Design of Financial Market Regulations against Large Price Fluctuations Using by Artificial Market Simulations *

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.


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.
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 market 3 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].Mi-zuta 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.

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 auction 4 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 We modeled an order price , by random variables of uniformly distributed in the interval 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 c 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.
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 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 t j 4 A continuous double auction is an auction mechanism where multiple buyers and sellers compete to buy and sell some financial assets, respectively, in the market, and where transactions can occur at any time whenever an offer to buy and an offer to sell match [12].
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, , i j is reset, random variables of uniformly distributed in the interval ,max i , 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   .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 t P P   000, .

Simulation Result
In this study, we set 1 n 

Verification the Model
In many previous artificial market studies 5 , the models are verified to see whether they can explain the stylized facts such as a fat-tail 6 , volatility-clustering 7 , 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 facts 8 .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 i   H i is used for the test of bubbles or crushes.

 
H i represents the conditional probability that a run ends at , given that it lasts until .Empirical studies show that i i

 
H i 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

 
H i declined increasing i , 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,

 
H i 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,

 
H i declined shallowly.These cases replicated big crushes like those occurring in actual markets.

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.
Figure 2 shows time evolution of prices in the fundamental value f P changed at from 10,000 to 7000, which was the new fun amental value.In the case 100, 000 t  d  bounds were investigated by previous studies [19].On the other hand, in the case with regulations, the price variation limit, the market price took longer to reach the new fundamental price   7000 f P  than in the case without regulations, the price went far 6000 beyond the new fundamental value, in other words, an overshoot occurred.After this overshoot, the price rebounded to the new fundamental value.Such the overshoots and re-without the regulations, but the bubble almost vanished.
In the case with the short selling regulation and the uptick rule, small bubbles and crushes occurred repetitively after the prices converged as same when the fundamental values were constants.The price variation limit prevented both such small bubbles and the overshoots.The most efficient market, an ideal market, prevents overshoots and achieves immediate convergence to the new fundamental value.However, this study shows that no market achieves both at once.

Switching Strategy
Figures 3 and 4 show a time series of fundamental strategy weights

 
in case of the sharp declining fundamental value, without regulations and with the price variation limit, respectively.Figure 3 shows that, during the crush from about to , the technical strategy weight increased and the fundamental strategy weight declined.This is consistent with empirical studies [20,21] that show that investors tend to switch from the fundamental to the technical strategy during a crush because the fundamental strategy looses during an overshoot, declining the price less than the fundamental value.Figure 4 shows that, in contrast to in the case without regulations, during the crush the technical strategy weight did not increase but the fundamental strategy weight did.The price variation limit restricted agents switching to the technical strategy and 100, 000 sprevented overshoots occurring 10 .

Survey Adequate Price Limitation
Next, we investigated optimization of , pl pl P t  .The price variation limit prevents overshoots but also causes converging speed of market prices to the fundamental value to go slower.The most efficient market, an ideal market, prevents overshoots and achieves immediate convergence to the new fundamental value.However, no market achieves both at once.To make amplitude of an overshoot smaller, market prices converge to fundamental prices more slowly.On the other hand, if you speed up converging speed of market prices to the new fundamental value, occurrence of an overshoot is unavoidable.Therefore, it is important to search for better parameters, , pl pl P t  , preventing overshoots and not trying to make the converging speed much slower. .When pl pl P t  is same, it seemed that the time evolution of market prices tend to be similar from Tables 2 and 3.However, in Figure 5, only in the case of pl , the market prices declined a little and did never reach to the new fundamental value by the simulation end time, .This implicated that even if .Table 4 shows that the inequality constraint Equation ( 6) was not related whether or not the price converged to the new fundamental value.An upper right area above the solid line shows P t  v  pl t p l is satisfied.In this area, the market price never converged to the new fundamental value.Therefore, not only Equation ( 6) but also a condition, is needed to effective price variation limits, which make markets efficiently.

Conclusion and Future Study
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.One future study is to a find way to recommend actually good parameters of the price variation limit to real stock exchanges.We cannot observe actual shown in Section 3.4.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].We can interpret that the special quote that has a shorter time span determines and the daily price limit that has a longer time span prevents bubbles.Thus, where parameters of the daily price limit are these price variation limits can prevent bubbles.However, the left term of Equation ( 8) should be not so smaller than the right term as we suggested in Section 3.4.For example, minutes, yen, minutes, and yen for stocks that have prices from 1000 yen to 1500 yen in Tokyo Stock Exchange.Therefore, Equation ( 8) is strongly satisfied but the left term of Equation ( 8) is much smaller than the right term.It is possible that the daily price limit is too tight ( minutes is too large or yen is too small) or the special quote is too loose ( minutes is too small or yen is too large).However, this study could not discuss such actual parameters.This is a future work.

Figure 2 .
Figure 2. A time evolution of market prices in case that the fundamental value is sharply declining .  f P 10,000 7000

Figure 3 .Figure 4 .
Figure 3.Time series of a fundamental strategy weight and a technical strategy weight in case without any regulations.

Table 2
In this sense, the market is less efficient.When we design the price variation limit, it is important that the parameters satisfy Equation (6) and the left term is not much smaller than the right term.Figure5showed that several in the same