Toward an understanding of market resilience: market liquidity and heterogeneity in the investor decision cycle

In this study, we build upon three distinct bases of literature: modeling methodologies, empirical findings for financial markets known as stylized facts, and the evolving set of agent-based models of financial markets.

In the general modeling literature, Balci (1997) defines an iterative approach for verification (i.e., tests to confirm whether we built the model correctly) and validation (i.e., tests to confirm that we built the correct model). As part of verification testing, Axtell et al. (1996) call for researchers to contribute to the cumulative foundation provided by previous models from previous studies. Therefore, the successful researcher first replicates the model from a previous study to ensure an independent verification of that study and its findings. Only then should the researcher undertake the extension of the replicated model to address the model characteristics required of the new study. Lastly, the extended model must be subjected to a rigorous set of validation and calibration tests (LeBaron 2001a) such as Axtell and Epstein’s (1994) framework for evaluating empirical relevance of agent-based models. See Tivnan et al. (2011) for an example of a study that follows the above approach.

The empirical targets for the above validation tests are known as stylized facts. While many researchers have conducted empirical analyses of financial markets, Mandelbrot (1963) was one of the first to identify that price returns do not necessarily follow a Gaussian random walk. Many have built upon Mandelbrot’s foundational work and Cont (2001) provides a comprehensive review of these empirical analyses which identify several stylized facts of financial markets. These stylized facts have been shown to be robust across the price time series of various markets over varying time intervals. While many of these stylized facts have been debated in recent years, three have largely been agreed upon and applied by the quantitative finance community. These accepted stylized facts include decaying autocorrelations over moderate time lags, clustering volatility, and fat tails in the distribution of returns.

Put simply, decaying autocorrelations indicate very small correlations in price movements over moderate to long time lags, clustering volatility indicates that price returns of similar absolute size tend to follow one another, and fat tails indicate that returns are much more peaked (that is, have higher kurtosis and fatter tails) than a Gaussian distribution would predict. As described by Cont (2001: 224), “these stylized facts are so constraining that it is not easy to exhibit even an (ad hoc) stochastic process which possesses the same set of properties and one has to go to great lengths to reproduce them with a model.” Stated differently, this set of stylized facts provides empirical evidence to guide the comprehensive, validation testing of market models.

While there has been much discussion of the potential for agent-based models to advance the study of financial markets (Bookstaber 2012), it is important to note the evolution of agent-based models of financial markets. Consistent with two of the prevailing design issues (LeBaron 2001a), the evolution of the set of agent-based models of financial markets follow two distinct paths, one focused on the representation of the agents and the other focused on the representation of the trading mechanism. The former trajectory began in the early 1990s with the Santa Fe Institute (SFI) market model (Palmer et al. 1994). Largely predicated on Holland’s (1977) genetic algorithms, the SFI market model was well received as a novel departure from equilibrium models, largely based on its qualitative depiction of agreement with empirical observations of market dynamics such as bubbles and crashes. Following the SFI market model, Lux and Marchesi (1999) introduced a model with a single trade type (i.e., market orders) that was the first to demonstrate clustered volatility, one of the stylized facts common to many markets. Similar to the Lux market with market orders, LeBaron (2001b) introduced a market model with agents that learn based on a neural network. In the LeBaron model, agents decided how much of their total wealth to invest; therefore successful agents can have a large impact in the market. Cont and colleagues (Ghoulmie et al. 2005) developed a model that qualitatively reproduces the three prevailing stylized facts described above. Of note, the traders in the Cont models have heterogeneous trading thresholds, and the traders, many of whom trade rather infrequently, adapt their thresholds based upon performance feedback. More recently, Johnson et al. (2013) developed a model that quantitatively reproduces so-called “Flash Crashes” in the market (i.e., dramatic price movements which are immediately reversed).

The trajectory of models focused on trading mechanisms began with Maslov (2000) nearly a decade after the SFI market model. Maslov’s model was quickly extended by Darley et al. (2001) and Farmer et al. (2005). The Farmer model (subsequently referred to as the ZIM for Zero-Intelligence Model) built a model of zero-intelligence traders active within the structure of a continuous, double auction placing two types of orders: market orders and limit orders. Market orders are orders that enter the market with an intent to buy or sell a certain number of shares and do not specify a particular price. Limit orders, on the other hand, enter the market with both a specified price and quantity of shares. As market orders do not specify a price they are executed immediately upon entering the market at the best available price. Limit orders, however, will accumulate in the market until their specified price is met or they are cancelled. The accumulation takes place in a prioritized queue by price and arrival time. This accumulation of limit orders is called the limit order book. It is this accumulation of limit orders that creates liquidity (i.e., the ability for market orders to be executed) in the market. In the ZIM, both market and limit orders arrive and are cancelled according to a Poisson process. Although not as rich as the Johnson et al. (2013) model which can replicate the dynamics of market events such as the 2010 Flash Crash, the ZIM is useful in exploring structural aspects of market dynamics.

A follower of the cumulative approach espoused by Axtell et al. (1996), Preis et al. (2006, 2007) first replicated and then extended the ZIM. Their model reproduced a leptokurtic distribution of returns, which the ZIM did not. Because of this, we use the Preis model as a building block from which we extend to explore market-relevant aspects of heterogeneity that are not contained within the Preis model.

As with the ZIM, the Preis market consists of a continuous double auction order book in one asset. Both market and limit orders can be placed in the order book. When a sell (buy) market order is placed, the order is executed immediately against a buy (sell) limit order with the specified highest (lowest) price and earliest time-stamp. We will refer to the highest buy price as the best bid (\(p_b(t)\)), and the lowest sell price as the best ask (\(p_a(t)\)).

The Preis model consists of two sets of agents, patient and impatient agents. Impatient agents place only market orders with rate \(\mu \) orders per trader per time step. When an impatient agent is activated, they buy with probability \(q_{taker}\) and sell with probability \(1 - q_{taker}\), where \(q_{taker}\) is a mean reverting random walk which starts at 0.5, takes steps of size 0.001, and is bounded between 0.45 and 0.55. In this light, we refer to \(q_{taker}\) as a trending variable, as prices will tend to drop when \(q_{taker} < 0.5\) and rise when \(q_{taker} > 0.5\). Patient agents place only limit orders with rate \(\alpha \). Patient agents place buy and sell orders with equal probability. Limit orders are priced away from the spread according towhere \(\lambda _{0} = 100\). Sell orders are placed at price \(p_b(t) + X\) and buy orders are placed at price \(p_a(t) - X\), where X is drawn from an exponential distribution. Note that X is independently drawn from the exponential distribution for each limit order placed. This equation ensures that as \(q_{taker}\) deviates from its mean of 0.5, traders will tend to place orders further from the spread. All traders place only orders of size \(\sigma \), which is kept constant at one throughout Preis’ experiments. Limit orders are cancelled randomly with rate \(\delta \) per order per time step. Figure 1 provides a graphical depiction of the limit order book from Preis’ model.

Using stylized facts for tests of model validation, we draw from Cont (2001) and require that kurtosis is greater than 5, and that autocorrelation of absolute returns is at least five times greater than autocorrelation of raw returns. Another important factor to consider when measuring stylized facts are what we refer to as “market collapse cases”. When a market order is placed and there are no limit orders on the opposite side of the book to fill, we call this a collapse. While collapses occur in real markets, they are rare, and in a valid market model we would not expect to see a collapse under normal conditions. In total, our use of this set of stylized facts represents a more restrictive approach to validation testing than previously applied to agent-based models of limit order books (e.g., Farmer et al. 2005; Preis et al. 2006).

To test for model validity at each design point, d, we define a “stylized-facts” pass rate, \(R_d\), to explicitly assess stochastic aspects of model behavior. \(R_d\) is calculated as:and n is the number of realizations of the model—aka “runs”—per design point (i.e., replicates, each instantiated with a unique pseudo-random number (Kleijnen et al. 2005), ACF is the autocorrelation function, CC is the clustering volatility constant, K is the kurtosis function, \(t(\text {ACF})\) refers to the p-value of a Student t test of the ACF, and \(r_{d_i}\) is a vector of returns for design point d and random seed i. If all 3 of the above conditions are true, the randomly seeded run is given a value of 1, and if any of the conditions are false, the run is given a value of 0. Any run which yields a market collapses is assigned a score of 0.

The Farmer model represents the case of complete homogeneity. Preis introduces one element of heterogeneity from Farmer, in that Preis uses a statistical distribution to emulate heterogeneous sensitivity to price for liquidity providers (i.e., traders placing limit orders).

Instead, we aim to take advantage of recent empirical analysis of a relevant market event (e.g., Kirilenko analysis of the Flash Crash) where the entire population of traders fall into only four distinct categories according to heterogeneous values of the following attributes:To maintain consistency with Preis and to allow docking with the Preis model, we retain the functional form of his effective limit order rate by coupling \(\lambda \) and \(\sigma \) together.

While our model allows us to instantiate a class of traders with any combination of the above trading attributes, we only allow for plausible combinations of the above parameters so as to maintain consistency with empirical evidence. For example,

We have built up the model step-by-step in order to illuminate the important phases of its development. For the final model, on which we run controlled experiments and analyze in Sect. 4, the following simulation procedure and parameter settings are used.

In the following section, we describe our controlled experiments to investigate the impact of heterogeneous decision cycles on market stability. In the first experiment, we vary \(\sigma \) to investigate the impact of heterogeneous order sizes and rates, which are attributes of classes of traders (i.e., short-term liquidity takers, short-term liquidity providers and long-term liquidity providers). In the second experiment, we vary the firms’ perception of the asset’s fundamental value, which may be an attribute of the individual firm. In our last experiment, we vary \(\sigma \) as well as introduce an exogenous, liquidity shock to investigate the effect of heterogeneous decision cycles during market crises. Together, these experiments demonstrate the ability to investigate dynamics at three levels: individual traders, classes of traders, and the market level where liquidity accumulates and prices emerge.

All of the experiments which are analyzed in this section are tested on the sample of 36 design points described above, each of which passed the relaxed stylized facts in at least 90 % of the replicates.

As before, we first varied the value of \(\sigma \), across five values: 1, 5, 10, 25, and 50. Figure 7 illustrates the stability of the stylized facts across various values of \(\sigma \), where pass rate is calculated according to our stricter standard test.

We focused on the median minimum price over a set of simulations in a given experiment. That is, for each run of the experiment, we looked at the lowest price after the price shock occurs, i.e., the trough of the resulting market event, and found the median across all of the runs. Depicting experiments for two design points with results representative of the remaining design points, Fig. 8 shows \(\sigma \) has large effects on the median minimum price; the median minimum price is lower as we increase \(\sigma \), though its effects start to dampen as it reaches values of 25 and 50. It is also apparent that in the worst-case scenarios, the value of \(\sigma \) has major effects on minimum price, but when the median of the minimum price is taken across all random seeds, the effects are almost negligible. This highlights an important aspect for financial markets, and complex systems in general, which is that in any fat-tailed distribution, the mean and median are not sufficiently reliable statistics to give accurate insight into the behavior of a system. From Fig. 7, we note that the average kurtosis does not increase significantly as \(\sigma \) increases, even though the range of minimum prices does increase. This is significant because as we have hypothesized, the effect of \(\sigma \) on median minimum price is not due to qualitatively different sizes of individual negative returns (i.e., larger, individual price drops) but due to the inability of short-term liquidity providers to meet the immediate needs of short-term liquidity takers between the arrival of long-term, liquidity providers.

The nonlinear effects of \(\sigma \) on minimum price as \(\sigma \) increases seem to be due to our cancellation methodology, which we derived from both Farmer and Preis. Small orders are more likely to be cancelled than larger orders. Thus, even if a firm’s order were randomly selected to be cancelled, if that cancelation will bring the total cancellations for that time step above \(\delta * L_{Total}\), then that order would not be cancelled. An area for future research and a potentially meaningful extension to the model would be a behavioral rule for order cancellation which is grounded in empirical research, hence the first demonstration of Level 3 Empirical Relevance in the Axtell and Epstein framework for an agent-based model of an order book.

For the next experiment, we tested the impact of firms’ perception of fundamental value on price. We considered three implementations of firms’ fundamental view of price: Constant and homogeneous, constant and heterogeneous (mean \(p_0\), standard deviation 5), and a heterogeneous random walk (starting at \(p_0\), step size 0.5). We do not see qualitatively different results across these three design points, which suggests that our results are robust to the firms’ independent views of fundamental price (Note: figure omitted due to space concerns). Thus, for the sake of simplicity, we used the constant and heterogeneous fundamental prices for our remaining DOE.

Lastly, we re-ran the \(\sigma \) experiment with a downward price shock to test how each market would respond (i.e., a stress test). Here we see similar, yet more pronounced results than the ones described above, see Fig. 9. Interestingly, we see less of a “dampening” effect as \(\sigma \) increases to 25 and 50 than we do in the un-shocked cases. Although the absolute value of the slope does decrease in both experiments, it decreases more slowly in the shocked case, lending more support for our hypothesis that markets are especially time-sensitive during fire-sale scenarios, and sometimes the only traders with sufficient capital to stabilize the market are the traders who have the longest deliberation times.

For these shocked experiments, we analyzed the effect of \(\sigma \) on market collapses, as depicted in Fig. 10. The rate of collapses increases monotonically up to \(\sigma = 25\), and then decreases at \(\sigma = 50\). This positive correlation between \(\sigma \) and collapse rate provides further evidence that in the market crisis scenarios, the long-term, liquidity providers were simply not responsive enough to provide sufficient liquidity to stabilize the market.

Overall, this set of results comports with our hypothesis that the timeliness of large firms is very important to market stability.

To summarize, we first ran experiments to vary the size of \(\sigma \) and found that increasing levels of \(\sigma \) contribute to lower prices in a given market, longer times for prices to recover and at slower rates. We then tested for potential biases to our results from particular representations of firms’ perception of fundamental value, but found no such biases. Lastly, we demonstrated the model as a stress testing platform by introducing a liquidity shock and analyzing the effect of \(\sigma \) on minimum prices and collapses. Again, we found evidence to support our hypothesis that markets are especially time-sensitive during crises, and sometimes the only traders with sufficient capital to stabilize the market are the traders who have the longest decision cycles.