An agent-based model for designing a financial market that works well
AAn agent-based model for designinga financial market that works well
Takanobu Mizuta ∗ SPARX Asset Management Co. Ltd., Tokyo, Japan
Abstract
Designing a financial market that works well is very im-portant for developing and maintaining an advancedeconomy, but is not easy because changing detailedrules, even ones that seem trivial, sometimes causesunexpected large impacts and side effects. A com-puter simulation using an agent-based model can di-rectly treat and clearly explain such complex systemswhere micro processes and macro phenomena interact.Many effective agent-based models investigating hu-man behavior have already been developed. Recently,an artificial market model, which is an agent-basedmodel for a financial market, has started to contributeto discussions on rules and regulations of actual finan-cial markets. I introduce an artificial market model todesign financial markets that work well and describe aprevious study investigating tick size reduction. I hopethat more artificial market models will contribute todesigning financial markets that work well to furtherdevelop and maintain advanced economies.
People have been able to develop advanced economiesby cooperating to exchange goods for money. Creationof any industry requires investment to first purchase orbuild tools to make goods. Thus, a financial marketthat enables smooth investment is obviously required.The economist John McMillan, who used game the-ory to investigate many markets, said “a market workswell only if it is well designed” [1]. Market design (reg-ulations, rules) determines whether a market workswell or badly. McMillan also concluded that “the econ-omy is a highly complex system. It is at least as com- ∗ [email protected], http://mizutatakanobu.com plex as the systems studied by physicists and biolo-gists.” The computer scientist Melanie Mitchell said“economies are complex systems in which the simple,microscopic components consist of people buying andselling goods, and the collective behavior is the com-plex, hard-to-predict behavior of markets as a whole,such as fluctuations in stock prices” [2]. A financialmarket is another highly complex system where a sim-ple summation of micro processes (trader behaviors)never explains macro phenomena (price formation).Changing detailed rules, even ones that seem trivial,sometimes causes unexpected large impacts and sideeffects. McMillan illustrated this nature as “both Godand the devil are in the details.” Designing a mar-ket well is very important for developing an advancedeconomy, but not easy. Separately investigating macro phenomena and microprocesses unclearly explains complex systems wheremacro phenomena and micro processes interact. Amathematical model and an empirical study cannot di-rectly treat or clearly explain the interactions. A com-puter simulation using an agent-based model, on theother hand, can directly treat and clearly explain theinteractions . An agent-based model includes agentsmodeling trader behaviors and shows macro phenom-ena as a result of their interactions. Agent behav-iors that are simple but affected by macro phenom-ena cause complex macro phenomena, which are nota simple summation of the agent behaviors. Thus, anagent-based model gives researchers new knowledge.An agent-based model, requiring no data, is a truecomputer simulation.Not only financial markets but also social systemsare complex system. Many studies using agent-basedmodels has already succeed to investigate social sys-tems . For examples, investigating effects of new cre-ation road ways to traffic jam and determining an evac- Sabzian et al. provide a comprehensive review of agent-based models for complex systems [3]. a r X i v : . [ q -f i n . T R ] J un ation route with terror and fire in a building. To solvesuch problems, researchers naturally use various ap-proaches: a mathematical model, an empirical study,and an agent-based model. Each approach has advan-tages and disadvantages and gives researchers variousviewpoints and knowledge to find unexpected side ef-fects. An agent-based model for such problems hasbeen as indispensable as a mathematical model andan empirical study. An artificial market model is an agent-based modelfor a financial market. Since the 1990s, many signif-icant artificial market models [4–6] have been devel-oped. Projects building generic artificial market mod-els have been conducted such as the U-mart project inJapan in the 2000s . These artificial market modelshave contributed to explaining the nature of financialmarket phenomena such as bubbles and crashes.An artificial market model, however, has rarely beenused to investigate the rules and regulations of a finan-cial market. After the bankruptcy of Lehman Broth-ers in 2008, some articles argued that traditional eco-nomics had not found ways to design markets thatwork well and anticipated an artificial market modelto do so. Indeed, in Science, Battiston et al. [8] ex-plained that “since the 2008 crisis, there has been in-creasing interest in using ideas from complexity the-ory (using network models and agent-based models)to make sense of economic and financial markets,” andin Nature, Farmer and Foley [9] explained that “such(agent based) economic models should be able to pro-vide an alternative tool to give insight into how gov-ernment policies could affect the broad characteristicsof economic performance, by quantitatively exploringhow the economy is likely to react under different sce-narios.”Financial regulators and exchanges, who deciderules and regulations, especially desire an artificialmarket model to design a market that works well.Indeed, the Japan Exchange Group (JPX), which isthe parent company of the Tokyo Stock Exchange, haspublished 30 JPX working papers including 9 papersusing an artificial market model as of April 2019 .Also in Europe, a three-year project (2014-2017)founded by the European Commission to integratemacro-financial modeling for robust policy design in-cluded a work package named bridging agent-basedand dynamic-stochastic-general-equilibrium modeling Kita et al. provide a comprehensive review [7]. approaches for building policy-focused macro-financialmodels . The Bank of England also published a work-ing paper investigating effects of passive funds in abond market using an artificial market model [10].Mizuta [11] reviewed other previous agent-basedmodels for designing a financial market that works wellthat are not mentioned above. Here, I will discuss features that an artificial marketmodel for designing a financial market should have.Such models aim not to accurately forecast but to de-sign a financial market that works well. To discusswhat a better design is, acquiring knowledge of whatmechanism affects prices is more important than repli-cating a real financial market.Such a model needs to reveal possible mechanismsthat affect price formation through many simulationruns, e.g., searching for parameters or purely compar-ing the before/after of changes. Possible mechanismsrevealed by these runs provide new knowledge and in-sights into the effects of the changes on price formationin actual financial markets. Other methods of study,e.g., empirical studies, would not reveal such possiblemechanisms.An unnecessary replication of macro phenomenaleads to models that are over-fitted and too complex.Such models would prevent us from understanding anddiscovering mechanisms that affect price formation be-cause the number of related factors would increase. In-deed, artificial market models that are too complex areoften criticized because they are very difficult to eval-uate [12]. A model that is too complex not only wouldprevent us from understanding mechanisms but alsocould output arbitrary results by over-fitting too manyparameters. It is more difficult for simpler models toobtain arbitrary results, so these models are easier toevaluate. An artificial market model should be builtas simple as possible and not intentionally implementagents to cover all the investors who would exist inactual financial markets.As Michael Weisberg mentioned, modeling is “theindirect study of real-world systems via the construc-tion and analysis of models. Modeling is not alwaysaimed at purely veridical representation. Rather, theresearchers worked hard to identify the features ofthese systems that were most salient to their investi-gations.” [13] Therefore, good models differ dependingon the phenomena being focused on. Thus, my modelis good only for the purpose of this study and may appened Will happen An empirical study
An artificialmarket model
Fig. 1: An artificial market model and an empiricalstudybe not good for other purposes. An aim of my studyis to understand how important properties (behaviors,algorithms) affect macro phenomena and play a rolein the financial system rather than representing actualfinancial markets precisely.Fig. 1 shows features of outputs of an artificial mar-ket model and an empirical study. Outputs of an em-pirical study are included in the area that has hap-pened in a real financial market. The advantage ofan empirical study is outputs exclude the all area nothappening in the past or future. The disadvantage,however, is outputs exclude any area happening in thefuture.The advantage of an artificial market model is out-puts include the part of the area happening in the fu-ture. The disadvantage, however, is outputs includethe part of area not happening in the past or future.An artificial market model just outputs “possible” re-sults to understand the mechanism of a market. Dis-cussing whether the results will occur or not needsother methods, e.g., an empirical study and a mathe-matical model.Discussing the outputs of an artificial market modelalways needs knowledge given by empirical studiesand mathematical models. A market that works wellshould be designed by not one but several methods (anartificial market model, empirical study and a math-ematical model), and the methods should collaborateto mutually compensate for their disadvantages.Many empirical studies, e.g., Sewell [14], haveshown that both stylized facts (fat-tail and volatility-clustering) exist statistically in almost all financialmarkets. Conversely, they have also shown that onlythe fat-tail and volatility-clustering are stably ob-served for any asset and in any period because finan-cial markets are generally unstable. This leads to theconclusion that an artificial market should replicatemacro phenomena existing generally for any asset and any time, fat-tail, and volatility-clustering. This is anexample of how empirical studies can help an artificialmarket model.
I introduce a paper investigating tick size reduction[15] (vol. 2, JPX working paper) as a typical studyinvestigating the design of a financial market using anartificial market model.The tick size is the minimum unit of a price change.For example, when the tick size is $1, order prices suchas $99 and $100 are accepted, but $99.1 ($99.10 cent)is not. Tokyo Stock Exchange used Y=1 as the tick sizeuntil 18 July 2014 and has used Y=0.1 (10 sen) since 22July 2014.More stock markets are now making full use of infor-mation technology (IT) to achieve low-cost operations,especially in the United States and Europe. Their mar-ket shares of trading volume have caught up with thoseof traditional stock exchanges. Thus, each stock istraded at many stock markets at once. Whether suchfragmentation makes markets more efficient has beendebated [16,17]. Many factors, such as tick size, speedof trading systems, length of trading hours, stabilityof trading systems, safety of clearing, and variety oforder types determine the market share of trading vol-ume between actual markets. A smallness of the ticksize is one of the most important factors to competewith other markets.Mizuta et al. [15] used an artificial market modelto investigate competition, in terms of taking marketshare of trading volume, between two artificial finan-cial markets that have exactly the same specificationsexcept for tick sizes and initial trading volume.
The model of Chiarella and Iori [18] is very simple butreplicates long-term statistical characteristics observedin actual financial markets: a fat tail and volatilityclustering. In contrast, that of Mizuta et al. [15] repli-cates high-frequency micro structures, such as execu-tion rates, cancel rates, and one-tick volatility, thatcannot be replicated with the model of Chiarella andIori [18]. Only fundamental and technical strategiesexisting generally for any market and any time areimplemented to the agent model.The number of agents is n . First, at time t = 1,agent 1 orders to buy or sell the risk asset; then at Many empirical studies found these strategies, which arecomprehensively reviewed by Menkhoff and Taylor [19]. undamental ExpectedreturnExpectedprice Market priceFundamental price Randomvariable
Technical
Historical return Noisej: agent numbert: time tjjt jhjtfji jit je urwPPwwr ,,2,1,, log1 j ttjht PPr /log , t P f P jt )exp( ,, jettjet rPP Fig. 2: An agent model t = 2 agent 2 orders to buy or sell. At t = 3 , , , , n ,agents 3 , , , , n respectively order to buy or sell. At t = n + 1, going back to the first agent, agent 1 ordersto buy or sell, and at t = n + 2 , n + 3 , , n + n , agents2 , , , , , n respectively order to buy or sell, and thiscycle is repeated. Note that t passes even if no dealshave occurred. An agent j determines an order priceand buys or sells by the following process. Agents usea combination of the fundamental value and technicalrules to form expectations on a risk asset return (Fig.2). The expected return of agent j at t is r te,j = 1Σ k w k,j (cid:18) w ,j log P f P t + w ,j r th,j + w ,j (cid:15) tj (cid:19) , (1)where w i,j is the weight of term i for agent j and isindependently determined by random variables uni-formly distributed on the interval (0 , w i,max ) at thestart of the simulation for each agent. P f is a funda-mental value and is constant . In addition, P t is themarket price of the risk asset, and (cid:15) tj is determinedby random variables from a normal distribution withaverage 0 and variance σ (cid:15) . Finally, r th,j is a historicalprice return inside an agent’s time interval τ j , where r th,j = log ( P t /P t − τ j ), and τ j is independently deter-mined by random variables uniformly distributed onthe interval (1 , τ max ) at the start of the simulation foreach agent .The first term of Eq. (1) represents a fundamentalstrategy: the agent expects a positive return when themarket price is lower than the fundamental value, andvice versa. The second term of Eq. (1) represents atechnical strategy: the agent expects a positive returnwhen the historical market return is positive, and viceversa. This enables focusing on phenomena in short time scales, asthe fundamental price remains static. When t < τ j , however, r th,j = 0. Orderprice
Determinedwith random
Sell (one share)
Buy (one share) )exp( ,, jettjet rPP t P jot P , djet PP , djet PP , Price
Expected price
Fig. 3: Scattering the order prices around the expectedprice.After the expected return has been determined, theexpected price is P te,j = P t exp ( r te,j ) . (2)The order price P to,j is determined by random vari-ables normally distributed with average P te,j and stan-dard deviation P σ , where P σ is a constant. (Fig. 3)Whether to buy or sell is determined by the magnituderelationship between P te,j and P to,j :when P te,j > P to,j , the agent places an order to buyone share, butwhen P te,j < P to,j , the agent places an order to sellone share .Scattering the order prices around the expectedprice enables the distribution of order prices in a realfinancial market to be replicated and a simulation torun stably.Agents always order only one share. The modeladopts a continuous double auction, so when an agentorders to buy (sell), if there is a lower price sell order (ahigher price buy order) than the agent’s order, dealingimmediately occurs. Such an order is called “marketorder”. If there is not a lower price sell order (a higherprice buy order) than the agent’s order, the agent’s or-der remains in the order book. Such an order is called“limit order”. The remaining order is canceled after t c from the order time. Agents can short sell freely. Thequantity of holding positions is not limited, so agentscan take any shares for both long and short positionsto infinity.The agents trade one stock at two markets: A andB (Fig. 4). The two stock markets have exactly thesame specifications except for the minimum unit of aprice change (tick size) per P f , ∆ P A , ∆ P B and initial When t < t c , however, to generate enough waiting orders,the agent places an order to buy one share when P f > P to,j , orto sell one share when P f < P to,j . gents Other casesDistributing orders on market shares
MarketA Market B
Market A: Large initial market share, Large tick size
Market B: Small initial market share, Small tick size
Market order and best prices differ
Ordering to the market with the best price is better
Fig. 4: A model selecting a market to which to order.share of trading volume, W A , W B . The agents shoulddecide to which market they order: A or B.The model of market selection is almost the same asan order allocation algorithm (Smart Order Routing,SOR) used in a real financial market. Each agent de-termines a market to which to order for every order.When the agent order is buy (sell), the agent searchesfor the lowest sell (highest buy) orders of each mar-ket. These prices are called “best prices.” When bestprices differ between two markets and the order will bea market order in least one of the markets, the agentorders to buy (sell) in a market in which the best priceis better, i.e., lower (higher) in the case of the buy(sell) order. In other cases, i.e., when the best pricesare exactly the same or the order will be a limit or-der in both markets, the agent orders to buy (sell) inmarket A with probability W A W A = T A T A + T B , (3)where T A is the trading volume of market A withinlast t AB , and the calculating span of W A and T B isthat of market B. To summarize, if the market orderand best prices differ, agents order to buy (sell) inthe market in which the best price is better than thatin the other market. In other cases, agents order tobuy (sell) in markets depending on the market shareof trading volume. Mizuta et al. [15] investigated the transition of marketshares of trading volume involving two markets. Thetwo stock markets (A and B) had exactly the samespecifications except for tick sizes per P f , ∆ P A , ∆ P B and initial market share of trading volume W A =0 . , W B = 0 .
1. They set n = 1000 , w ,max =1 , w ,max = 10 , w ,max = 1 , τ max = 10000 , σ (cid:15) =0 . , P σ = 30 , t c = 20000 , P f = 10000, and t AB =10000 (5 days). Simulations are ran to t = 10000000.
100 200 300 400 M a r k e t s h a r e o f m a r k e t A daysΔPB=0.01%ΔPA=0.01%ΔPA=0.1% Fig. 5: The time evolution of market shares of tradingvolume for tick sizes that are not too small. M a k e t s h a r e o f m a r k e t A days ΔPB=0.0001%
ΔPA=0.0001%
ΔPA=0.001%
Fig. 6: The time evolution of market shares of tradingvolume for tick sizes that are too small.Fig. 5 shows the time evolution of market shareof trading volume of market A, where ∆ P A =0 . , .
01% and ∆ P B = 0 . P A = ∆ P B =0 .
01 the market shares slightly moved. In ∆ P A = 0 . P B = 0 .
01, market Btook market share of trading volume from market A.On the other hand, Fig. 6 shows the case in which∆ P A = 0 . , . P B = 0 . P A also became 1/100.Market B could not take market share despite ∆ P B being 1/10 of ∆ P A . Therefore, competition under ticksizes that are too small does not affect the taking ofmarket share of trading volume.The relationship between tick size and taking mar-ket share of trading volume is investigated. Fig. 7shows the market shares of trading volume of W A at500 days for various ∆ P A and ∆ P B . The following two5 .001% 0.002% 0.005% 0.01% 0.02% 0.05% 0.1%0.001% 90% 92% 94% 97% 99% 100% 100%0.002% 89% 91% 93% 97% 99% 100% 100%0.005% 84% 87% 92% 96% 99% 100% 100%0.01% 77% 78% 83% 92% 98% 100% 100%0.02% 54% 54% 59% 70% 93% 100% 100%0.05% 5% 5% 5% 6% 23% 93% 100%0.1% 0% 0% 0% 0% 0% 0% 94%Market share ofmarket A at 500 Tick sizes of market BTick sizesof marketA Fig. 7: The market shares of trading volume for various∆ P A and ∆ P B . . % . % . % . % . % M a r k e t s h a r e o f m a r k e t A a t d a y s V o l a t ili t y Tick size of market AVolatility (left)Market share of market A at 500 days (right)
Fig. 8: Tick sizes, volatility and market share of trad-ing volume.borderlines, ∆ P A ≤ ∆ P B (dashed line) , (4)∆ P A < σ t (cid:39) .
05% (solid line) , (5)are drawn where σ t is the standard deviation of returnfor one tick, which was small enough.In the region in which at least Eq. (4) or (5) issatisfied, market share of trading volume of market Ais rarely taken. In the region in which neither Eq. (4)nor (5) is satisfied, under the dashed and solid lines,market share of trading volume of market A is rapidlytaken. This shows that when the tick size of market Ais smaller than σ t , market share of trading volume ofmarket A is rarely taken even if the tick size of marketB is much smaller than that of A.Fig. 8 shows the standard deviation of return forone tick (volatility), σ t , and market share of tradingvolume of market A at 500 days, W A for various ∆ P A where ∆ P B = 0 . P A in Fig. 8. The horizontaldotted line is σ t = σ t . On the left side, σ t equals σ t and σ t does not depend on ∆ P A . This means that thedifference in tick size does not affect price formationswhere tick sizes are smaller than σ t . time Impossible to trade at market A Market priceMarket A never represents movement of market prices. ⇒ Orders move to market B time ⇒ not need market B
Market A represents them.
Tick size of market A< ΔP A t > ΔP A t Enough resolution at market A Market price
Tick size of market A
Fig. 9: Mechanism taking market share of trading vol-ume. . % . % . % . % M a r k e t s h a r e o f P T S V o l a t ili t y Tick size of Tokyo stock exchange
Volatility (left)Market share of PTS (right)
Fig. 10: An empirical analysis for tick sizes, volatility,and market share of trading volume.On the right side, ∆ P A is larger, σ t is larger. Thisimplies that the prices normally fluctuate less than∆ P A ; however, price variation less than ∆ P A is notpermitted. Thus, price fluctuations depend on ∆ P A .In this case, market share of trading volume rapidlydeceases in accordance with increasing ∆ P A . On theleft side, however, the shares are stable.Fig. 9 summarizes the above discussion. When ∆ P A is larger than σ t (Fig. 9 top), if ∆ P B is smaller than∆ P A , there is a large amount of trading in market Binside ∆ P A . Thus, market B takes market share oftrading volume from market A. When ∆ P A is smallerthan σ t (Fig. 9 bottom), even if ∆ P B is very small,price fluctuations cross many widths of ∆ P A and suf-ficient price formations occur only in market A. Thus,market B can rarely take market share of trading vol-ume from market A.6 .4 An empirical analysis comparingwith the simulation results Next, Mizuta et al. [15] analyzed empirical data andcompared them with the simulation results shown inFig. 9, using Japanese stock market data. The dataperiod included all business days in the 2012 calendaryear. The number of stocks analyzed is 439, whichwere selected by TOPIX 500 over the entire indexdata period, had the same minimum unit of a pricechange for every month end, and were traded everybusiness day.Fig. 10 shows the standard deviations for 10 secondsof each stock (volatility), σ t (triangles), which is theaveraged standard deviation of return for 10 minutesexcept for opening prices for every day, and ∆ P ismarket share of the trading volume of the ProprietaryTrading System (PTS) of each stock (circles) withtick sizes per averaged prices at the end of every monthof the Tokyo Stock Exchange . The right vertical axisis upside down to easily compare with Fig. 9. Thehorizontal dotted line is σ t = σ t .On the left side, σ t , which equaled σ t , did not muchdepend on ∆ P . On the right side, ∆ P and σ t werelarger. These results are similar to those in Fig. 9.The market share of trading volume of PTS deceasedalong with ∆ P . When ∆ P was larger, PTS more easilytook market share of trading volume, and σ t tended toincrease along with ∆ P . A market having a tick size larger than volatility willlose market share of trading volume to other markets.In contrast, a market having a tick size smaller thanvolatility, even if the tick size is larger than those ofother markets, will rarely lose market share to othermarkets. A tick size smaller than volatility rarely af-fects competition of market share between financialmarkets, whereas a tick size larger than volatility en-larges the volatility and prevents adequate price for-mation. TOPIX 500 is a free-float capitalization-weighted index thatis calculated on the basis of the 500 most liquid and highlymarket capitalized domestic common stocks listed on the TokyoStock Exchange first section. Electric trading systems outside stock exchanges are calledPTSs in Japan. A PTS is very similar to an Alternative TradingSystem (ATS) and Electronic Communications Network (ECN)in other countries. Mizuta et al. used the data from the Tokyo Stock Exchangeto calculate ∆ P and σ t . They used Bloomberg data to calculatethe market share or trading volume of the PTS, which is itsentire trading volume divided by those of Japans traditionalstock exchanges and PTS, where PTSs are Japan Next PTS J-Market, Japan Next PTS X-Market, and Chi-X Japan PTS, andwhere Japans traditional stock exchanges are the Tokyo, Osaka,Nagoya, Fukuoka, and Sapporo stock exchanges and JASDAQ. This simulation study is the first to discuss an ad-equate tick size . An empirical study cannot investi-gate tick sizes that have never been used in an actualfinancial market or isolate a direct effect on price for-mation affected by many factors. In contrast, an arti-ficial market model can isolate the pure contributionsof changing a tick size to price formation and simu-late a tick size that has never been used. An artificialmarket model has these advantages over an empiricalstudy.After the simulation study of Mizuta et al. [15], themathematical model of Nagumo et al. [21] achieved thesame result. In this way, an artificial market model canindicate new problems that studies using mathemati-cal models and empirical analysis should solve. In this paper, I introduced an artificial market model,which is an agent-based model for a financial market,to design a financial market that works well. An artifi-cial market model has recently started to contribute todiscussions on rules and regulations of actual financialmarkets such as tick size reduction. The contributionhas not been great yet but will become greater soon.Some readers may think tick size reduction is a triv-ial matter for a financial market. This is, however,important and should not be underestimated. Chang-ing detailed rules sometimes causes unexpected largeimpacts and side effects. John McMillan illustratedthis nature as “both God and the devil are in thedetails” [1]. Detailed design can determine whethera financial market develops or destroys an advancedeconomy. Designing a market well is very importantfor developing and maintaining an advanced economy,but not easy.I hope that more artificial market models will con-tribute to designing a financial market that works wellto further develop and maintain advanced economies.
Disclaimer
Note that the opinions contained herein are solely those of theauthors and do not necessarily reflect those of SPARX AssetManagement Co., Ltd. Darley and Outkin [20] investigated tick size reduction us-ing an artificial market model when NASDAQ, a stock exchangein the U.S., was planning tick size reduction. They showed,for example, a market can be unstable when some investmentstrategies increase. However, their model had too many param-eters and they focused on which investors earned more, whichprevented them from discussing a good design of a financial mar-ket. eference [1] J. McMillan, Reinventing the bazaar: A naturalhistory of markets . W. W. Norton & Company, 2002.[Online]. Available: https://books.wwnorton.com/books/Reinventing-the-Bazaar/[2] M. Mitchell,
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