make decisions in an extremely complex environment, but in the end these decisions are reduced to the simple actions of placing and cancelling trading orders. The data that we study contain tens of millions of records of trading orders and prices, allowing us to reconstruct the state of the market at any instant in time. We have a complete record of decision making outcomes in the context of the phenomenon we want to study, namely price formation. Within the domain where this model is valid, this allows us to make a simple but accurate model of the statistical properties of prices. 1.2. Goal Our goal here is to capture behavioral regularities in order placement and cancellation, i.e. order flow, and to exploit these regularities to achieve a better understanding of liquidity and volatility. The practical component of this goal is to understand statistical properties of prices, such as the distribution of price returns and the bid-ask spread. We will use logarithmic returns r(t)=πm(t)-πm(t-1), where t is order placement time2 and πm is the logarithmic midprice. The logarithmic midprice View the MathML source, where pa(t) is the best selling price (best ask) and pb is the best buying price (best bid); on the rare occasions that we need a price rather than a logarithmic price, we will use p=exp(πm). We are only interested in the size of price movements, and not in their direction. We will take the size of logarithmic returns |r(t)| as our proxy for volatility. Another important quantity is the bid ask spread s(t)=logpa(t)-logpb(t). The spread is important as a benchmark for transaction costs. A small market order to buy will execute at the best selling price, and a small order to sell will execute at the best buying price, so someone who first buys and then sells in close succession will pay the spread s(t). Our goal is to relate the magnitude and the distribution of volatility and the spread to statistical properties of order flow. The modelling task is to understand which properties of the order flow are important for understanding prices and to create a simple model for the relationship between them. 1.3. Liquidity The model we develop here describes the endogenous dynamics of liquidity. We define liquidity as the difference between the current midprice and the price where an order of a given size can be executed. Previous work has shown that liquidity is typically the dominant determinant of volatility, at least for short time scales (Farmer et al., 2004, Weber and Rosenow, 2006 and Gillemot et al., 2006). Periods of high volatility correspond to low liquidity and vice versa. Here we model the dynamics of the order book, i.e. we model fluctuations in liquidity, and use this to predict fluctuations in returns and spreads.3 Thus understanding liquidity is the first and principal step to understanding volatility. 1.4. The zero intelligence approach to the continuous double auction Our model is based on a statistical description of the placement and cancellation of trading orders under a continuous double auction. This model follows in the footsteps of a long list of other models that have tried to describe order placement as a statistical process (Mendelson, 1982, Cohen et al., 1985, Domowitz and Wang, 1994, Bollerslev et al., 1997, Bak et al., 1997, Eliezer and Kogan, 1998, Tang and Tian, 1999, Maslov, 2000, Slanina, 2001, Challet and Stinchcombe, 2001, Daniels et al., 2003, Chiarella and Iori, 2002, Bouchaud et al., 2002 and Smith et al., 2003). For a more detailed narrative of the history of this line of work, see Smith et al. (2003). The model developed here was inspired by that of Daniels et al. (2003). The model of Daniels et al. was constructed to be solvable by making the assumption that limit orders, market orders, and cancellations can be described as independent Poisson processes. Because it assumes that order placement is random except for a few constraints it can be regarded as a zero intelligence model of agent behavior. Although highly unrealistic in many respects, the zero intelligence model does a reasonable job of capturing the dynamic feedback and interaction between order placement on one hand and price formation on the other. It predicts simple scaling laws for the volatility of returns and for the spread, which can be regarded as equations of state relating the properties of order flows to those of prices. Farmer et al. (2005) tested these predictions against real data from the London Stock Exchange and showed that, even though the model does not predict the absolute magnitude of these effects or the correct form of the distributions, it does a good job of capturing how the spread varies with changes in order flow. The predictions for volatility are not quite as good, but are still not bad. Despite these successes the zero intelligence model is inadequate in many respects. Because of the unrealistic assumptions that order placement and cancellation are uniform along the price axis, to make comparisons with real data it is necessary to introduce an arbitrary interval over which order flow and cancellation densities are measured, and to assume that they vanish outside this interval. This assumption introduces arbitrariness into the scale of the predictions and complicates the interpretation of the results. In addition it produces price returns with non-white autocorrelations and a thin-tailed distribution that do not match real data. 1.5. Regularities in order flow The model here has the same basic elements as the zero intelligence model, but each element is modified based on empirical analysis. The model for order placement is developed in the same style as that of Challet and Stinchcombe (2001).4 In order to have a complete model for order flow we must model three things: 1. The signs of orders (buy or sell) – see Section 3. 2. The prices where orders are placed – see Section 4. 3. The frequency with which orders are cancelled – see Section 5. In the course of modelling each of these we uncover regularities in order placement and cancellation that are interesting for their own sake. For order placement we show that the probability of placing an order at a given price relative to the best quote can be crudely approximated by a Student distribution with less than two degrees of freedom, centered on the best quote. We also develop a crude but simple cancellation model that depends on the position of an order relative to the best price and the imbalance between buying and selling orders in the limit order book. The strategic motivation behind these regularities in each case are not always obvious. Particularly for items (2) and (3), it not clear whether the regularities we observe are driven by rational equilibrium or irrational behavior. We do not attempt to address this question here. Instead we work in the other direction and construct a model for volatility. Nonetheless, our studies illustrate interesting regularities in behavior that provide a intermediate milepost for obtaining any strategic understanding of market behavior. 1.6. Method of developing and testing the model This model is developed on a single stock and then tested on 25 stocks. The tests are performed by fitting the parameters of each component of the model on order flow data alone, using a simulation to make a prediction about the distribution of volatility and spreads, and comparing the statistical properties of the simulation to the measured statistical properties of volatility and spreads in the data during the same period of time. When we say ‘prediction’, we are using it in the sense of an equation of state, i.e. we are predicting contemporaneous relationships between order flow parameters on one hand and statistical properties of prices on the other. 1.7. Heavy tails in price returns Serious interest in the functional form of the distribution of prices began with Mandelbrot's (1963) study of cotton prices, in which he showed that logarithmic price returns are far from normal and suggested that they might be drawn from a Levy distribution. There have been many studies since then, most of which indicate that the cumulative distribution of logarithmic price changes has tails that asymptotically scale for large |r| as a power law of the form |r|-αr, where (Fama, 1965, Officer, 1972, Akgiray et al., 1989, Koedijk et al., 1990, Loretan and Phillips, 1994, Mantegna and Stanley, 1995, Longin, 1996, Lux, 1996, Muller et al., 1998, Plerou et al., 1999, Rachev and Mittnik, 2000 and Goldstein et al., 2004), but this remains a controversial topic. The exponent αr, which takes on typical values in the range 2<αr<4, is called the tail exponent. It is important because it characterizes the risk of extreme price movements and corresponds to the threshold above which the moments of the distribution become infinite. Having a good characterization of price returns has important practical consequences for risk control and option pricing. For our purposes here we will not worry about possible asymmetries between the tails of positive and negative returns, which are in any case quite small for returns at this time scale. From a theoretical point of view the heavy tails of price returns excite interest among physicists because they suggest non-equilibrium behavior. A fundamental result in statistical mechanics is that, except for unusual situations such as phase transitions, equilibrium distributions are either exponential or normal distributions.5 The fact that price returns have tails that are heavier than this suggests that markets are not at equilibrium. Although the notion of equilibrium as it is used in physics is very different from that in economics, the two have enough in common to make this at least an intriguing suggestion. Many models have been proposed that attempt to explain the heavy tails of price returns (Arthur et al., 1997, Bak et al., 1997, Brock and Hommes, 1999, Lux and Marchesi, 1999, Chang et al., 2002, LeBaron, 2001, Giardina and Bouchaud, 2003, Gabaix et al., 2003, Gabaix et al., 2006 and Challet et al., 2005). These models have a wide range in the specificity of their predictions, from those that simply demonstrate heavy tails to those that make a more quantitative prediction, for example about the tail exponent αr. However, none of these models produce quantitative predictions of the magnitude and functional form of the full return distribution. At this point it is impossible to say which, if any, of these models are correct. 1.8. Bid-ask spread In this paper we present new empirical results about the bid-ask spread. There is a substantial empirical and theoretical literature on the spread. A small sample is (Demsetz, 1968, Stoll, 1978, Glosten, 1988, Glosten, 1992, Easley and O’Hara, 1992, Foucault et al., 2005 and Sandas, 2001). These papers attempt to explain the strategic factors that influence the size of the spread. We focus instead on the more immediate and empirically verifiable question of how the spread is related to order placement and cancellation. 1.9. Organization of the paper The paper is organized as follows: Section 2 discusses the market structure and the data set. In Section 3 we review the long-memory of order flow and discuss how we model the signs of orders. In Section 4 we study the distribution of order placement conditioned on the spread and in Section 5 we study order cancellation. In Section 6 we measure the parameters for the combined order flow for order signs, prices, and cancellations on all the stocks in the sample. In Section 7 we put this together by simulating price formation for each stock based on the combined order flow model, and compare the statistical properties of our simulations to those of volatility and spreads. Finally in the last section we summarize and discuss the implications and future directions of this work.

We have built an empirical behavioral model for order placement that allows us to study the endogenous dynamics of liquidity and price formation in the order book. It can be viewed as an agent based model, but it differs from most agent-based models in that the specification of the agents is quite simple and each component of the model is quantitatively grounded on empirical observations. For the low volatility, small tick size stocks in our sample (which we call Group I), measurements of a small set of parameters of order flow give accurate predictions of the magnitude and functional form of the distribution of volatility and the spread. Our model suggests that there is an equation of state linking the properties of order flow to the properties of prices. By this we mean that there are constraints between the statistical properties of order flow and the statistical properties of prices, so that knowing one set of parameters automatically implies the other. To see why we say this, please refer to Fig. 10, where we plot the volatility predicted by the model against the actual volatility. The prediction of volatility varies because the order flow parameters in Table 3 vary. The fact that there is agreement between the predicted and actual values for Group I shows that for these stocks the order flow parameters are sufficient to describe the volatility. Of course, at this stage the equation of state remains implicit – while the model captures it, we do not know how to explicitly write it down. This model shows how market microstructure effects, such as long-memory in the signs of orders, and heavy tails in the relative prices of orders in an auction, can generate heavy tails in price returns that closely match the data. As discussed in the previous section, this reinforces the hypothesis that large returns asymptotically scale as a power law. It also means that in order for the tail exponents of price returns to have a universal value near three, as previously hypothesized (Liu et al., 1999, Gabaix et al., 2003 and Gabaix et al., 2006), there must be constraints on the microstructure properties that enforce this. The methodological approach that we have taken here can be viewed as a divide and conquer strategy. We have tackled the problem of price formation by starting in the middle. Rather than trying to immediately derive a model based on strategic motivations, we have empirically characterized behavioral regularities in order flow. From here one can work in two directions, either working forward to understand the relation between order flow and price formation, or working backward to understand the strategic motivations that give rise to the regularities in the first place. Here we have addressed the much easier problem of going forward, but our results are also potentially very useful for going backward. It is always easier to solve a problem when it can be decomposed into pieces. Going all the way from strategic motivations to prices is a much bigger step than moving from strategic motivations to regularities in order flow. By empirically observing regularities in order flow we have created intermediate mileposts that any theory of strategic motivations should explain; once these are explained, we have shown that many features of prices follow more or less immediately. At this point it is not obvious whether these regularities can be explained in terms of rational choice, or whether they represent an example of irrational behavior, that can only be explained in terms of human psychology.