We built an artificial market model and compared effects of price variation limits, short selling regulations and up-tick rules. In the case without the regulations, the price fell to below a fundamental value when an economic crush occurred. On the other hand, in the case with the regulations, this overshooting did not occur. However, the short selling regulation and the up-tick rule caused the trading prices to be higher than the fundamental value. We also surveyed an adequate limitation price range and an adequate limitation time span for the price variation limit and found a parameters’ condition of the price variation limit to prevent the over-shorts. We also showed the limitation price range should be bigger than a volatility calculated by the limitation time span.
Artificial Market; Multi Agent Simulation; Financial Crash; Price Variation Limit; Short Selling Regulation; Up-Tick Rule1. Introduction
Financial exchanges sometimes employ a “price variation limit”, which restrict trades out of certain price ranges within certain time spans to avoid sudden large price fluctuations. For example, Tokyo Stock Exchange employs two kinds of price variation limits that adopt different time spans: one is “daily price limit” which restricts price fluctuations within a single trading day, and the other is “special quote”, which restricts within three minutes [1]. Most Asian stock exchanges (Tokyo, Taipei, Shanghai, Shenzhen, Seoul, and so on) employ the price variation limits, but many American and European stock exchanges do not because there is a debate over whether the price variation limit makes financial market more efficient or not. On financial markets, other regulations, “short selling regulation”1, “up-tick rule”2 and so on, also are debated over whether makes more efficient or not.
3Excellent reviews are [2,3].
Because so many factors cause price formation in actual markets, an empirical study can not isolate a pure contribution of these regulations to price formation. Therefore, it is very difficult to discuss about a pure effect of these regulations only by results of empirical studies. An artificial market3 which is a kind of an agent based simulation will help us to discuss about this very well. There are several previous studies to discuss about regulations of financial market using artificial market simulations. Yagi et al. investigated effect of short selling regulations induce bubbles [4,5]. Westerhoff discussed effectiveness of transaction taxes, central bank interventions and trading halts [6]. Thurner et al. showed that as leverage increases price fluctuation becomes heavy tailed and display clustered volatility [7]. Kobayashi and Hashimoto showed that circuit breakers play an important role in controlling price fluctuations, while they also reduce the trading volume [8]. Yeh and Yang investigated the effectiveness of price variation limits and showed that price variation limits help to reduce volatility and price distortion [9]. Mizuta et al. discussed effectiveness of price variation limits and argued that an artificial market model testing such regulations should be implementing a learning process to replicate bubbles, and showed that a hazard rate enables verification of whether the models can replicate a bubble process or not [10]. However, no simulation studies have investigated up-tick rules, and compared effects of the short selling regulation, the up-tick rule and the price variation limit using an artificial market model. These regulations are expected to be especially effective to prevent bubbles and crushes, so simulation studies investigating these regulations should use artificial market models replicating bubbles and crushes.
We built an artificial market model and compared effects of price variation limits, short selling regulations and up-tick rules. In the case without the regulations, the price fell to below a fundamental value when an economic crush occurred. On the other hand, in the case with the regulations, this overshooting did not occur. However, the short selling regulation and the up-tick rule caused the trading prices to be higher than the fundamental value. We also surveyed an adequate limitation price range and an adequate limitation time span for the price variation limit and found a parameters’ condition of the price variation limit to prevent the over-shorts. We also showed the limitation price range should be bigger than a volatility calculated by the limitation time span. The paper is structured as follows; in section 2, we explain details of our artificial market model. In section 3, we show results of simulations. The paper’s conclusions are presented in section 4.
2. Artificial Market Model
We built a simple artificial market model on the basis of the model of [11]. The model treats only one risk asset and non-risk asset (cash) and adopts a continuous double auction4 to determine a market price of the risk asset. The number of agents is. At first, at time, an agent 1 orders to buy or sell the risk asset; after that at an agent 2 orders; at, an agent orders respectively. At, going back to the first, the agent 1 orders, and at, an agent orders respectively, and this cycle is repeated. Note that time passes even if no deals are done. An agent determines an order price and buys or sells by the following process. Agents use a combination of fundamental value and technical rules to form expectations on a risk asset returns. An expected return of the agent is
where is weight of term of the agent, and is determined by random variables of uniformly distributed in the interval at the start of the simulation independently for each agent. is a fundamental value that is constant. is a market price of the risk asset at time. (When the dealing is not done at, remains at the last market price, and at,). is a noise determined by random variables of normal distribution with an average 0 and a variance. is a historical price return inside an agent’s time interval, and. is determined by random variables uniformly distributed in the interval at the start of the simulation independently for each agent. The first term of Equation (1) represents a fundamental strategy: an agent expects a positive return when the market price is lower than the fundamental value, and vice verse. The second term of Equation (1) represents a technical strategy: an agent expects a positive return when historical market return is positive, and vice verse. After the expected return has been determined, an expected price is
We modeled an order price by random variables of uniformly distributed in the interval , where is a constant. A minimum unit of a price change (tick size) is, we round off a fraction of less than. Buy or sell is determined by a magnitude relation between the expect price and the order price, that isWhen, the agent orders to buy one share.
When, the agent orders to sell one share.
Agents always order only one share. Our model adopts the continuous double auction, so when an agent orders to buy (sell), if there is a lower price sell order (a higher price buy order) than the agent’s order, dealing is immediately done. If there is not, the agent’s order remains in the order book. The remaining order is canceled at after the order time. Agents can short selling freely. The quantity of holding positions is not limited, so agents can take any shares for both long and short positions to infinity.
We also developed a model implementing a learning process of agents. Every agent learns just before every ordering. If there is only the first term (representing a fundamental strategy) or second term (representing a technical strategy) in Equation (1), an expected return at time of an agent is
respectively. We define where is a constant evaluation term. is changed when both and are the same signs,
On the other hand, when both and are opposite signs,
where is random variables of uniformly distributed in the interval (0,1) for each, is constant. Besides this process, is reset, random variables of uniformly distributed in the interval, occurring with small probability,. In this way, agents learn better parameters and switch to the investment strategy that estimates correctly: the fundamental strategy or technical strategy.
We investigated effectiveness of price variation limits, short selling regulations and up-tick rules. In this study, we modeled these regulations as follows. Price variation limits were modeled that any agents can freely order a price from to, where is a constant time span, and is a constant price. Any order prices of buy above are changed to, and any order prices of sell under are changed to. This prevents trading that prices outside. Short selling regulations were modeled that agents which do not have the risk asset can not order to sell. Any agents have initially one unit risk asset. Up-tick rules were modeled when an agent which do not have the risk asset tries to order to sell a price not greater than, the order price is changed to.
3. Simulation Result
In this study, we set , , . We also investigated two cases: the fundamental value was fixed to 10,000 (constant fundamental value); was 10,000 until and changed to 7000 after (sharp declining fundamental value). We ran simulations to .
3.1. Verification the Model
In many previous artificial market studies5, the models are verified to see whether they can explain the stylized facts such as a fat-tail6, volatility-clustering7, and so on. Table 1 lists stylized facts in each case. We used returns for 100 time units’ intervals to calculate the statistical values for the stylized facts8. In all runs, we can find that both kurtosis and autocorrelation coefficients for square returns with several lags are positive, which means that all runs replicate stylized facts.
In the actual financial markets including bubbles (crushes), the probability that a run, sequence of observations of the same sign, of positive (negative) returns will end should decline with the length of the run [17,18]. A hazard rate is used for the test of bubbles or crushes. represents the conditional probability that a run ends at, given that it lasts until. Empirical studies show that decline with the length of run if observation data include bubble or crush phenomena [17,18]. This means that the bubble (crush) returns tend to continue to be positive (negative) and this tendency becomes stronger as runs of positive (negative) returns become longer. In the Table 1, represents the conditional probability that a negative return run ends at, given that it lasts until. When hazard rates declined increasing, the simulation can replicate a significant crush like those occurring in actual markets. In the case of the constant fundamental value with a short selling regulation, declined. This indicates that some small crushes occurred even though there was no crush-inducing trigger. In the case of the sharp declining fundamental value, except the case of implementing the price variation limit, declined shallowly. These cases replicated big crushes like those occurring in actual markets.
3.2. Time Evolutions of Prices
Figure 1 shows time evolution of market prices in the case of the constant fundamental value. In the case without regulations and with the price variation limit, the market prices oscillated around the fundamental value, 10,000. This indicates that the market was efficient. On the other hand, in the case with the short selling regulation, bubbles and crushes occurred repetitively, and the prices were almost higher than the fundamental value. This result is consistent with previous studies [4]9. In the case with the up-tick rule, the prices were almost higher than the fundamental value although the amplitudes are less than those of the case with short selling regulation.
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. Buy or sell is determined by a magnitude relation between the expect price 
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where 
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is a constant price. Any order prices of buy above 
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, and any order prices of sell under 
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. This prevents trading that prices outside
. Short selling regulations were modeled that agents which do not have the risk asset can not order to sell. Any agents have initially one unit risk asset. Up-tick rules were modeled when an agent which do not have the risk asset tries to order to sell a price not greater than
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. We also investigated two cases: the fundamental value 
was fixed to 10,000 (constant fundamental value); 
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[17,18]. A hazard rate 
is used for the test of bubbles or crushes. 
represents the conditional probability that a run ends at
, given that it lasts until
. Empirical studies show that 
decline with the length of run 
if observation data include bubble or crush phenomena [17,18]. This means that the bubble (crush) returns tend to continue to be positive (negative) and this tendency becomes stronger as runs of positive (negative) returns become longer. In the 
represents the conditional probability that a negative return run ends at
, given that it lasts until
. When hazard rates 
declined increasing
, the simulation can replicate a significant crush like those occurring in actual markets. In the case of the constant fundamental value with a short selling regulation, 
declined. This indicates that some small crushes occurred even though there was no crush-inducing trigger. In the case of the sharp declining fundamental value, except the case of implementing the price variation limit, 
declined shallowly. These cases replicated big crushes like those occurring in actual markets.