A Speculative Futures Market with Zero-Intelligence

Eastern Economic Journal, 2008, 34, (518–549) r 2008 EEA 0094-5056/08 www.palgrave-journals.com/eej

SYMPOSIUM ARTICLE

A Speculative Futures Market with Zero-Intelligence Leanne J. Usshera,b a

Department of Economics, Queens College, CUNY, Flushing, New York 11367, USA. Multi Agent Systems Lagrange Laboratory, ISI Foundation, Turin 10133, Italy. E-mail: [email protected] b

This paper investigates the price formation of an artificial futures market with zerointelligence traders. It extends the zero-intelligence model to speculative agents trading for immediacy on a futures exchange with open outcry, margin constraints, and real-time settlement. Like prior studies it finds that the imposition of scarcity, not intelligent optimization, is surprisingly good at producing allocative efficiency. The double auction trading mechanism even with open outcry and real-time settlement anchors prices to a dynamic Walrasian equilibrium, even when it is not unique. This study supports zero-intelligence agent-based methodology as a tool to isolate the impact of market microstructure, as opposed to information, on price formation. Eastern Economic Journal (2008) 34, 518–549. doi:10.1057/eej.2008.34 Keywords: agent-based model; zero-intelligence model; margins; double auction; futures market JEL: C63; D44; D61

INTRODUCTION The continuous double auction (CDA) was demonstrated in experimental markets by Smith [1962] to promote a speedy convergence, of price and quantity allocations, to the competitive equilibrium. This auction mechanism, which limits bilateral exchange to only the highest bid or the lowest ask, quoted up to that point, was even more dramatically acclaimed as efficient and Pareto optimizing in an agent-based simulation where profit optimizing agents were replaced with so called zerointelligence (ZI) agents [Gode and Sunder 1993, 1997]. This paper will take a similar approach, formulating a ZI speculative model of a futures market with margin trading, transaction costs, short selling, and real-time settlement (RTS). This market, while much more volatile than traditional ZI models, also has Pareto optimizing tendencies due to the double auction (DA) mechanism and the imposition of scarcity via margin accounts held at the futures exchange, and not from rational competition between agents. This paper will elucidate the potential use of ZI agent-based models to simulate the impact that financial market architecture has on price dynamics. Gode and Sunder [1993] (hereafter GS), and then again more robustly Gode et al. [2004] (hereafter GSS), simulated financial market trading in an agent-based model, but replaced profit optimizing strategies and rational competition with random but constrained trading.1 Despite a market populated with ZI traders, GS and GSS were still able to attain results where more than 90 percent of the gains from trade were quickly exploited and prices converged to the proximity of the theoretical equilibrium price. They concluded that intelligent architecture, such as the DA mechanism, and the constraint of scarcity, were alone enough to force trading prices and allocations to

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reach the competitive equilibrium.2 This result was surprising within the economics profession, where the maximization of utility is thought to be a necessary assumption for the invisible hand to work, and hence the ZI model became widely cited.3 This paper will attempt to replicate these results in an institutionally rich environment which is liquidity constrained. It will also attempt to resolve the criticisms of the GS model by offering an alternative ZI model where ZI speculators (rather than buyers or sellers) trade in a futures market, and the imposition of a budget constraint and the marking-to-market of each trader’s balance sheet in real time allows traders to lose equity when prices move against them. Market structure and trading rules cannot effectively be studied with assumptions of perfect rationality in analytical equilibrium models. Alternative methods include the use of intelligent traders in mechanism design using game theoretic techniques, or empirically via experimental markets or statistics. Mirowski [2007] has recently advocated the use of ZI models to study market structure. Such a platform appears to be a particularly useful alternative. Overall, the ZI traders are a tool to isolate and understand the effect of market rules on market outcomes. Understanding the effects of market rules and other social institutions is crucial because rules are observable and controllable, while individual strategies are inherently private and not directly controllable. Theories based on the effect of market rules are therefore easier to test. The ZI model provides a benchmark of the ‘‘structural’’ effect of market rules. The traditional strategic model in which traders respond fully to changes in market rules [and price outcomes] is another benchmark. The two benchmarks bracket the range in which human behavior lies [Gode et al. 2004, p. 2]. A significant empirical step forward in the ZI literature came when Farmer et al. [2005] was able to simulate, for a basket of stocks on the London Stock Exchange, 96 percent of the variance of the bid-ask spread, and 76 percent of the variance of the price diffusion rate (variance of price over time). The price dynamics in their ZI model were driven by the CDA and the gaps in the limit order book.4 This empirical success by Farmer et al. makes the case for simulating institutionally rich ZI models that are applicable to specific financial markets.5 The contribution of this paper is its novel reformulation of the standard ZI spot market DA model ala GS and GSS into a speculative DA futures market with open outcry and retrading.6 Each agent trades on margin, obeying exchange margin requirements and settlement rules. Chowdhry and Nanda [1998, p. 181] in an analytical model showed that ‘‘the rigidity of margin requirements is precisely what leads to price instability’’ via multiple equilibria. In a study on settlement frequencies, Farmer et al. [2004] found that a higher settlement frequency can have a positive impact on price volatility. Our model tries to combine both these insights. Since a margin requirement is only relevant with a settlement period, this model combines margin requirements and RTS. The marking-to-market of margin accounts in real time promotes the interaction of agents producing feedbacks into the volatility of returns. Interaction and feedback is at the heart of most agent-based models, but in this case it is through wealth and quantity effects rather than expectations and information. The risk neutrality of the ZI speculators and RTS characterizes a study of an illiquid market setting. Changes in prices can produce forced liquidations and simulations that show clustered volatility in returns. Preliminary findings suggest Eastern Economic Journal 2008 34


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that leverage increases the likelihood of multiple equilibria that can lead to sustained price volatility. Leverage also multiplies wealth transfers, which also has an impact on price dynamics. The ZI speculative model simulates returns that are characteristic of real markets, not possible in the simpler GS and GSS models that converged to a single Pareto optimal price and trading stopped. In this model, despite retrading, prices can always move away from the Walrasian equilibrium creating new price dynamics. This model is able to produce endogenous liquidity shocks, volatility clustering, and fat tails in returns. And yet despite the presence of realistic return moments during the trading process, the price series remains anchored by the Walrasian equilibrium at any one point in time analogous to the original GS and GSS models. In the next section, there is a basic introduction to the GS and GSS model vs the paper’s speculative ZI model of a futures market. This section is followed by the specific details of the model and an analytical derivation of a ZI speculator’s demand function. The paper then presents results from ZI speculator simulations across three different levels of leverage and concludes.

A SPECULATIVE ZI MODEL The ZI literature that began with GS was inspired by classroom experiments using a CDA. It was explained by GS using Herbet Simon’s [1955, 1996] distinction between global rationality and individual satisficing. Using Smith [1982] one can define three categories which determine the performance of a micro system: institutional structure, environment, and behavior. By removing intelligent or rational behavior a ZI model is created. In a typical micro system the institutional structure is defined by the market protocol and the rules that govern trading. The environment is represented by wealth endowments and technology, which includes the number of goods, markets, and traders. Demographics, gene pool, and the distribution of wealth are also environmental factors. Behavior is defined as instincts, preferences, trader tastes, risk profile, and trading strategy. Intelligence is required for learning or adapting behavior to optimize utility based on the information a trader obtains from the environment, institutions, other traders, and past prices. In standard models simple learning is typically termed bounded rationality and perfectly consistent learning is a rational agent. ZI is characterized by no learning — behavior is static except for some random element that is independent of experience. GS and GSS partitioned their ZI traders into buyers or sellers with fixed reservation prices drawn from a probability (uniform) distribution. Agents chose random mark-ups on their reservation price, and as a result they always moved higher up the utility hill through incremental bilateral trades, optimized by the DA rules rather than individual behavior. These ZI traders might have been better characterized as random satisfying agents. Counter to their claims the original GS traders had no budget or trading constraint and their results were determined by the distribution of reservation prices which allowed their claims to be falsified by Cliff and Bruten [1997] who argued that the DA was not sufficient to bring trading to the competitive equilibrium. The mean trading price for the ZI traders was only close to this theoretical Walrasian equilibrium price when the reservation prices averaged around the equilibrium price and the supply and demand curves were symmetric. When the demand and supply functions were not symmetric, they showed that trading did not converge a market clearing equilibrium. Eastern Economic Journal 2008 34


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The author here believes that this problem was rectified in the GSS working paper of 2004 through the proper inclusion of trader wealth: effectively adding a budget constraint that was not a determinant of individual rationality. With scarcity properly imposed the distribution of expectations, even weighted by wealth, are no longer the same as the Walrasian equilibrium. This result will also be highlighted in the model below. While scarcity was properly incorporated by GSS, it had the limiting assumption that the utility functions of agents were represented by replication. In effect they had two representative agents with two indifference curves in a closed two goods Edgeworth-Bowley box, with fixed endowments. Each group of traders (buyers or sellers) had a well-defined Cobb-Douglas utility function, showing the declining marginal utility for stocks and money with risk aversion. Another issue is that in both GS and GSS all placements of random limit orders by buyers and sellers, for unit size one, go into a limit order book which improves the stability of a DA and aids convergence to a Pareto optimal that is typically in close proximity to the Walrasian outcome prior to trading. When a transaction occurs the limit order book is emptied and the process of filling starts again such that several trading rounds occur producing a series of transaction prices.7 The clearing of the limit order book and the resampling of traders have also been criticized as inherent biases that promote convergence to the competitive equilibrium beyond the DA mechanism [see LiCalzi and Pellizzari [2008]]. Another problem, which has already been stated, is the impossibility for the ZI agents in both GS and GSS to lose. The guaranteed cost and resale price (or reservation price) endowed on each trader in GS. In GSS each limit order was a quote at a random price for one unit of the security, within the set of Pareto improving prices. In both papers the ZI agents would randomly climb the utility hill. Gjerstad and Shachat [2007] argue that in effect individual rationality has been reintroduced by these rules and hence these models are not truly ZI. These problems are resolved one way or another in this paper’s formulation of a speculative ZI futures market with a budget constraint on all traders, no limit order book, and RTS. Marking-to-market or RTS means losses can occur since the ZI reservation price is subjective rather than objective or informed which would otherwise guarantee a gain from trade every time. Listed below are the major areas in which the framework departs from the GSS ZI mode, and this ZI speculative model offers an alternative way in which to model a ZI agent. Details of each are explained in the next section. (1) (2) (3) (4) (5)

The DA is open outcry rather than a limit order book. Short selling and retrading is allowed. Margin rules and transaction costs apply. Traders are risk neutral rather than risk adverse. Trader limit orders are ‘‘truth telling’’ rather than a random markup, thus randomness is limited to the order queuing function. (6) Quantities are rational numbers rather than being limited to order size 1. The first three modifications are drawn from real futures markets, and future research will compare simulated results to returns in actual futures markets during periods of tight liquidity. This may highlight the role specific institutional structure has for price dynamics in a market. Eastern Economic Journal 2008 34


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THE MODEL A ZI model of n risk neutral speculators, who have the same initial endowment of money, mi0, and futures contracts, xi0 ¼ 0, for all i traders. All cash is kept in a margin account that cannot be added to exogenously. Each speculator or trader will buy low and sell high to make a profit in a futures contract x, on a non-storable underlying commodity, liquidated prior to the spot date.8 Each speculator has their own subjective valuation about the future spot price of the commodity, pi,y, at some date in the future beyond the period of study. This valuation is drawn from a symmetric Beta [2, 2] distribution with finite boundaries. The valuation does not change: characteristic of our ZI agents who have no learning or evolving animal spirits. Simulated time t represents the sequence of discrete bilateral transactions in futures contracts at price pt. At transaction time t, the sale or purchase of futures contracts Dxit by speculator i, at either the best ask pat , or best bid pbt , according to the rules of an open-outcry DA mechanism which makes the highest bid and the lowest ask the standing prices at which a crossing limit order trades. There is no restriction on short selling: trader i’s contract position at any one time, xit, can be positive (long contracts) or negative (short contracts). In this model the futures contract size and price are real values and perfectly divisible, that is, there is no pre-specified tick, and unlike most models the order size is allowed to be different from 1. Since there is no limit order book, the size of each trade is the minimum of the desired amount between the two matching traders, and there is no direct price impact as in markets with limit order books, where large orders ‘‘walk up the book.’’ But large orders can deplete the desire to trade by the market makers, those traders with the best current bid or ask, who drop out. This will be explained in the next section and allows for large price shocks. The number of futures contracts is endogenous to the trading process, but as in all such markets futures contracts always sum to zero Xt ¼ Sixit ¼ 0 for all t. The futures exchange charges a percentage transaction cost, opt|xtxt1|, incurred on each oneway trade (long or short). This creates a threshold around pi,y for which demands for changing one’s contract position is zero. In other words, speculator i wants to hold their current position for all |(pi,ypt)|popt. In contrast, speculator i will want to take on a longer position when prices are expected to rise by more than the tax: pi,y>(1 þ o)pt; and a shorter position when prices are expected to fall by more than the tax: pi,yo(1o)pt. Note that a hold decision can still be represented as a limit order by a speculator in the order flow, but for a quantity of zero futures contracts. Also, a speculator is not forward thinking enough to consider the tax incurred for the future liquidation of the contract position. Order size, or demand, is dependent on the speculator’s collateral or wealth position and the leverage limit placed on traders, by the exchange. A risk neutral ZI speculator, i, uses all his wealth at time t to take on a futures position. We can plot a demand function from basic principles of linear optimization. Since futures are promises in a derivatives market, rather than a realized exchange in goods, there is no trade-off between holding both futures and cash, there is only a margin requirement that is held as cash in a margin account. For simplicity there is no interest paid on cash deposits. In a model with no taxes, the budget set is a rectangle in a two goods space, mit and xit, with a positive or negative net amount of futures xit. Figure 1 shows such a budget set, where cash is on the vertical axis and contracts on the horizontal axis. Instead of the horizontal line at mt we have two sloping lines to accommodate the shrinkage in capital due to transaction taxes on Eastern Economic Journal 2008 34


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Figure 1. Bullish speculator i’s budget set and expected wealth objective. ((pypt)>0; xit1>0; 0ooo1; k ¼ 1)

changing ones position from yxt1 to xt. In this figure, the speculator is carrying over a long futures position xt1from the previous price at t1. Important in this model is the RTS of margin accounts or wealth positions, at every t by the exchange. All traders must meet a variation margin requirement on all positions with RTS.9 In practice, the margin requirement allows the exchange to cover all counterparty risk, that is, removing the risk of default. The inverse of the margin requirement is also known as the leverage ratio. If the exchange reduces the margin requirement mit/ptxit, from 50 to 25 percent, then they also double the leverage ratio, ptxit/mit, from 2 to 4. If prices rise and a trader has a long position then the budget set will expand through mark-to-market and margin payments. If prices fall and a trader has a negative position, then the budget set will also increase. As in all futures markets, for every price change there is a winner and a loser between traders with contract positions. This symmetry, along with the symmetry in expectations, and the zero default risk, allows for a neutral testing ground for the impact of mark-to-market or RTS on prices, void of other business cycle impacts such as expectations. Unlike GSS where agents have constant absolute risk aversion, here agents are risk neutral as in Chan et al. [1998]. Each speculator has a monotonic (linear) objective function for money, shown in Figure 1 as pale gray negatively sloped dashed lines. By speculating in a future market they can generate more (expected) money. This objective function is represented by equation (1), and its slope is determined by whether the speculator expects prices to go up or down. Given the chance to trade, a risk neutral speculator will either do nothing, plunge long, or plunge short: investing as much as possible. The amount that a speculator will place as a limit order for a given price can be derived as the speculator’s demand curve at time t, which is not only affected by transaction costs and the margin requirement, but also by the settlement frequency since the current mid-price is used to mark-to-market wealth and this also determines the amount a trader can afford. There are also margin calls transferring money from a losers account to a winning account. Adjusting the capital base m every t for RTS valued at pm t . Eastern Economic Journal 2008 34


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We use all these constraints to solve for each ZI speculator’s demand function for futures xit by speculator i from the following objective to maximize expected future wealth pte þ 1: Maximize: petþ1 ¼ ðpy pt Þxt þ mt

ð1Þ Subject to: ð2Þ

m pt xt X kððpm t pt1 Þxt1 þ mt1 opt ðxt xt1 ÞÞ

ð3Þ

m pt xt X kððpm t pt1 Þxt1 þ mt1 þ opt ðxt xt1 ÞÞ

ð4Þ

m pt xt Xkððpm t pt1 Þxt1 þ mt1 opt ðxt xt1 ÞÞ

ð5Þ

m pt xt Xkððpm t pt1 Þxt1 þ mt1 þ opt ðxt xt1 ÞÞ

ð6Þ

m mt pðpm t pt1 Þxt1 þ mt1 opt ðxt xt1 Þ

ð7Þ

m mt pðpm t pt1 Þxt1 þ mt1 þ opt ðxt xt1 Þ

ð8Þ

m 0pðpm t pt1 Þxt1 þ mt1 opt ðxt xt1 Þ

ð9Þ

m m 0pðpm t pt1 Þxt1 þ ðpt pt Þðxt xt1 Þ þ mt1 þ opt ðxt xt1 Þ

ð10Þ

mt X0

This produces a demand function for trader i: ð11Þ

m xit ðpt ; pi;y ; xit1 ; mit1 ; pm t ; pt1 ; k; o; eÞ

where pt is the price at tick t, which must be at either a bid pbt , or an ask pat , pi,y the price valuation of the next futures price pt þ 1 (and long run expected spot price), xit1 the previous contract position, mit1 the previous cash position in margin account following last transaction, pm t the current mid-price, used by exchange to mark-to-market position xit under RTS, pm t1 the previous mid-price, used by exchange to calculate position i change Dpm t xt under RTS, 1/k the percentage margin requirement (where k>1) of m futures position xitpm t valued at pt , o the percentage transaction tax on a one-way trade (i.e. paid each way), e the interpolation parameter to solve for indeterminacy. In summary, the risk-neutral speculator maximizes next periods expected wealth (1). The first four boundary constraints represents the limit on a speculator’s investment by the margin requirement when one is short in futures, (2) and (3), vs the extent to Eastern Economic Journal 2008 34


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which futures can be long, (4) and (5). These are the black dashed vertical lines in Figure 1. We have two each of these restrictions to take into account the one-way tax on both buying and selling opt|(xtxt1)|. If the transaction tax is positive then one of these boundary constraints will be slack for both short or long positions. The tax also effects the marginal change in ones position (xtxt1) creating a maximum point for cash at the hold position, (6) and (7). The bankruptcy conditions, (8)–(10), stop money wealth from voluntarily going below zero. There is an indeterminate solution when the objective or iso-wealth function is parallel to the budget set, pt ¼ pi,y/(17o). To solve this we use an interpolation method that multiplies the slope of the iso-wealth function by a very small amount (17e) to interpolate (x*, m*) by drawing a line from the origin, at xt1 and cash position minus maximum potential taxes, to the intersection of these two modified iso-wealth functions. This solution is shown diagrammatically in Figure 2. In our simulations below we use e ¼ 0.001 to produce an interpolation gap that is very small, a range for pt of less than 0.0003 on either side of expectations. This retains agents to be effectively risk neutral but removes the discontinuity. The speculative RTS demand schedule is drawn in Figure 3. Usually a futures contract demand curve is represented as a smooth downward sloping line from the top of quadrant II to the bottom of quadrant I in the two dimensional R2 space. Here, the speculator’s demand curve is piecewise due to risk neutrality and the transaction tax o, and dominated by corner solutions: either buy, sell, or hold. It asymptotically approaches the price and quantity axis due to the income effect on fixed cash holdings mit. The last seven terms of the demand function in equation (11) are out of the ZI speculator’s control. Input into his decision is only based on the current price and his valuation to make his decision to dive into a position or do nothing. The last seven terms are in effect outside of his behavior — they are environmental and institutional factors. All money is kept as collateral in the margin account by each speculator.10 Each speculator can leverage their absolute futures to cash position to a maximum ratio of i i kX|pm t xt|/mt, where kX1 and set by the exchange. The margin requirement is assessed in real time at the midpoint price. If collateral is less than the real-time

Figure 2. Bearish speculator using the e interpolation method to solve for xt. (xt1 ¼ 0; k ¼ 1) Eastern Economic Journal 2008 34


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Figure 3. Speculator i’s demand for futures xit as a function of pt. (xit1 ¼ 0)

Figure 4. Speculator i’s demand curve, with the dashed line doubling o or k. (e ¼ 0.3 and xit1 ¼ 0)

i margin requirement, that is mito|pm t xt|/k, then speculator i will be forced to liquidate their position with an offset, purchase, or sale, of the required amount to avoid a margin call, or injection of new capital.11 The competitive bid-ask spread is determined by the difference in expectations between the best bidder and the best offerer, plus the transaction tax approximating 2(1 þ o)pm t if valuations near the best bid and ask are far apart. It is possible for one speculator to offer both sides of the market with a spread 2opm t but even this can be narrowed with two traders if valuations are closer than this distance. An increase of the transaction tax would possibly widen the bid-ask spread as shown by a single demand function in Figure 4. Increasing the allowed leverage

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ratio, k, will stretch the middle of the single trader’s demand curve out, also shown in Figure 4. Marking-to-market in real time Marking-to-market positions in real time is the duty of the exchange. At each tick time t the exchange will simultaneously redistribute profits and losses across all traders valued at the mid-price, which is the average of the best bid and ask: b a pm t ¼ (pt þ pt )/2. The profit or loss is calculated with price changes of the mid-price from one tick to m i the next, and it shows up in our budget conditions (1)–(9): (pm t pt1)xt1. If a i speculator currently has a prior futures position, xt1a0, then RTS may lead to forced liquidation on losing position to lower the leverage to the maximum, if prices move against expectations. This is the possibility of a backward bending demand function, as in Figure 5. This is typical of markets where collateral that underlies demand for x, is priced in the same market.12 While the margin requirement determines how big a position a speculator can take to reinforce its expectations, it can also force a speculator to liquidate a portion of a losing position, placing an order that is in opposition to the trader’s expectations. To ease exposition of RTS all traders have their positions valued at the midpoint at each time t. When an exchange occurs between two traders at the bid or ask, this change in their position is immediately revalued to the midpoint. This simplifies the updating of m positions in real time for traders who are not trading, from one pm t1, to the next pt . It m i i also explains the (pt pt)(xtxt1) element ofequation (9) for the trading agent. If one were to aggregate the individual demand curves together, along Walrasian lines where all traders could trade simultaneously, then it is easy to see how we can end up with multiple equilibria (see Figure 6). This result is very similar to the analytical model presented by Chowdhry and Nanda [1998] which can have multiple prices at which the market can clear due to margin requirements. Given closure in this zero sum model, we always have existence of at least one equilibrium.

BIDDING BY ZI SPECULATORS The above derivation of demand only explains the desired quantity exchanged at a transaction price. What limit order price a trader submits is based on their nonoptimizing, satisfying, static behavior rules. Which price is ultimately used for a bilateral trade is determined by the rules of the DA mechanism. The auction itself consists of a sequence of bidding between each transaction. Speculators are selected randomly, without replacement in each round, such that every trader has a chance to enter the market and place their limit order every round. Speculators trade if their bid crosses or is crossed by another limit order according to the DA rules. Quantities traded are the minimum between the bilateral parties and their transaction prices are registered at each time t if the amount is non-zero. The DA replicates open-outcry on the floor of a futures exchange where ‘‘a quote is good only as long as the breath is warm’’ [Erenburg et al. 2003, p. 7]. Only the best bid and best ask prevail at any one time.13 This is formally known as a limit order book of size one, as there is one quote on either side for a certain quantity. Since there is no accumulation of verbal limit orders between transactions there is no need for a tally book and effectively no limit order book.14 Eastern Economic Journal 2008 34


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Figure 5. Backward bending demand curves for short or long positions for a single trader. (mt1 ¼ 5000, py ¼ 150, k ¼ 2, o ¼ 0.03, e ¼ 0.001, and pm t ¼ pt)

No order book means that non-competitive limit orders expire instantly and traders are characterized as demanding immediacy: the desire to trade immediately by posting values close to their reservation prices. LiCalzi and Pellizzari [2007, p. 3573] would call these traders ‘‘almost truth telling’’ agents. The random trading element among these ZI traders is the sequence of the order flow and the valuation distribution of the initial trading population. The bidding algorithm for ZI speculators is drawn from Chan et al. [1998]. They use an endogenous variable st ¼ (pat pbt )/2, which is the half spread at time t, to determine the limit order when the ZI valuation pi,y is between the best bid and ask. This asynchronous bilateral bidding process allows two or three traders to participate at any one time: offering, or bettering, limit order quotations or carrying out market order trades. The example below presents four different scenarios where speculator k has the best current bid, speculator j has the best current ask (k and j could be the same trader), and the new ZI speculator is entrant i. b ZI speculator i enters the market scenario with bid:ask (ptj, a:pk, t ) and makes a quote choice: k,b Scenario 1: (Figure 7) The ask, pj,a t , and bid, pt , currently exist with non-zero offers, at time t.

1. If pi,y>pj,a t speculator i will post a market order and buy at this ask price — this is termed as lifting the ask quote. 2. If pi,yopk,b t , speculator i will post a market order and sell at this bid price — this is termed as hitting the bid quote. j,a k,b j,a i,y 3. If pk,b t pp ppt and o(pt þ pt )/2, speculator i will post a sell limit order at a i,y price of (p þ st) and thus quote his own ask, replacing agent j. i,y j,a k,b j,a 4. If pk,b t pp ppt and X(pt þ pt )/2, speculator i will post a buy limit order at i,y a price of (p st), and thus quote his own bid, replacing agent k. If there is a zero quantity standing bid, then effectively there is no standing bid, and an entering speculator will announce their own non-competitive limit order, often increasing the spread considerably. The algorithm used here allows the speculator to quote their reservation price plus or minus a small margin, as in Chan et al. [1998], which is some multiple of the transaction fee or So. If S is greater than 1 then the new limit order will guarantee the possibility that a new crossing limit will Eastern Economic Journal 2008 34


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execute this price since the demand is different from zero. For example: k’s demand to Scenario 2: (Figure 8) Only the best ask, ptj,a, exists; that is, at pk,b t go long is zero as (xkt xkt1)p0. 1. If, pi,y>pj,a t speculator i will post a market order, buy at this ask price. i,y i,b 2. If pi,yppj,a t , speculator i will post a buy limit order pt at a price of (1So)p , i i but only if excess demand at this price is (xtxt1)>0. j,a Scenario 3: (Figure 8) Only the best bid, pk,b t exists; that is, at pt j’s demand to go j j short is zero as (xtxt1)X0. 1. If pi,yopk,b t , speculator i will post a market order and sell at this bid price. i,y i,a 2. If pi,yXpk,b t , speculator i will post a sell limit order pt at a price of(1 þ So)p , i i but only if excess demand at this price is (xtxt1)o0. Scenario 4: If no bid or ask effectively exists; that is at the ask quote pj,a t , k k (xjtxjt1)X0, and at the bid quote pk,b t , (xt xt1)p0. 1. The new entrant speculator will post both a buy and a sell limit order at (1So)pi,yand (1 þ So)pi,y, respectively, as long as his bid is quoted for a buy of greater-than-zero contracts, and the ask is to sell greater-than-zero contracts. If this is not the case then the current bid-ask remains, even though both traders have zero demand, and entrant i exits to join the queue to trade again later.

Figure 6. Aggregate demand function at time t, showing multiple equilibria.

Figure 7. Scenario 1, both bid and ask exist (i.e. have positive order size) prior to new entrant. Eastern Economic Journal 2008 34


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Figure 8. Scenario 2 in which an ask but no bid exists prior to new entrant and Scenario 3, in which a bid but no ask exists prior to new entrant.

In this model, under Scenario 2 (Scenario 3) the speculator tendering the best bid (ask) might have had prices move against him; for example, if he is long (short) and prices fell (rose). They may remain offering a bid (ask) price to buy (sell), but at a quantity of zero. Now he wants to offset his position and sell (buy) so that excess demand is less (greater) than zero: k k k,b Scenario 2: (xkt [pk,b t ]xt1)p0 where xt is a function of pt j j j,a Scenario 3: (xjt[pj,a t ]xt1)X0 where xt is a function of pt

Effectively under Scenario 2 (Scenario 3) agent k (agent j) falls silent, as he can no longer buy, and will eventually be replaced by a new entrant, as long as the new entrant i i,y i has pi,yopj,a (has pi,y>pk,b t t ) and as long as [xt(1 þ So)pt xt1]>0 (as long as i i,y i [xt(1So)pt xt1]o0) otherwise agent k (agent j) will remain. When agent k (agent j) is replaced and exits the market he will be given the chance to satisfy margin requirements by liquidating his position with a market order, in turn in the random trading round.

SIMULATIONS A simple Monte Carlo study of 100 runs exemplifies the types of results attained from this model for just 20 traders.15 While GS use reservation prices from a uniform distribution this speculation model uses hump-shaped distributions to better represent the distribution of valuations in a speculative market, although with agents who do not learn. Each speculator’s reservation valuation is drawn from a Beta(2,2) distribution bounded between 100 and 130. This distribution is very similar to a Normal distribution: N(115,7.5); but with finite and hence thinner tails. The global parameters for all simulations are: n=20 t=2000 mi0=$10,000 for all i pi, yB100+30*Beta(2,2)


k=1, 2, or 4 o=0.001 S=8 e=0.001 b 5, a p7, 0 :p0 =(120:130)


531

The simulations cover three leverage ratios k ¼ {1, 2, 4}, alternatively known as margin requirements 100, 50, and 25 percent. Each k scenario has 100 simulations: 10 different realization sets for price valuations pi,y for the 20 traders (shown in Table 1); and 10 differently seeded sequences of random order flow. Each DA run is initialized by trader i ¼ {7, 5} with an exogenous bid (120) and ask (130) quote and their desired order size for this price (which could be zero). The transaction tax is kept to a small amount to simplify the results. Each single run simulates a series of bilateral transactions for a price and quantity at time t. Wealth, or margin account balances and contract positions, are also tracked for all traders at time t, settled at pm t , which is the real-time mark-to-market valuation for all traders. After each simulation run, we solve ex post the Walrasian equilibrium solution p* t for the desired demand for all traders at each t, using Mathematica’s fixed point algorithm, which uses Newton’s Method. The proximity of the actual price to this fixed point price is used as a measure of efficiency. The drawback of this method is that this finds only one fixed point when there may be multiple. Nevertheless the single fixed point solution remains a relevant, though hypothetical, benchmark to compare the ‘‘efficiency’’ of the price series since in most cases only a single point exists. Efficiency in this context is where aggregate supply equals aggregate demand for a given price. At this point a Pareto optimum prevails since gains from trade across all agents are exhausted. Another measure of increasing efficiency or liquidity is the narrowing of the spread between the best bid and ask.16 Description of a single run Different valuation dispersions and order flow produces different price paths. This is to be expected given the bilateral non-taˆtonnement trading process [Negishi 1961], which produces prices that depend on when a trader enters the auction. When wealth remains fairly constant these prices appear to be anchored by a steady Walrasian equilibrium price at each time t. If RTS was not adjusting wealth and creating forced liquidations, or if wealth was unlimited with a fixed order size, then trading and retrading would produce prices that would converge and stay at a constant Walrasian equilibrium for all t, since one or two traders closest to the equilibrium would dominate the market-making process indefinitely. However, liquidity or wealth constraints stop this from happening and produce price movements that can create distortions away from and then back to the current Walrasian equilibrium, which changes over time due to wealth redistributions. Unlike other ZI data, the simulated time series in this model can produce price return patterns, or moments, that are similar to actual financial data. One example is plotted in Table 2 of a single run that was quite erratic with a leverage ratio of 4, from valuation set 1. The logged raw data shows the bid-ask bounce and spread which narrows until there is an event that could be called an endogenous liquidity shock, where a market maker (i.e., a speculator with the best bid or ask) drops out of the bidding process because of a wealth constraint. At this point a new entrant can come in and the spread can suddenly blow out. If we take the log of returns we see volatility clustering that resembles actual times series, at least more so than other ZI models. The ACF plot of this data indicates that it has a negative autocorrelation coefficient, and could be represented by an MA(3), ARIMA process. This is also similar to financial tick data where an MA(1), process is often used to model the bidask bounce, and where there is generally no autoregressive term in financial markets that are considered efficient. Eastern Economic Journal 2008 34


Table 1 10 Realizations of sample valuations for 20 ZI speculators

532


Table 2 Transaction price time series: single run with k=4 (valuation set 1, order flow 4) Log transaction price data

Log differenced transaction price data

ACF

PACF


533



534

Table 3 shows a single run of prices for the same distribution of valuations (set 2 in Table 1), the same sequence of limit order quotes, but different leverage ratios k ¼ {1, 2, 4}. Each graph plots the transaction price pt with its characteristic bid-ask bounce, and the dynamic Walrasian equilibrium p*. t This is called dynamic in the sense that it is the equilibrium solution for simultaneous multi-lateral trade recalculated for each t prior to the bilateral trade or transaction price at pt. As in most runs, the benchmark Walrasian price changes very little over the trading period, as there is only limited re-distribution of wealth and positions through trading and margin calls in this scenario. What is striking is the degree to which the hypothetical Walrasian price, even at t0 anchors the DA bilateral trading sequence. Despite no limit order book (greater than 1), risk neutral traders, short selling, margin trading, and RTS, the price series is relatively stable and the Walrasian price is between the bid and ask quotes for most of the trading periods (see bottom row of Table 7). While exceptions exist, one can conclude that the Walrasian price anchors the DA price series, similar to the GSS results. In these simulations, despite re-trading, trading never stops although volume may fall to a very small amount. Without a limit order book large price shocks exist: a market-maker that offers the best standing bid or ask or both, with a desired zero quantity (that is for xio70.001), will be replaced by a new limit order entrant with a single quote of either pymst. The lack of an effective second-best limit order waiting to counter the new incoming quote can produce a sudden decline in market liquidity.17 This new best bid or ask limit order will only turn into a transaction price if the next entrant does not narrow the spread through competing limit order quotes. The first plot in Table 3 shows a few price spikes that could be thought of as such misprices or trades far from ‘‘equilibrium’’ prices, similar to prices in real data. These misprices occur relatively infrequently in the scenario in Table 3. When they do occur they instigate a flurry of trading (as seen in row 2 of Table 3) by winners and losers, since RTS will mark-to-market margin balances at the new pm t . In the case where there is no leverage, k ¼ 1, there is no involuntary trading or margin calls and the market quickly converges back to the equilibrium price under valuations for set 2. However, in the case where leverage is four times, k ¼ 4 then a similarly large price movement causes a much larger increase in trading volume and even larger price movements. This slows convergence time back to the equilibrium, and there even occurs overshooting of prices. In the second column where leverage is just twice the wealth, k ¼ 2, the scenario is locked into a stable trading pattern where the traders with valuations very close to the Walrasian equilibrium act as market makers and buy and sell either side of pt*. In the valuation set 2 there are three traders who all have a valuation of 117, which is also the Walrasian equilibrium for most of t. As a result these traders act as market-makers and buy low and sell high, small amounts (just over 2 units xt) adding to their wealth each time. This allows them to maintain purchasing power. The smaller plot in the right-hand corner of row 2 column 2 in Table 3 is at the end of the simulation period and shows a close up of the absolute trading volume. The amount purchased is equivalent to the gains made by the trader through buying at the bid and selling at the ask. The amount bought is always a little more than the amount sold, hence the bounce in the volume series in this plot. The consolidation of valuations such as in set 2 appears to be an important factor in determining the stability of our price formation. Among the different sets of valuation realizations, set 3 and 9 have the fattest tails and greatest dispersion. Runs with these valuation sets have the most volatility in prices. An example is shown in Table 4, where the only change from Table 3 is the use of set 3 valuations in Table 1. Eastern Economic Journal 2008 34

Table 3

Double auction transaction prices vs Walrasian equilibrium, and trade size at time t (valuation set 2, order flow 1)

— DA transaction prices; - - - Walrasian equilibrium


Trade size and total trading rounds

535



536

The second row in Table 4 plots the hypothetical aggregate demand function at a specific time t. One can see that the potential for multiple Walrasian equilibria exist for all three scenarios, since there are backward bending components of the aggregate demand schedule that have been plotted for RTS at time t. In the case of no leverage this backward bend is due to a wealth effect: as prices rise less can be purchased with a unit of cash. Risk neutral speculators have no cash buffer to make these payments and positions must be liquidated at their next turn to trade. The greater the leverage the steeper the backward bend for both the wealth effect and adverse price movements. Leveraged speculators with non-zero positions will incur margin calls when prices move against them, as in Figure 5. This perverse aggregate demand function will exist over a larger price range when expectations are dispersed. Comparing across all the simulations for a given k there appears to be a smaller likelihood that a multiple equilibria would occur when k ¼ 1. In general, this likelihood of multiplicity increases as leverage increases.18 But in some cases, when expectations are dispersed, a lower leverage can retain a multiplicity of equilibria in the trading series for a longer time. For all the plots in row 1 of Table 4, we see a move from one stable Walrasian price to the next: from 112 to 116 midway through the simulations. During this price transition there are usually multiple equilibria in existence. This multiplicity is not represented in the time series of p*, t the dashed line in row 1 of Table 4, calculated with Mathematica’s fixed point solution. At a single t the multiplicity in equilibria is shown by the aggregate demand functions plotted in row 2 in Table 4. The existence of multiple equilibria produces price volatility. When the series return to a dominant single equilibria then the bid-ask spread also stabilizes. In the plot where k ¼ 4, the times where the bid-ask bounce has stabilized are periods of single equilibria. Where prices are unstable there is multiple equilibria. This is the same for k ¼ 2. While RTS, diverse valuations, and leverage raise the existence of multiple equilibria, it seems that leverage can also remove this instability. When leverage is high a new single equilibrium comes about more quickly through trading, than when leverage is low. In the k ¼ 1 scenarios, while the equilibria are more likely to be singular, in those rare cases when there is a multiple equilibria the price series remains erratic for longer. Low leverage means less transfers of wealth and hence the valuations that in part create these perverse dynamics remain relevant between traders. Thus, while higher trading leverage increases the likelihood of multiple equilibria, it also appears to lead to faster convergence to a situation of a single equilibria as traders have larger wealth redistributions and the same market-makers start to dominate prices more quickly. This does not occur in the k ¼ 1 scenarios in Tables 4 and 5. If we take a different sequence of trader order flow, but the same valuation set 3, then we see a similar outcome in Table 5. Again the dispersion in expectations along with RTS creates the situation for multiple equilibria. Leverage appears to defuse the multiple equilibria through the redistribution of large losses and wins to create a single equilibrium environment. Our conclusion from looking at all 300 single runs of ZI risk neutral speculators, is that the open outcry DA mechanism produces prices that are anchored by the Walrasian equilibrium. Despite a small population of traders the Walrasian equilibrium remains steady and unique when reservation valuations are consolidated and wealth transfers are limited. Wealth transfers are greater when trading is leveraged and valuations dispersed. Multiple equilibria is a cause for price instability. Under RTS price shocks can create more volatility and trading especially when markets are leveraged since this intensifies the perverse slope of individual and Eastern Economic Journal 2008 34

Table 4 Double auction transaction prices vs Walrasian equilibrium, aggregate demand, and trade size, for single runs (valuation set 3, order flow 1) — DA transaction prices; - - - Walrasian equilibrium


Trade size and total trading rounds

537


538

Double auction transaction prices vs Walrasian equilibrium, and aggregate demand schedules for single runs (valuation set 3, order flow 10)

— DA transaction prices; - - - Walrasian equilibrium

Walrasian aggregate demand schedules at time t={700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500} for each k in k ¼ 1, k ¼ 2 and k ¼ 4



Table 5


539

aggregate demand functions. However, while RTS and leverage together produces more multiple equilibria events, in this ZI speculative model, the initial volatility and higher leverage can more quickly reduce the diversity of opinion through large wealth transfers from losers to winners, which removes the multiplicity of equilibria and stabilizes prices. If we characterize our market as having ‘‘known fundamentals’’ when there is a single equilibrium, vs ‘‘unknown fundamentals’’ when there are multiple equilibria, then our results lend support to notions by Chowdhry and Nanda [1998] and Brunnermeier and Pedersen [2007] that higher margins or lower leverage can produce higher volatility when the fundamentals are unknown. Although in this case it is not due to knowledge, but rather that higher leveraged markets knock out other equilibria more quickly and speed convergence to a single equilibrium. Description of the average over 100 runs To generalize the above results we compare statistics over 100 runs for different leverage ratios k. Table 6 top left, takes the standard deviation of the mid-price across 100 runs for time t, and then smoothes this outcome using a moving average of 100 t. The variation in price outcomes across runs is greatest when trading is first initialized. The DA mechanism reduces this variation through trading as prices converge to the mean midpoint price across runs. The higher leverage trading scenarios, where k ¼ 4, has much greater variation between price paths. Table 6 top right plots the Gini coefficients over time. There is quite a bit of variation across the different valuation sets and order flows. In this plot the average Gini across 100 runs for each k is bounded by one standard deviation on either side of the mean by a dashed line of the same shade. Leverage dramatically increases the possible gains and losses from trade between agents. In addition, those runs with the greatest transfers of wealth and greatest inequality for a given k, are for trader populations that have dispersed valuations, such as valuation sets 3 and 9 in Table 1. One measure of liquidity typically used in financial markets is the size of the bidask spread. The narrower the spread the lower is the cost of a market order relative to the midpoint price. Usually the midpoint is considered to be the objective fundamental market valuation in an efficient market. In our case, we use the bid-ask spread as a measure of how efficient the DA rules are in promoting competition between traders to narrow the bid-ask spread and reduce the marginal profits to market-makers.19 From Table 6 bottom left, the DA rules replicates competitive behavior and narrows the spread over time, especially in the first 500 transactions. The lower leveraged markets tend to have a lower spread on average, less than 1 price unit, than the higher leveraged market. In terms of the variation of this spread between markets, the higher leveraged market has much greater extremes, with at times very narrow spreads and at other times quite wide spreads. This can be seen in the bottom right of Table 6 which plots the standard deviation of the spread across 100 runs for different k. Among the single runs, the multilateral Walrasian price could quickly jump to a dramatically different price, but it was generally followed by the bilateral DA price. In contrast, the average valuations of traders, weighted by wealth, moved in a slower and more independent fashion from midpoint prices. The first row in Table 7 compares the average absolute spread between the midpoint price and the expected price (weighted by wealth) across k, and the spread between the midpoint price and Eastern Economic Journal 2008 34


540

the Walrasian equilibrium, which is the much lower and darker line in all three graphs. The midpoint is much closer on average to the Walrasian price. This goes against Cliff and Bruten’s [1997] critique of GS that equilibrium prices were equivalent to the average valuation price. In the second row of Table 7 these same spread values, call them pxt , are subtracted x from the bid-ask half spread: ((pat pbt )/2|pm t pt |) for each run, ranked and then averaged across 100 runs. The solid line is where pxt ¼ p*t and represents the averaged ranked difference between the half bid-ask spread and the equilibrium spread. The dashed line is the averaged ranked difference between the bid-ask spread and the distance of the valuation of the trader population weighted by wealth from the P midpoint, where pxt ¼ i20¼ 1 pi,t y/20. This is plotted for each k. From these graphs we can see how often the Walrasian equilibrium was between the bid-ask spread, vs how often the weighted expectation was between the bid-ask spread. We see that in 86, 80 and 66 percent of the time the Walrasian equilibrium was between the spread for k equal to 1, 2 and 4, respectively. The averaged expected price (weighted by wealth) was only ever between the bid and ask 10–15 percent of the time. We conclude that the Walrasian equilibrium price anchors the midpoint price better in lower leveraged scenarios on average. The difference between the Walrasian price and the midpoint could be thought of as a measure of price efficiency. In Table 7 in the first row we see that this distance declines most in the first 250 periods of trading. The lower the leverage the closer the midpoint is on average to the Walrasian equilibrium price. Thus, for no leverage it falls to below 0.5 price units on average. For a leverage ratio of 2 it does not decrease

Table 6 Mid-price variation, wealth redistribution and bid-ask spread across 100 runs for each k


Table 7 Measure of efficiency, convergence to Walrasian equilibrium across 100 runs P i, y x P20 i, y Average absolute spread between midpoint price and pxt where: — pxt =pt* Walrasian price; and y pxt = 20 i=1pt /20 pt = i=1pt /20 wealth weighted average valuation.


P i, y Half bid-ask spread minus spread between mid-price and pxt where: — pxt =pt* Walrasian price; and y pxt = 20 i=1pt /20 wealth weighted average valuation.

541



542

as fast but it again settles to just below the 0.5 price units. For the quadruple leverage, this distance does not go below 0.5 on average across the 100 runs. From this measure and the spread, it appears that the lower leveraged market produces more price efficiency on average, with faster convergence to the Walrasian equilibrium. Looking at a less aggregated summary, Table 8 plots the average for each valuation set in Table 1, averaging across 10 runs. The 2 sets with the highest distances between the midpoint and the Walrasian equilibrium, over a 100 t moving average, were typically set 3 and set 9, for all leverage ratios, which are the two most uniform or more dispersed distributions in Table 1. Despite the cases shown in Tables 4 and 5 for particular order flows, in general the lower leverage market appears to be more efficient across 10 runs of different order flows. One conclusion, is that a fixed lower leverage is more efficient than a fixed higher leverage which increases the spread, on average, between the midpoint and the equilibrium as shown in Table 8, for the same dispersion of fixed valuations. However, there is coincidental evidence for the Brunnermeier and Pedersen [2007] conclusion that raising the possible leverage rates during times of multiple equilibria and instability (times of ‘unknown fundamentals’), would allow for a bigger wealth reallocation and a quicker move towards a market with a single equilibria, returning the market to a stable bid ask spread around the equilibrium price, which may otherwise not happen when there is a tendency for multiple equilibria, as shown in Tables 4 and 5. While trading and re-trading aids convergence to the equilibrium it is not a guarantee of price stability or the removal of trading opportunities unlike GS or GSS. Looking at Table 9, average trade size is plotted for at each t, at the same valuation set, then smoothed over a moving average of 100 t. Trading volumes, even for zero leverage, remain above 1 unit per trade on average for the duration of the simulations. It is substantially more for traders with more dispersed valuations. Set 3 has the higher trading values across all leverage ratios, as expected since valuation dispersion allows for larger trading volumes. The graphs in Table 9 are scaled according to the leverage ratios. Taking careful note of the vertical axis, we naturally see that average trade sizes are smaller in the zero leveraged market. But the addition of leverage does not simply mean a scaling up of order sizes in k ¼ 1. While the smallest size trades do remain on average, multiplies of 2 and 4 for k ¼ 2 and k ¼ 4, respectively, trading volumes are quite often scaled up more than the leverage ratio would suggest. The distribution of the log returns of transaction prices for all runs in each leverage category is presented in Table 10. We find that the most common return is zero, which is where a transaction is done at the same price as the previous tick price. This is also common in tick data of financial time series. The next most common return is the bid-ask bounce, with a fairly constant size. These distributions have fat tails that can be fitted with a power law exponent that is either 1.8., 1.9, or 1.7 for the zero, two, and four leverage ratio cases, respectively. Volatility clustering and fat tails in our returns can be explained as characteristics of an illiquid market. The distance between trader price valuations, especially those around the equilibrium of the series, determines the size of the bid-ask spread and the jump from misprices where there is no limit order book. Other liquidity constraints, such as wealth constraints, frequency of margin calls, order-flow, trade capacity, the number of traders, and transaction costs all produce price and quantity feedbacks, especially in a market with leveraged risk neutral speculators. In this risk neutral ZI speculative futures market with DA trading, lower leveraged markets appear to be more efficient and more liquid in terms of a smaller spread, and better resiliency — faster convergence to the Walrasian Eastern Economic Journal 2008 34

Table 8 Average spread between mid-point to Walrasian equilibrium for the same valuation set Averaged across 10 runs and smoothed by moving average over 100 t


543


544

Averaged across the 10 runs and smoothed by moving average over 100 t



Table 9 Average trade size for same trader valuation set

Table 10 Log returns of all transaction prices, 100 runs k=2 log returns

k=4 log returns

Log returns of 199,900 tick transaction prices Mean 0 Min 0.18 Max 0.18 S.D. 0.015 Skew 0.15 Kurt 16.51 Power law tail, Exponent 1.8 with est R2 0.78

Log returns of 199,900 tick transaction prices Mean 0 Min 0.23 Max 0.20 S.D. 0.016 Skew 0.05 Kurt 19.02 Power law tail, Exponent 1.9 with est R2 0.80

Log returns of 199,900 tick transaction prices Mean 0 Min 0.23 Max 0.19 S.D. 0.017 Skew 0.04 Kurt 17.53 Power law tail, Exponent 1.74 R2 0.77


545


k=1 log returns


546

equilibrium — when valuations are consolidated. However, in some cases, low leverage combined with RTS and risk neutrality can produce multiple equilibria and evoke long periods of instability in prices. This model may be useful in analyzing the impact of relaxing margin requirements or adjusting settlement frequencies to help stabilize prices in times of tight liquidity. The frequency of settlement could also be a policy lever by exchanges or clearing houses. Many banks during the 2007 sub-prime crisis kept losses off their books because they thought that market prices were not correct hence reducing their speed of settlement. This may have stabilized markets or slowed the efficiency process. This ZI model has offered preliminary results that in circumstances of multiple equilibria, higher leverage increases efficiency and convergence of prices by removing multiple equilibria. But if higher leverage brings about stabilization through a massive redistribution of wealth, this can cause other political problems and may not be the best result for ensuring market competition.

CONCLUSION In this paper, we model a futures market with impatient and non-adaptive speculators who trade on margin. The futures exchange determines the institutional structure in which speculators interface with each other. The exchange imposes a DA mechanism with standard open outcry rules: only the highest bid and lowest ask can persist as standing limit orders at any one time. The exchange also charges a transaction tax on all one-way trades, imposes margin requirements, and settles margin accounts in real time. This paper extends the GS and GSS ZI models, using speculative traders in a futures market with re-trading to elucidate the characterization of ZI modeling and supports the notion that market microstructure and scarcity are important structures in the production of not only price convergence, and allocative efficiency, but also volatility clustering, fat tails in returns, and liquidity price shocks. The price volatility that exists in these markets is due to the distribution of reservation valuations, the rules for trading, wealth constraints, and the liquidity of the market. This is distinctly different from the price volatility that comes from new information or adaptive expectations, which are omitted from this study. For example, when speculators are on their budget constraints, they will liquidate some of their position when prices move against them in order to stay within their margin requirements. This creates backward-bending demand functions and multiple equilibria that can lead to spikes or large shifts in the DA transaction price series. We find that multiple equilibria are more prevalent in highly leveraged markets, even though stability or convergence is increased to one of these equilibria since opposing valuations get knocked out of the market very quickly. This model may provide an alternative way of looking at how the impact of a change in the margin requirement affects prices, as opposed to the analytical studies by Chowdhry and Nanda [1998] and Brunnermeier and Pedersen [2007]. It is thought that this model provides a fertile ground for studies in the area of how microstructure and leverage impact liquidity and trading prices. Extensions of this work would include the study of asymmetric distributions of speculator valuations to see if this makes convergence to the Walrasian equilibrium more difficult to obtain. This is effectively the Cliff and Bruten [1997] critique. From our finding that the Walrasian equilibrium dominates the DA price series over the Eastern Economic Journal 2008 34


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weighted average expectation, it is our hypothesis that asymmetries will not make any difference to the convergence conclusion, and this follows on from preliminary studies by Ussher [2008]. Other extensions include the study of other parameters of the model, such as the frequency of settlement, the transaction tax, an adjustable margin requirement, the size of the trader population, and the addition of other ZI traders such as hedgers and scalpers. While most research in agent-based finance focus on the modeling of information and its use by utility optimizing but boundedly rational agents, ZI models offers a different orientation that attempts to disentangle market microstructure from adaptive behavior. Of particular value is that these tools may give insights into how wealth and liquidity constraints may impact price volatility for given market rules. The contribution of intelligent or adaptive behavior to price formation should be distinguished from the institutional contribution. Agent-based modeling is particularly good at separating and highlighting the mechanical interface or financial architecture that is inherent in real financial markets. These institutions are not amorphous, and should be understood in their impact on price formation with both ZI and intelligent agents. Formulating more strategic trading behavior with adaptation is an important extension of this research. The ZI model is a useful method to benchmark such extensions.

Acknowledgements I thank Duncan K. Foley, Robert Axtell, Salih Neftci, Alan G. Isaac, and two anonymous referees for comments and suggestions. The paper benefited from discussions with participants in the New York City Computational Economics and Complexity Workshop, the 2007 Eastern Economic Association annual meeting, and seminars at the Institute for Scientific Interchange Foundation in Turin. All mistakes are my own. This work was supported in part by E.C. contract no. 018474-2 DAPHNet, and The City University of New York PSC-CUNY Research Award Program.

Notes 1. GS ZI agents trade randomly but are constrained by their own fixed reservation price at which buyers could not buy above and sellers could not sell below. 2. The term DA is used here for a simulated auction where as a CDA exists in real markets where sequential trades are time stamped in continuous time. A simulation with a sequence of transactions, but no relevant time stamp, is considered to be a sequential double auction in discrete time. 3. See footnote 5 in Gjerstad and Shachat [2007] and a review of the ZI literature can be found in Duffy [2006], along with extensions to models with near-ZI agents. 4. A limit order is a request to buy or sell a certain quantity at a certain price. It may be time stamped and entered into a limit order book, or order queue, good until it is cancelled or executed by a market order. A market order is an action to sell or buy a specified amount at the best buy or sell limit order, respectively. A crossing limit order or executable limit order is effectively a market order, that is, when a limit order comes in to buy at a price of 120 and crosses a standing limit order to sell at 100, this is the same as a market order to buy. The sequence of these orders is called order-flow. For more definitions of all the financial terms used in this article see Harris [2003]. 5. This is similar to the ZI approach by LiCalzi and Pellizzari [2007]. 6. Retrading is when one sells then buys, or vice versa, the same commodity. In the GS and GSS models ‘‘retrading makes allocative efficiency a moot issue because with sufficient time retrading ensures 100 percent allocative efficiency’’ [Gode and Sunder 2004, p. 1713]. In the model of speculation presented Eastern Economic Journal 2008 34


548

7.

8.

9.

10. 11. 12. 13.

14.

15. 16. 17. 18.

19.

here retrading does not ensure stability of prices because of the forced liquidation of positions that produces multiple equilibria. In real life electronic exchanges, the limit orders (bids and asks) in the book stay there until canceled by their owner, or executed in the form of a transaction. They are also not limited to single units, and prices are discrete, with a minimum tick, price change, for example, 1 cent. A non-storable commodity has a futures valuation equivalent to the expected final spot price of that commodity. Commodities that are storable would be complicated by cash and carry pricing formula from arbitrageurs. The maturity date of the futures contract is considered to be in the future beyond our analysis and does not figure into any calculations. A model with interest rates and spot markets was considered in previous work [Ussher 2005a]. Maintenance margin is the minimum margin requirement and typically some fraction of the initial margin that the trader must maintain to avoid a margin call. If the maintenance margin is equivalent to the initial margin then the margin call is the variation margin. In real markets this is typically held as Treasury bonds earning the risk free rate, hence it is often said that there is no opportunity cost in holding futures. Deposits of new cash to satisfy margin calls are ruled out as they require additional cash placed into the account from outside the model. These demand functions are a slightly simplified version from Ussher [2005b] as the interest on money holdings here is zero. In a continuous DA open-outcryfutures market, as described by Silber [1984], all bids and offers must be announced publicly to the pit through the outcry of buy or sell orders. Strict priority is kept, where the highest bid price and the lowest offer take precedence, known as the inside spread. Lower bidders must keep silent when a higher bid is called out, and higher offers are silenced when a lower offer is announced. All bids and offers, even non-competitive ones, are called limit orders. For every limit order there is a quantity attached. The current market bid and ask remain until the quantity they want is filled. In the United States the New York Mercantile Exchange, the Chicago Mercantile Exchange, the Chicago Board of Trade, the Chicago Board Options Exchange, the Minneapolis Grain Exchange (MGEX), and in the United Kingdom the London Metal Exchange still makes use of open outcry. The market size is similar to that in Gode and Sunder [1993] and Gode et al. [2004]. A GS or GSS efficiency measure of consumer surplus is complicated by the backward bending demand functions. If there was a limit order book, as in GS and GSS with depth greater than one, then prices would be constrained to remain around the midpoint at the next best limit order from past entries. For k ¼ 1, 24 percent of the 100 runs had at least one multiple equilibria, and there was a jump in the Walrasian price |p*p *t1|>1 over all t, 0.02 percent of the time. For k ¼ 2, 26 percent of the runs had t at least one multiple equilibria, but occurred with twice as much frequency over all t than in the no leverage case. For k ¼ 4, 36 percent of the runs had at least one multiple equilibria, and there was a jump in the Walrasian price greater than 1, over all t, of 0.1 percent. This jump in the Walrasian price underestimates the existence of multiple equilibria as it only reflects the events when Mathematica’s fixed point algorithm found a significantly different equilibrium result in a row. Market-makers are speculators who buy at the bid and sell at the ask.

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