Comparing effects of price limit and circuit breaker
in stock exchanges by an agent-based model ^††thanks: Note that the opinions contained herein are solely those of the authors and do not necessarily reflect those of SPARX Asset Management Co., Ltd.

We introduce agents for modeling a general investor so as to replicate the nature of price formation in actual financial markets. The number of agents is $n$ and they can short sell freely. The holding positions are not limited, so agents can take an infinite number of shares for both long and short positions. Time $t$ increases by one when an agent places an order or when a circuit breaker in operation skips placing an order by an agent.

Agents always order only one share at a time. First, agent $1$ places an order to buy or sell a stock, and then agent $2$ places an order to buy or sell. After that, agents $3,4,,,n$ each place orders to buy or sell. After the final agent $n$ places an order, going back to the first agent, agent $1$ places an order to buy or sell, and at agents $2,3,,,,n$ each place orders to buy or sell, and this cycle is repeated.

An agent determines the order price and buys or sells using a combination of fundamental and technical analysis strategies to form an expectation of the stock return. The expected return of agent $j$ at $t$ is calculated as

where $w_{i,j}$ is the weight of term $i$ for agent $j$ and is independently determined by random variables uniformly distributed on the interval $(0,w_{i,max})$ at the start of the simulation for each agent. $\ln$ is the natural logarithm. $P_{f}$ is a fundamental value and is constant. $P^{t}$ is a mid-price (the average of the highest buy-order price and the lowest sell-order price) at $t$, and $\epsilon^{t}_{j}$ is determined by random variables from a normal distribution with average $0$ and variance $\sigma_{\epsilon}$ at $t$. $\tau_{j}$ is independently determined by random variables uniformly distributed on the interval $(1,\tau_{max})$ at the start of the simulation for each agent⁵⁵5When $t<\tau_{j}$, the second term of Eq. (5) is zero..

The first term in Eq. (5) represents a fundamental strategy: the agent expects a positive return when the market price is lower than the fundamental value, and vice versa. The second term represents a technical analysis strategy using a historical return: the agent expects a positive return when the historical market return is positive, and vice versa. The third term represents noise.

After the expected return has been determined, the expected price is

Order prices are scattered around the expected price $P^{t}_{e,j}$ to replicate many waiting limit orders. An order price $P^{t}_{o,j}$ is

where $\rho^{t}_{j}$ is determined by random variables uniformly distributed on the interval $(0,1)$ at $t$ and $P_{d}$ is constant. This means that $P^{t}_{o,j}$ is determined by random variables uniformly distributed on the interval $(P^{t}_{e,j}-P_{d},P^{t}_{e,j}+P_{d})$.

Whether the agent buys or sells is determined by the magnitude relationship between $P^{t}_{e,j}$ and $P^{t}_{o,j}$. Here⁶⁶6When $t<t_{c}$, to generate enough waiting orders, the agent places an order to buy one share when $P_{f}>P^{t}_{o,j}$, or to sell one share when $P_{f}<P^{t}_{o,j}$. ,

• when $P^{t}_{e,j}>P^{t}_{o,j}$, the agent places a buy order, and

• when $P^{t}_{e,j}<P^{t}_{o,j}$, the agent places a sell order.

The remaining orders are canceled $t_{c}$ after the order time except when a circuit breaker is in operation.

We utilized the same model of erroneous orders as Mizuta et. al.[6]. Erroneous orders start at time $t_{ms}$ and finish at time $t_{me}$ (see also Fig. 1 (left)). Within that period, with a constant probability $p_{m}$, each order of the agents is changed to one share sell at the highest buy-order price listed in the order book. The changed sell order is immediately executed matching the highest buy order. Increasing such erroneous sell orders is what makes market prices fall.

The agent $j$ starts to place stop-loss orders when $P^{t}$ becomes under $P_{f}\exp{\epsilon^{t=0}_{j}}-P_{l,j}$, where $P_{l,j}$ is independently determined by random variables uniformly distributed on the interval $(P_{lmin},P_{lmax})$ at the start of the simulation for each agent $j$. $t_{ls,j}$ is the start time of placing stop-loss orders for each agent. As shown on Fig. 1 (right), within the stop-loss period with a constant probability $p_{l}(t_{ls,j}+t_{l,j}-t)/t_{l,j}$, each order of an agent is changed to one share sell at the highest buy-order price, where $t_{l,j}$ is independently determined by random variables uniformly distributed on the interval $(t_{lmin},t_{lmax})$ at the start of the simulation for each agent $j$ and $p_{l}$ is constant.

Each agent estimates $P_{f}\exp{\epsilon^{t=0}_{j}}$ as a fair price and has to re-estimates a fair price when market prices fall significantly below that price. The agent needs a long time for the re-estimation and places stop-loss orders during this process. In this study, because erroneous orders cause prices to fall significantly, the fair price is not changed. The agent learns this fact after the re-estimation. Therefore, the probability of a stop-loss order decreases as time progresses in this model, and the agent stops a stop-loss order after the re-estimation.

We utilized the same model of a price limit as Mizuta et al.[6]. In the case of adopting a price limit, as shown Fig. 2 (left), any order prices of sell under $P^{t-tr}-Pr$ (of buy above $P^{t-tr}+Pr$) are changed to $P^{t-tr}-Pr$ ($P^{t-tr}+Pr$), where $tr,Pr$ are constant parameters to determine the nature of the price limit. We simulated the model to investigate the nature of price formations when these parameters were changed. As Mizuta et al. [6] demonstrated, the price limit prevents large price variations because agents cannot place orders far from $P^{t-tr}$ when the price limit is in place.

In the case of adopting a circuit breaker, as shown Fig. 2 (right), when $P_{t}$ reaches under $P^{t-tr}-Pr$ or above $P^{t-tr}+Pr$, the circuit breaker starts, and after starting, placing or cancelling orders are stopped during $tr_{2}$. Actually, the circuit breaker limits orders (unlike the price limit) and just waits for time to pass; however, the erroneous order period and stop-loss order period continue while the circuit breaker is activated. This leads to a decrease in the erroneous orders and stop-loss orders and, like the price limit, prevents large price variations. We also introduced a price limit version two in which any order prices of sell under $P^{t-tr}-Pr$ or buy above $P^{t-tr}+Pr$ are canceled. In an actual financial market, investors will place orders exactly at the $P^{t-tr}\pm Pr$ with such a price limit because they know orders will be canceled outside the $P^{t-tr}\pm Pr$. Therefore, in real financial markets version two does not cause any changes and the result is exactly the same as the normal version. The reason we introduce version two is that we want to investigate the effect of multiple orders existing at $P^{t-tr}\pm Pr$, so we use agents that do not care whether their orders are cancelled even though this is unrealistic in actual financial markets.

An artificial market model is an agent-based model for a financial market. There are thorough great reviews on such model [17, 18, 19, 20, 21, 4, 2, 3, 22]. Numerous significant artificial market models [23, 24, 25, 26] have been developed since their first appearance in the 1990s. Because of their generality, they are often qualitative. In particular, artificial markets have been used to investigate the mechanism by which stylized facts⁷⁷7A stylized fact is a term used in economics to refer to empirical findings that are so consistent (for example, across a wide range of instruments, markets and time periods) that they are accepted as truth[27]. (fat-tails, volatility-clustering, and so on) emerge [23, 26].

Here, let us briefly examine the nature of financial market phenomena such as bubbles and crashes, as described in [24, 25]. It is said that micro-macro feedback loops have played very important roles in bubbles and crashes, and artificial market models can directly treat such loops. A number of projects have built generic artificial market models, such as the U-mart project in Japan in the 2000s. (Kita et al. provide a comprehensive review of the U-mart project[28].) These projects have helped to explain the nature of financial market phenomena and the mechanism by which stylized facts emerge.

Artificial market models, however, have rarely been used to investigate the rules and regulations of financial markets. After the bankruptcy of Lehman Brothers in 2008, some researchers argued that traditional economics had not found ways to design markets that work well but suggested that an artificial market model could do so. Indeed, in Nature, Farmer and Foley [29] explained that “such (agent based) economic models should be able to provide an alternative tool to give insight into how government policies could affect the broad characteristics of economic performance, by quantitatively exploring how the economy is likely to react under different scenarios.” Richard Bookstaber, an expert on risk management who has worked for investment banks and hedge funds, wrote a book [30] that “provides a nontechnical introduction to agent-based modeling, an alternative to neoclassical economics that shows great promise in predicting crises, averting them, and helping us recover from them.” In 2010, Jean-Claude Trichet, then President of the European Central Bank (ECB) [31], stated that “agent-based modelling dispenses with the optimization assumption and allows for more complex interactions between agents. Such approaches are worthy of our attention.”

Financial regulators and exchanges, who decide rules and regulations, are especially interested in using artificial market models to design a market that works well. Indeed, the Japan Exchange Group (JPX), which is the parent company of the Tokyo Stock Exchange, has published 40 JPX working papers, including 12 on using artificial market models, as of December 2022⁸⁸8https://www.jpx.co.jp/english/corporate/research-study/working-paper/index.html.

Mizuta [4] reviewed other previous agent-based models for designing a financial market that works well.

An artificial market model, which is an agent-based model for financial markets, can be used to investigate situations that have never occurred, handle regulation changes that have never been made, and isolate the pure contribution of these changes to price formation and liquidity[2, 3]. These are the advantages of an artificial market simulation. However, the outputs of this simulation would not be accurate or be credible forecasts of the actual future. The simulation needs to reveal possible mechanisms that affect price formation through many simulation runs, e.g., searching for parameters or purely comparing the before and after states of changes. The possible mechanisms revealed by these runs provide new intelligence and insight into the effects of the changes in price formations in actual financial markets. Other methods of study, e.g., empirical studies, would not reveal such possible mechanisms.

Artificial markets should replicate the macro phenomena that exist generally for any asset at any time. Price variation, which is a kind of macro phenomenon, is not explicitly modeled in artificial markets. Only micro processes, agents (general investors), and price determination mechanisms (financial exchanges) are explicitly modeled. Macro phenomena emerge as the outcome of interactions from micro processes. Therefore, the simulation outputs should replicate existing macro phenomena to generally prove that simulation models are probable in actual markets.

However, it is not the primary purpose for an artificial market to replicate specific macro phenomena only for a specific asset or period. Unnecessary replication of macro phenomena leads to models that are overfitted and too complex. Such models would prevent us from understanding and discovering mechanisms that affect price formation because the number of related factors would increase. In addition, artificial market models that are too complex are often criticized because they are very difficult to evaluate[19]. A model that is too complex not only would prevent us from understanding mechanisms but also could output arbitrary results by overfitting too many parameters. It is more difficult for simpler models to obtain arbitrary results, and these models are easier to evaluate.

Therefore, we constructed an artificial market model that is as simple as possible and does not intentionally implement agents to cover all the investors who would exist in actual financial markets.

Such simplicity is very important not only for artificial market models but also generally for agent-based models. Gilbert argued that there are three types for agent-based models, an abstract model, a middle-range model and a facsimile model[13]. Gilbert said ”the aim of abstract models is to demonstrate some basic social process that may lie behind many area of social life.” Axelrod mentioned that to understand mechanism abstract models should be as simple as possible because needless complex model preventing to understand the mechanism, and called the principle as KISS (keep it simple stupid)[14]. For example, Thomas Schelling, who received the Nobel Prize in economics, used an agent-based model to discuss the mechanism of racial segregation. The model was built very simply compared with an actual town to focus on the mechanism[32]. While it was not able to predict the segregation situation in the actual town, it was able to explain the mechanism of segregation as a phenomenon. Of course, the model was not calibrated to empirical data because of not leading to make it more complex to prevent understanding. For such abstract models, we should be cautious complex modeling by calibrations using empirical date, and this is a fundamental for modeling[33]. Indeed, many artificial market models were validated by replicating fat-tails and volatility-clustering that are very famous stylized facts in financial markets[17, 19, 2, 3]. On the other hand, the aim of facsimile models is to replicate some specific situation and ”predict” future exactly. So, facsimile models are needed to be calibrated by empirical data at least and should replicate existing social phenomena exactly.

As Michael Weisberg mentioned[33], “Modeling, (is) the indirect study of real-world systems via the construction and analysis of models.” “Modeling is not always aimed at purely veridical representation. Rather, they worked hard to identify the features of these systems that were most salient to their investigations.” Therefore, effective models are different depending on the phenomena they focus on. Thus, our model is effective only for the purpose of this study and not for others. The aim of our study is to understand how important properties (behaviors, algorithms) affect macro phenomena and play a role in the financial system rather than representing actual financial markets precisely.

The aforementioned discussion holds not only for artificial markets but also for agent-based models used in fields other than financial markets. For example, Thomas Schelling, who received the Nobel Prize in economics, used an agent-based model to discuss the mechanism of racial segregation. The model was built very simply compared with an actual town to focus on the mechanism[32]. While it was not able to predict the segregation situation in the actual town, it was able to explain the mechanism of segregation as a phenomenon.

Michael Weisberg studied what mathematical and simulation models are in the first place and cited the example of a map[33]. Needless to say, a map models geographical features on the way to a destination. With a simple map, we can easily understand the way to the destination. However, while a satellite photo replicates actual geographical features very well, we cannot easily find the way to the destination.

The title page of Michael Weisberg’s book[33] cited a passage from a short story by Jorge Borges[34], “In time, those Unconscionable Maps no longer satisfied, and the Cartographers Guilds struck a Map of the Empire whose size was that of the Empire, and which coincided point for point with it…In the Deserts of the West, still today, there are Tattered Ruins of that Map, inhabited by Animals and Beggars.” The story in which a map was enlarged to the same size as the real Empire to become the most detailed of any map is an analogy to that too detailed a model is not useful. This story give us one of the most important lessons for when we build and use any model.

In many previous artificial market studies, the models were validated to determine whether they could explain stylized facts, such as a fat-tail or volatility clustering[17, 19, 2, 3]. A fat-tail means that the kurtosis of price returns is positive. Volatility clustering means that square returns have a positive autocorrelation, which slowly decays as its lag becomes longer.

Many empirical studies, e.g., that of Sewell[27], have shown that both stylized facts (fat-tail and volatility clustering) exist statistically in almost all financial markets. Conversely, they also have shown that only the fat-tail and volatility clustering are stably observed for any asset and in any period because financial markets are generally unstable. This leads to the conclusion that an artificial market should replicate macro phenomena that exist generally for any asset at any time, fat-tails, and volatility clustering. Other stylized facts should be replicate only when a purpose of study relates them to prevent to make a model needless complex. In the case of this study, our model should replicate only universally established stylized facts about time evolution of market prices which are fat-tails and volatility clustering to prevent to make a model needless complex because we focus generally and universally impacts to market prices by the rules. We did not dare to replicate other stylized facts that are unstably observed because, as we mentioned, the simplicity of the model is very important for this study because unnecessary replication of macro phenomena leads to models that are overfitted and too complex. This is an example of how empirical studies can help an artificial market model.

The kurtosis of price returns and the autocorrelation of square returns are stably and significantly positive, but the magnitudes of these values are unstable and very different depending on the asset and/or period. Very broad magnitudes of about $1\sim 100$ and about $0\sim 0.2$, respectively, have been observed[27].

For the aforementioned reasons, an artificial market model should replicate these values as significantly positive and within a reasonable range. It is not essential for the model to replicate specific values of stylized facts because the values of these facts are unstable in actual financial markets.

Table VI lists the statistics showing the stylized facts, kurtosis of price returns for $100$ tick times ($\ln(P^{t}/P^{t-100})$), and autocorrelation coefficient for square returns for $100$ tick times without the erroneous orders. Note that without the erroneous orders no stop loss is happened and no regulation is triggered. This shows that this model replicated the statistical characteristics, fat-tails, and volatility clustering observed in real financial markets.