The Effect of Microstructure in (Artificial) Financial Markets Nicol´as Garrido∗ January 29, 2008

Abstract In this paper we explore two mechanism of price generation; a continuous auction market with limit orders and the standard mechanism of price formation through excess demand. The investors are the same in both markets, they use the same forecasting and decision rule to make their investment decisions. We made the comparison according to the properties generated by the simulated data on stable distributions.

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Introduction

Market microstructure study the trading mechanism that produces the prices for financial securities. In this paper we compare two mechanisms for computing the price of a security: a continuous auction market with limit orders and the standard mechanism of price formation through excess demand. In order of making the comparison useful we keep the behavioral specification of investors constant across both institutions. This allows us to explore what are the effects of the mechanism on the market price, leaving constant the behavior of the agents. The financial market that we explore in this paper, are impossible to be solved analytically. As mentioned by LeBaron (2006) we have interacting group ∗ This paper was developed thanks to the financial support given by the University of Trento. [email protected].

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of learning, bounded-rational agents. Thus, computer simulation will be the tool used for understanding the dynamic of these markets. The investors in our model are heterogeneous in two senses: first, they have different forecast windows, and second they use different learning rules. All the agents use the same choice functions for deciding their portfolio composition and all the investors began the simulations with the same wealth. The time series produced by real financial markets contain many curious empirical puzzles. Among these are the overall level of volatility and long swings around fundamentals, the large amount of trading volume and the persistence of volatility showing repeatedly switches between periods of relative calm and periods of relative turmoil. Closely related to volume and volatility persistence is the issue of fat tails, or excess kurtosis. At frequencies of less than one month the unconditional returns are not normally distributed. They usually display a distribution with too many observations near the mean, too few in the mid range and again, too many in the extreme left or right of the distribution. There is a continuous debate about the exact shape of the tails of return distributions. Most of the literature of econophysics remarks the presence of power laws in the distribution of returns, which reflect of the multiple interaction environment and it characterize the frequency of the events. Indeed, the efficient market hypothesis, claims that prices reflects all the news coming into the markets. Therefore, statistical characteristics of financial returns are mere reflection of same characteristics of the news arrival process. Therefore, news arrival is also

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clustered in time. Mandelbort(1963) and Fama(1965) brought the fat tails characteristics on the returns of the financial time series. Since stable distributions can accommodate the fat tails and asymmetry, they often give a good fit to empirical data (see Borak et al. (2005)). We will use stable distributions to characterize the simulated time series returns produced by our models. An interesting result observed out of the simulations is that as the risk aversion of the investors increases, the data generated by the simulations get close to the results observed in four financial markets in LatinAmerica. Moreover, the institutional mechanisms produces different patterns of data. The remainder of this paper follows with the exposition of the model of agents and institutions. Next, we present the parameters of the simulations and their meaning. Next we explain how do we estimate the parameters of the stable distribution, and finally we presents the results with the conclusions.

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Model

In our model there is a central bank, an stock market, and a group of N managers operating in the market. The central bank hold the deposits of the managers, and give loans to them. The stock market collects the demands for selling and buying of the managers. We assume that there is a single risky asset with variable return R1 (t) and a single riskless asset with constant return R0 . The risky asset is assumed to be the market portfolio or the unit beta portfolio and the safe asset can be thought

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of as government securities free of risk. Every manager chooses the leverage position x on the risky asset, according to his risk preferences. Thus x = 1 means fully invested in risk asset, x > 1 means leveraged investment and x < 1 means that the mangers has the proportion x − 1 invested in the safe asset. The managers decide their leverage position to maximize their risk adjusted net return. Following Friedman and Abraham (2007), the gross return obtained by a manager with leverage x is R(x) = xR1 + (1 − x)R0 . The managers cost is given by the risk free rate R0 plus a risk cost C(x). The risk free rate can be interpreted as the opportunity cost, and the risk cost function reflect the standard concavity of investors’ utility function. Assuming negligible trading cost we specify the risk cost function as C(x) = 2c x2 . We assume that all the mangers have the same risk cost, or what is the same, c is constant and equal for all the agents in the market. Thus the net return enjoyed by a managers choosing a leverage x is

c R(x) = x(R1 − R0 ) − x2 2

(1)

We assume that the manager payoff function φ(x) is represented by the net return obtained with the leverage x. The return of the risky asset R1 depends on the decision made by all the managers and the market microstructure as it will be explained below. Therefore, the managers revise their leverage periodically to have the portfolio with the highest return adjusted by the risk.

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The managers use the expected value of the return of the risky asset to compute their optimal portfolio. Therefore the payoff function is computed as,

c R(x, t) = x(Et [R1 (t + h)|Ωt ] − R0 ) − x2 2

(2)

where Et [1 (t + h)|Ωt ] is the expected return computed at t, Ωt is the information available for all the managers at the beginning of the period t and R1 (t + h) is the return of the risky asset after h periods. The market is populated by heterogeneous traders. Agents are heterogeneous because they trade with different time horizon. Basically there are three different set of agents. First, there is a set of traders operating every day, who are taking advantages of pieces of information unknown for us as observers. Moreover, the information used by one of these agents is not available for all the agents in the market. Thus, their behavior for the observer might be interpreted as a noise trader, selling and buying without clear explanations. Second, there is a set of traders operating every day according to the fundamental value of the stock market. Finally, the third group are managers who made portfolio composition to hold it for during a longer periods of time. The second group represent institutional investors, making investment decisions on behalf of a set of customers. Their differences in their investment horizons make them to forecast different price behavior. However, all the agents make their trading decision to maximize

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their payoff function R(x, t). Thus, the rule for deciding the optimal composition of the portfolio for a manager is given by the first order condition of (2)

x∗ =

(Et [R1 (t + h)|Ωt ] − R0 ) c

(3)

The difference between this optimal leverage and the current leverage xi for every manager ∆xi = x∗i − xi determines the type of operation that the managers will made during the current period. Every manager keeps record of their current asset and liability positions. The assets of a manager is composed by their deposits, riskless investment, and the risky asset bought or sold during previous periods.

2.1

Price forecasting

In order to compute the expected return in equation (3), agents make forecast of the risky asset price. Price forecasting is important for deciding the type of operations that the agents will made. In our model agents use three type of forecasting rule. There exist agents making decisions on the bases of a rule that is not observable for the external observer, there are agents following the fundamental price of the risky asset and finally agents who follows the market price trend. The first group, or noise traders, forecast the evolution of the market price on the bases of information not available for the observer. Therefore we are not able to explain its behavior. However in order to avoid any bias in the

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market we assume that there exist in mean, the same number of these traders forecasting an increase on the price than a decrease on the price. Thus an agent i of the first group, forecast the one period price according to,

pet = pt−1 + ²

(4)

where ² is a number draw from a normal distribution with mean 0 and standard deviation σ. The second group, the fundamentalists, predict that the price of the risky asset will return to its fundamental value. Thus, a fundamentalist forecast that the price will be

pet = pt−1 + (pft−1 − pt−1 )

(5)

where pft−1 is the fundamental price of the risky asset at period t − 1. Finally the agents in the third group use a linear autoregressive process to forecast the price. Agents in this group, use all the information available up to period t − 1 to forecast the price for the period t. First, these technical traders, fit an autoregressive AR(q) process. They use Akaike information criteria to identify the optimal value of q. With this information, the agent simulate the estimated process to forecast the price for the next period. Thus,

pet = AR(q) 7

(6)

2.2

Agents Accounting

Every traders keep tracks of its operations, and the balance of his counting. Basically the active is equal to its loans plus equities. This basically means that every agent keeps the following identity in every period of time,

mt + at ∗ pt = lt + et

(7)

where mt is the liquidity available during the period t, at is the amount of risky asset in the portfolio, pt is the current price of the asset, lt is the amount of loan asked and finally et is the equity of the agent. On the basis of this equation we can define precisely what the leverage of every agent is,

xt =

at pt et

(8)

The equity of an agent change in every period as consequence of the looses and profits obtained from his position. Moreover, agents have mt as deposits in the central bank. Therefore the bank pays the free risk return R0 to the amount mt hold during the hold period. We assume that when an agent ask for a loan, the unique condition that the bank check before of giving it to him is that xt < 2. In another words, if the risky position of an agent is twice his equities, the bank will not give to him more loans.

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2.3

Microstructures: price determination

We compare two mechanism of price formation: on the one hand, we have a continuous auction market with limit orders and on the other hand we have the standard mechanism of price formation through excess demand. In the continuous auction market at the beginning of every period the agents might book a bid limit order or an ask limit order. In both cases, the limit order has information about size, price and the identification of the agent who established the ask or bid. The market maker arrange the bid in descending order and the ask in ascending order. According to that, the market maker match first the orders with the biggest spread, splitting the price half the spread between the two orders. For instance, if an ask order for the risky asset is for the price pa,1 and the bid price is pb,1 with pb,1 > pa,1 , so the price of the transaction will be p = pa,1 + 0.5 ∗ (pb,1 − pa,1 ). If a bid is too low it might never found an ask to match with. This is also the case for asks having a very high price. Thus, orders which stay during more than τ periods in the market are canceled. Moreover our double auction mechanism has an additional constraints. If a dealer has booked a bid order, he can not put in the book another bid until either of two events occurs: first, the previous bid order match an ask or second, the order is canceled because after τ periods it did not match an ask. This is also valid for ask orders. If a dealer has booked an ask order, he can not put in the book another ask until the previous order match a bid, or until the order is canceled because after τ periods it does not match any bid.

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The market price is determined at the end of every period. Given this microstructure, two possible situations might occur. First, given the bids and the asks of the period, its might happens that there is no matching, therefore the market volume is zero. In this case the market price for period t is given by the market price of the previous period. The second case occurs when at least one bid and ask order match. In this case, the market price during the current period, will be given by the last match between a bid and an ask order. Notice that this means, that the market price is very close to the equilibrum price of every period. The second mechanism uses a slow price adjustment process where the market is never really in equilibrum. The price is determined by the excess of demand as in Day and Huan (1990). In this case the market announces a price and agents submit demands to buy and sell at this price. The sum of the asks for every agent composes the aggregate supply side in the period t, St and the sum of the bids for all the agents composes the aggregate demand side of the market during the period t, Dt . If there is an excess of demand the price is increased, and if there is an excess supply the price is decreased. Thus in every period, the price is computed according to,

pt = pt−1 + ρ (Dt − St )

(9)

where ρ is the price speed of adjustment to the excesses of demands in every period t and the aggregates are defined as Dt =

PN i=1

di,t and St =

PN

i=1 si,t .

The lower case represent the individual demand or supply for agent i, and N is

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the number of dealers in the market. Dealers close their operations at the price obtained at the end of every period, pt . Due to the excess of demand or supply, there are bid or ask orders which are not fully satisfied. The short side of the market, decide the proportion or orders that is satisfied. For an agent i who submit demands to sell of size si , its volume of operation at time t will be given by, ¹ vs,t = si

si min{St , Dt }

º (10)

whereas for an agent who submit demand to buy of size di ¹ vd,t = di

di min{St , Dt }

º (11)

where the symbol bxc takes the integer side of the real variable x. For instance b4.2c = 4. If during a period, there is no demand or supply the market does not operate and the price for the period is equal to the price during the previous period. Finally, for both mechanisms we simulate the fundamental value of the risky asset using the following process,

pf,t = θ0 + θ1 pf,t−1 + ²f

(12)

where ²f is an independent and identically distribuited normal process N (0, θ2 ).

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3

Simulations

The model has two group of parameters: there are mechanisms parameters related to the properties of the mechanism making the price of the risky asset, and behavioral parameters which are related to the decisions make by the agents, like the risk adversion parameter in . Some of the mechanism parameters are: the speed parameter ρ in equation (9) in the standard mechanism of adjustment through excess demand or the criteria to compute the price when two orders are matched in the continuous auction market with limit orders. We are interested in comparing the effects of different institutions on the price of a risky asset. We are not interested in studying the sensitivity of a single mechanism. Therefore, our simulations will explore the effect of different behavioral parameters. The agents decide according to their risks aversion preference. The parameter c captures the risk aversion of the agents. In (3), when the risk aversion increases (decreases), the investors reduce (increase) their exposition. During our simulation we will fix the mechanism parameteres and we will explore the effect of the risk aversion in the range [0.3; 1]. In our simulations there are N = 100 traders, buying and selling the risky asset during 500 periods of time. Each trader start his operations with m = 1000 monetary units (or riskless asset) and a = 100 units of risky asset. The speed of adjustment for the excess demand mechanism in equation (9) is ρ = 0.0005. In the auction market with limit orders, the orders wait for another 12

order during τ = 3 periods. The parameters for the fundamental price process in equation (12) are θ0 = 5, θ1 = 0.4 and finally θ2 = 0.05. The heavy tail or leptokurtic distributions of returns of financial asset has been repeatedly observed in various markets (Carr et al. 2002). In response to these empirical observations Mandelbort (1963) and Fama (1965) proposed stable distributions as an alternative model to capture the empirical properties of data. According to this, we will use stable distributions for capturing the properties of the two microstructure models as we change the risk aversion.

3.1

Stable Laws

Although there are many family of distributions with heavy tails, stable distributions have some properties that make them interesting for modeling financial data. An important property of stable law is that every sum of independent stable variates with a given characteristic exponent α has a stable distribution with the same characteristic exponent α. Stable distributions are supported by the generalized Central Limit Theorem, which states that stable laws are the only possible limit distributions for properly normalized and centered sums of independent, identically distributed random variables. The alpha stable distribution -also called stable Paretian, or L´evy stable require four parameters for complete description: an index of stability α ∈ (0, 2] also called the tail index, tail exponent or characteristic exponent, a skewness parameter β ∈ [−1, 1], a scale parameter σ > 0 and a location parameter µ ∈ R. The tail exponent determines the elasticity at which the tail of the distri13

bution taper off. When α = 2, it represents a Gaussian distribution, whereas for α < 2 the tail become thinner slowly and the distribution exhibit power law behavior. When α < 2 the variance of the distribution is infinite. For financial returns, this means that eventually it is possible to obtain big loses or gains in the market. When tail index is closer to zero, there are more probabilities of observing extreme events. The side where it is more likely to observe the extreme events is given by the skewness parameter β. If the β is negative (positive) it is more likely to observe big losses (gains). If β = 0 then the distribution is symmetric about the mean µ. When α ≥ 1 there exist the first moment of the distribution and it is equal to the location parameter µ. In general, the n moment of a stable distribution exist if and only if α ≥ n. Finally the scale parameter σ determines the width of the distribution. The standard stable distribution has µ = 0 and σ = 1. Due to the lack of a closed form formulas for density of stable distributions, there are many difficulties for estimating the parameters of the distribution. In this paper, we will use the procedure suggested by Koutrouvelis (1980). This procedure is fast enough for computing the parameters obtained from the Monte Carlo experiments.

3.2

Results

We explore the behavior of the mechanisms as the risk aversion of the agents is changed. For each value of the parameter c = 0.03; 0.05; 0.07; 0.09 we run 500 simulations, during 1200 periods of time. With the time serie of price generated 14

by every simulation we use the procedure suggested by Koutrouvelis (1980) for estimating the parameters of the stable law. In Figure (3.2) and (3.2) we represents the stable law parameters estimated for each one of the four variations of c generated by the continuous auction mechanism and the excess demand mechanism. Throughout both mechanism, we can see that as the investors are more risk adverse, the are less chances of registering extreme events. Notice that as the value of α is greater the stable law becomes closer to a normal distribution, and the flat tails become thinners. However both mechanism produce different patterns. For instance in figure (3.2) where we plot the data produced by the continous auction mechanism, the graph in the first raw and column shows the parameters computed for the risk aversion parameter c = 0.03. In the y-axes are the values of alpha, in the x-axes the values of σ and finally in the z-axes are the values of β. Every point represent the values of the parameters estimated at the end of a simulation. Notice that the values of α and σ do not vary. For low values of the risk aversion parameter, there is only variation on the skewness of the returns, the β, obtained by the simulations. There are some clear patterns produced by the continuous auction mechanism: when the risk aversion increases, the values of α increases as well. This means that as investors become more risk adverse, there are less chances of registering extreme events (gains or losses). Moreover, as the risk aversion increases the skewness of the distribution is toward the left, i.e. β becomes negative. This means that extreme losses are more probable than extreme gains.

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0.03

0.05

Beta

1

1

Beta

0

−1 1

0

0.005

2

1 0 0

Sigma

x 10

0.09

0.07 1

1

0.5

Beta

2

Alpha

1

0 0 Sigma

Alpha

−10

2

Alpha

−1 0.01 0.5

0.5

Beta

0 −0.5

0 −0.5

−1 0.01

Sigma

2

1

0 0

−1 0.02

Alpha

0.005

0.01 0

0

1

Sigma

Figure 1: Continous Auction Mechanism

0.03

0.05

2

2

1.5

1.5

0.005

0

Beta

Beta

0.5

Alpha

1 1 0 −1 0.01

1

0

1 0 −1 0.02

0.5 Alpha 0.015

Sigma

1 0 −1 0.02 0.015

0

2

2

1.5

1.5

1

1

0.5 Alpha 0.01 0.005 Sigma

0

0.09

Beta

Beta

0.07

0.01 0.005 Sigma

0

0

1 0 −1 0.03

0.5 Alpha 0.02

0.01 Sigma

Figure 2: Excess Demand Mechanism

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0

0

For the case of the excess demand mechanism (3.2) as the risk aversion increase, the probability of observing extreme events increase as well. However, the skewness is toward the right. This means that with the excess demand mechanism it is more probable to observe extreme gains.

3.3

Real Data

In order to compare the results obtained by markets with real data, we estimate the parameters of the stable law distribution for the index stock markets of four Latinamerican countries: Argentina (Merval), Brazil (Ibovespa), Mexico (Mxy) and Chile (Ipsa). We take 2793 diary prices for Argentina, for Brasil 3645 prices, for Chile 1887 prices and finally for Mexico 3393 prices. In figure (3.3) we observe that the estimated value of α for the returns of the four markets is greater than one. Moreover the the values of σ estimated are below the value of 0.015. Notice that the simulations where the investor are risk more risk adverse approximate the patterns produced by the real data.

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Conclusions

It seems obvious that microstructure factors ought to affect asset prices. The literature (O’Hara (1995)) identify two main sources of friction for changing the market price. On the one hand there is the valuation effect of real friction, such as the cost of processing orders or searching for counter parties. Its effects is clearly to reduce the asset value. On the other hand there is the valuation effect of informational friction which effect is less clear. Informational friction arises if one investor is better informed

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0.2 México

Beta

0

Brasil

−0.2 Argentina −0.4 Chile

−0.6 0.015 1.85 1.8

0.01

1.75 1.7

Sigma

0.005

1.65 1.6

Alpha

Figure 3: Data from Argentina, Brazil, Mexico and Chile

than another. The informed investor with good news will bid up asset prices to the disadvantage of the uninformed investor who sells the shares. Similarly, when disposing of shares, the informational investor receives a better price than the uninformed investor. The presence of informed investors disadvantages uninformed investors and redistributes income from the uninformed to the informed. Informational frictions introduce distributional uncertainty, which may make some investors reluctant to buy an asset, thereby lowering its market price. However the final effect of the information friction us uncertain. The microstructures presented in this paper produce mainly information friction, producing uncertain effect on the price time series. However, when the data generated by each market microstructure is compared, using the parameters estimation of stable law distributions, the continuous auction mechanism produces a negative skewness whereas the mechanism with excess demand pro-

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duces positive skewness. Looking at the parameters estimated using the stable distributions, there is a negative relationship between the risk aversion of the investor and the heaviness of the tail of the return distribution. As the investor are more risk adverse, there are less extreme events or the tail becomes thinner. This property is common to both mechanism of price formation. Finally, when compared the results of our model with the parameters estimated from the real data there are two interesting conclusions: first in order to approximate the real data investor has to be risk adverse, and second the behavior of the data from the chilenean stock market seems to be easier to approximate with continuous auction market, whereas the data from Argentina, Brasil and Mexico are closer to the excess demand mechanism.

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[4] Fama, E. F. (1965). The behavior of stock market prices, Journal of Business 38: 34-105. [5] Friedman, Daniel and Ralph Abraham (2007), Bubbles and Crashes: Escape Dynamics in Financial Markets, Mimeo. [6] LeBaron, Blake (2006), Agent-based Computational Finance, in The Handbook of Computational Economics, vol II edited by K.L.Judd and L. Tesfatsion. [7] Levy, Moshe, Haim Levy and Sorin Solomon (1994) A microscopic model of the stock market. Cycles, booms and crashes. Economics Letters 45 (1994) 103-111. [8] Koutrouvelis, Ioannis A. (1980) Regression-Type Estimation of Parameters of Stable Laws, Journal of American Statistical Association, Vol. 75, No. 372, pp 918-928. [9] Fama, Eugene and Richard Roll (1971) Parameter Estimates for Symmetric Stable Distributions, Journal of American Statistical Association, Vol. 66, No. 334-38. [10] Carr, P., Geman, H., Madan, D. B., and Yor, M. (2002). The Fine structure of asset returns: an empirical investigation, Journal of Business 75: 305-332. [11] Mandelbrot, B. B. (1963). The variation of certain speculative prices, Journal of Business 36: 394-419.

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[12] O’Hara, Maureen (1995) Market Microstructure Theory, Blackwell, Cambridge, MA.

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