-This work proposes a double auction artificial stock market based on the Santa Fe market structure. Our market tries to shed light on some facts that usually arise in real stock markets, specially the creation of technical figures in price series. The origin of these figures is believed to be caused by the self-fulfilling prophecy effect, which will be investigated with the proposed market.
The main purpose of the agent-based simulation of an
artificial stock market (ASM) is to reproduce, in a controlled
environment, some properties of real stock markets. In that
way, ASMs are a suitable tool to analyze and understand
market dynamics. See [
Many market models has been developed in order to
reproduced those properties, like Genoa Stock Market [
These behaviors, which are usually know as technical
patterns, cannot be explained from the fundamental analysis
perspective. Despite this fact, they are used by many investors
and also by chief dealers, as is reported in [
Technical analysts, also known as chartist investors, base their expectations in historical price patterns that are expected to appear again at some future point. They try to predict future extreme prices in order to buy assets when the value is under those limits, and sell them when the price is close to a bounce zone. Moreover, technical traders usually follow price trends. A common example is the use of the moving averages crosses to set their trading strategies. This method provides buy and sell signals when a short run moving average crosses a long run moving average price.
Chartism is one of the two main investment approaches that
can be usually found in stock markets [
This article proposes an ASM based on the SFM structure that exhibits technical figures and that reproduce the selffulfilling prophecy effect. The proposed ASM modifies some important features of the SFM with the aim of being more realistic and reproducing stylized facts. The most relevant changes are introduced in the next section.
II. DESCRIPTION OF THE DOUBLE AUCTION ARTIFICIAL
STOCK MARKET PROPOSED
The SFM [
The SFM shows, amongst other features, the theoreticallypredicted rational expectations behavior, with low overall trading volume, uncorrelated price series. However, it is difficult to find realistic market behavior such as high trading volume, time-varying volatility clustering (periods of swings followed by periods of relative calm), bubbles and crashes and market patterns such as supports, resistances, channels, etc. One of the main reasons is that the auction mechanism is not realistic as there is an auctioneer that takes into the account the demand of shares and the fixed supply of shares (25 shares) to set the price that clears the market.
In our ASM, a continuous double auction system is
implemented. In continuous double auction markets, agents place
buy or sell orders at any time in an asynchronous manner.
In this kind of auctions, a public order book lists the bids
in descending order and the sell offers in ascending order.
When a new order matches with the best waiting order of
the opposite type then a trade is made, otherwise the new
order remains waiting. Once the transaction is carried out, both
orders are removed from the order book. This kind of auction
is implemented in stock markets such as NYSE or AMEX.
This kind of auction has been already implemented as an ASM
[
In our market, agents are rationally bounded, , which means that their rationality is limited by the information they have. They make price bids (offers to buy) and/or price asks (offers to sell) subject to a budget constraint and using the information they have about the state of the market. Our market allows agents to place both market-price and fixed-price orders. The ASM does not allow fixed-price orders. However, this kind of orders are used in real-life stock markets. In an artificial market that allows this orders it is possible to observe technical patterns. If a group of traders set fixed-price orders close to a certain price value, then supports or resistances lines may appear in the resulting price time series. Also, cascade effects could emerge behind certain price limits obtained using technical analysis.
In our ASM, agents tune the fixed-price orders using a system of forecasting rules similar to that used in the SFM, but based on support and resistance lines. In doing so, they will fix the price taking into account the support and resistance lines and the length of the trading horizon (it denotes if is a short-, mid- or long-term trade). The mechanism will be described below.
Our market follows the basic structure of the SFM model, but implementing a double auction market. Another noteworthy difference is the number of agents. Instead of the 25 agents used in the SFM, in our market there are 512 agents that will make possible to have enough trade operations in the market and a great variety of behaviors. It is important to remark that our aim is not related with the rational expectations equilibrium theory, but with the study of real-life phenomena such as resistance and support lines. These changes affect not only the auction mechanism but also the equations that determine the wealth and the classifier rules. More details will be given in the next subsections.
As in the SFM, our market has two assets: a risk free bond in infinite supply with constant interest rate (r = 0.1) and a risky asset. The price of the risky asset in t, pt is endogenously determined by the market. The risky asset considered does not pay dividends, in contrast to the SFM’s risky asset.
The trading strategy consists of buying (or selling, if the agent goes short) risky assets at the current price of the market and at the same time placing a fixed price stop-profit order. The price of the stop-profit order is determined by taking into account the agent’s resistance line (or the agent’s support line, if the agent goes short). Both orders are sent at time ta. In addition, the agent also estimates the price of a stop-loss order using the support line (or the resistance line, if the agent goes short). This order will be placed as a market-price order only if the risky asset reaches the stop-loss price.
As any market-price operation has its corresponding stopprofit order, the total number of stocks M is always available in the market, providing the necessary liquidity to supply the possible demands of other investors. The system restrictions ensure the liquidity of the market and consequently market price orders are executed at the time when they are placed.
In ta the stop-profit order is booked in its corresponding priority queue of awaiting orders, depending on if the trading agent goes long or short. The agent determines the stop-profit price using a future stock price that is forecasted with the help of a support line (or a resistance line if the investor is going short) that is drawn by the agent using three parameters determined by the activated forecast rule j (more details about the forecast rules are given in the next subsection). The parameters are: the number of local maximum (resp. minimum) points used to draw the resistance (resp. support) lines ai,j , the length of the sliding window used to look for the local maxima and minima bi,j , and the length of the trading horizon ci,j . A support (resp. resistance) line is drew joining at least two minimum (resp. maximum) price values. These parameters allow agents to operate to different time horizons.
If at time ci,j the price of the risky stock has not been
matched the agent close its position and cancel the stop-profit
order that was previously submitted. This is an interesting
feature because, as is reported in [
The behavior of the trader is determined by the classifier
system they use to set their trade orders. The classifier system
consists of a set of rules that are triggered when some market
conditions are present. The classifier system implemented that
follow our agents is based on the one used in the SFM, which
is described in detail in [
The agents have to set of rules one for ”going long” and other for ”going short”. The rules of both sets have the same structure, which consists of two parts. The left part of the rule is a string of 30 conditions, where each string position represents a state of the market. The possible values of each position are 1, 0 or ♯ . The 1 means that the state have to be present, the zero that the state have not to be present while the ♯ is a wildcard that matches either. The right part of the rule consists of three parameters (ai,j , bi,j and ci,j ) that are used to draw the resistance and the support for the trade operation. If the agent is going long, it will use the resistance to set the stop-profit order and the support for the stop-loss order, while if the agent is going short it will do it the other way round. The idea is that each rule matches a state of the system, where the agent invest in the risky stock asset at market price. The position will be held until the asset reaches either the stopprofit or the stop loss prices. If the rule is matched at ta, the agent expects to reach the stop profit price at ta + ci,j .
The parameters are initially set to random values distributed uniformly. As in the SFM the rules are not static. Each agent is allowed to learn by changing its set of rule. The learning process is implemented by means of a genetic algorithm where the poorly performing rules are substituted by new ones. Rules are selected for rejection and persistence based on a accuracy measure that takes into account both the errors in the price and in the forecast horizon.
When a forecast rule j of agent i is activated in ta, the agent will forecast the future stock returns for time horizon ta +ci,j , instead of ta +1 as in the SFM. Once the operation is done, the agent will hold its position until instant ta + ci,j or until the price of the stock reaches a stop-profit or a stop-loss value at an undetermined time tb > ta, whatever comes first. In the second case, the operation may not be closed at that undetermined time, which will be denoted as tb. It happens when there are not enough buy (resp. sell) orders to sell (resp. buy) all the stocks at the stop price in tb. Figure 1 illustrates the process for the case where the agent goes long.
Once the transaction is completed (i.e. once the ordered stocks have been bought and then sold or viceversa for short positions), the investor’s wealth has to be updated. As said before, the transaction is completed either when stocks reach the stop-profit price at tb or when the holding time period is expired ta + cj . For the sake of simplicity, we will refer to these periods as te. The wealth equation must take into account that the sold of stocks can require more than one period. The number of extra periods will be denoted as f . While agent traders can not execute several operations at the same time, and also can not modify their current trading strategy, wealth value is updated when a trading operation has concluded. Given that, the wealth of agent i in te+f is
Wte+f ,i = Wtrai→sktye+f + Wtfar→eete+f + Wtfer→eete+f . (1) The equations of these three terms are explained next.
The term Wtrai→sktye+f ,i represents the changes from ta to te+f in the wealth invested in the risky asset. Its equation is slightly different depending on the way stocks are sold. If they are sold because they reach the stop-profit price, then te = tb is the period when that price is reached and the wealth is f Wtrai→sktye+f ,i = pte X xtoeu+tl,i, l=0 where xout te+l,i is the number of stocks sold at time te+l by f agent i and satisfying Pl=0 xout
te+l,i = xta,i.
On the other hand, if stocks are sold because the maximum holding time ends, which happens at te = ta+cj , then all the stocks are sold at that time (which means that f = 0). However, it may happen that not all the stocks are sold at the price pte . This happens when the demand of the awaiting order does not cover the whole sell. If this is the case, other awaiting orders, possibly with a different price are required. The wealth Wtrai→sktye+f ,i in this case is estimated as v Wtrai→sktye+f ,i = X xtoeu,tl,ipte,l.
l=1 As all the stocks are sold at the same price, the price pte,l and the stocks number xout
te,l,i both depend on a parameter l = 1, ..., v that represents the number of awaiting sell order matched.
The second term, Wtfar→eete+f ,i, represents the evolution of the capital invested in the free risk asset since ta. It is given by k Wtfar→eete+f ,i = (1 + r)tb+f −ta Wta,i − X pta,lxitna,i,l , (4) l=1 where r is the constant interest rate of the free-risk asset and the pair of values (pta,l, xin ta,i,l) represents the awaiting sell order matched at time t with the market price order. It means that xin
ta,i,l stocks have been bought at price pta,l with Plk=1 xitna,i,l = xta,i.
Finally, the term Wtfer→eet∗e+f ,i represents the evolution of the capital of the stocks sold between te and tef , because during this period this capital is invested in the free risk asset. Thus, the term is only taken into account if f > 0. It is given by f Wtfer→eet∗e+f ,i = X(1 + r)tb+f −tb+l (ptb+l xtobu+tl,i).
l=1 (5)
In our market as in the SFM, investors use a simply constant
relative risk aversion preference for stock demands [
This paper represents the current state of a work-in-progress. An example of a candlestick time series with the trading volume generated by the ASM is shown in Figure 2. The model is being satisfactorily programmed using Nvidia CUDA technology, which allows to drastically reduce the simulation time. This parallel programming technology also allows to scale the number of agents without increasing the simulation time. Detailed information about trading strategies, agent behavior and evolution, the rule system and auction mechanism will be explained in the subsequent extended paper along with the simulation results and validation.
Regarding validation it will consist of analyzing two issues. First the usual econometric properties, i.e. the stylized-facts, that are present in real-life stock market time series such us fat tails and leptokurtic properties in return distributions, excess of kurtosis, no significant autocorrelation and volatility clusters. Second, it will be shown that technical patters, familiar to professional technical analysts, do appear in the time series of the prices as a result of the self-fulfilling prophecy effect.
The authors acknowledge support from the project Agentbased Modelling and Simulation of Complex Social Systems (SiCoSSys), supported by Spanish Council for Science and Innovation, with grant TIN2008-06464-C03-01. We also thank the support from the Programa de Creaci o´n y Consolidaci o´n de Grupos de Investigaci o´n UCM-BSCH, GR58/08