Download Heterogeneous Beliefs under Different Market Architectures

Heterogeneous Beliefs under Different Market
Architectures
Mikhail Anufriev1 and Valentyn Panchenko2
1
2
CeNDEF, University of Amsterdam, Amsterdam [email protected]
CeNDEF, University of Amsterdam, Amsterdam [email protected]
Summary. The paper analyzes the dynamics in a model with heterogeneous agents trading
in simple markets under different trading protocols. Starting with analytically tractable model
of [4], we build a simulation platform with the aim to investigate the impact of the trading
rules on the agents’ ecology and aggregate time series properties. The key behavioral feature
of the model is the presence of a finite set of simple beliefs which agents choose each time
step according to a fitness measure. The price is determined endogenously and our focus is
on the role of the structural assumption about the market architecture. Analyzing dynamics
under such different trading protocols as Walrasian scenario and ’order-book’ mechanism, we
find out that market architecture affects the ecology of traders, i.e. distribution of different
beliefs across population. In addition, behavioral and structural assumptions are intertwined
in defining the aggregate time series properties.
1 Introduction
The paper contributes to the analysis of the interplay between behavioral ecologies
of markets with heterogeneous traders, on the one hand, and institutional market setting, on the other hand. The investigation is motivated by the willingness to explain
inside a relatively simple and comprehensible model those numerous “stylized facts”
which are left unexplained in the limits of classical financial market paradigm (see
e.g. [3]). Since the dynamics of financial market is an outcome of a complicated interrelation between behavioral patterns and underlying structure, it seems reasonable
to start with analytically tractable model based on realistic behavioral assumptions
and simulate it in a more realistic market setting. This is our main goal.
The first generation of agent-based models of financial markets followed so
called bottom-up approach. The models were populated by an “ocean” of boundedly rational traders with adaptive behavior and were designed to be simulated on
the computers. The Santa Fe artificial market (AM) model [1, 9] represents one of
the most known examples of such approach. See also [10] and reviews in [7] and [8].
An inherent difficulty to interpret the results of simulations in a systematic way led
many researchers to build the models with heterogeneous agents which can be rigorously analyzed by the tools of the theory of dynamical systems. The achievements
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Mikhail Anufriev and Valentyn Panchenko
of the latter approach are summarized in [6]. In particular, evolutionary model in [4]
follows some ideas of the Santa Fe AM in that the traders repeatedly choose among
a finite number of predictors of future price according to the past performance of the
predictors. It turns out that when the intensity of choice is high, the price time series
may deviate from a fundamental benchmark in a systematic way, become chaotic
and exhibit excess volatility and volatility clustering (see [5]).
All the models mentioned so far, both simulation and analytic, are built in a
simple framework with mythical Walrasian auctioneer clearing the market. The real
markets are functioning in completely different way, of course, and many recent
models try to capture this fact. For instance, [11] show that artificial market with
realistic architecture, namely order-driven market under electronic book protocol, is
capable to generate satisfactory statistical properties of the prices (like leptokurtosis
of returns distribution) even in the presence of the homogeneous agents. [2] analyze
the market with heterogeneous agents under different market protocols and show that
the architecture does bear a central influence on the statistical properties of returns.
The purpose of the current investigation is to address the question about interrelation between market architecture and behavioral assumptions. It is in the spirit of
approach of [2], but differs from it in one important respect. Indeed, the population
has been “frozen” in the latter model, i.e. agents did not update their behaviors. Instead, our analytic benchmark constitutes the model of [4] where the key feature is
evolution of the agents’ population. We build a simulation platform to compare the
aggregate properties of the model under two different trading set-ups: a Walrasian
auction, and an “order-book” mechanism.
2 Model Structure
The core of our setting is inspired by an evolutionary model in [4]. There are two
assets in the market. A riskless asset is perfectly elastically supplied at gross return
R = 1 + rf . A risky asset pays dividend yt at the beginning of each trading period,
which is an i.i.d. random variable with mean ȳ. The time is discrete and index t
denotes the time period. The price with which the market closes is denoted as pt .
Traders have heterogeneous expectations about one-period return and are divided
into H types accordingly. The expectations determine the traders’ demand derived
from a mean-variance optimization, and affect the current price. Let us denote the
expectations of traders of type h as Eh,t [pt+1 + yt+1 ] and assume that these expectations are shared by fraction nh,t of traders.
Expectations depend both on the past price history and on the economic fundamentals. In the simplest case there are two types of agents capturing, in a very
stylized way, two different attitudes observed in the real markets. Fundamentalists believe that price will be equal to the fundamental value pf = ȳ/rf , so that
E1,t [pt+1 + yt+1 ] = pf + ȳ. Instead, trend followers believe that deviations from pf
can be persistent and expect E2,t [pt+1 + yt+1 ] = (1 − g)pf + gpt−1 + ȳ with some
positive or negative g. The traders are referred as trend chasers in the former case
and as contrarians in the latter case.
Heterogeneous Beliefs under Different Market Architectures
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The fractions of different types are evolving according to the relative success of
types, which in turn depends on the performance measure. Thus, the population is
endogenously changing. Fractions nh,t are determined as follows. After the trade of
previous period one can compute the average profit πh,t−1 obtained by agents of type
h for assets’ possession between periods t − 2 and t − 1. The performance measure
Uh,t−1 of strategy h is an accumulated profit weighted with positive parameter η
representing “memory strength”, so that Uh,t−1 = πh,t−1 + η Uh,t−2 . Finally, the
fraction is given by the discrete choice probability so that
X
nh,t = exp[βUh,t−1 ]/Zt−1 , where Zt−1 =
exp[βUh,t−1 ] . (1)
h
The key parameter β measures the intensity of choice, i.e. how fast agents switch
between different prediction types. If this intensity of choice is infinite, the traders
immediately switch to the most successful strategy. On the opposite extreme β = 0,
and agents are equally distributed between different types independent of the past
performance.
3 Different Market Designs
The remaining ingredient of the model is the mechanism for price determination. In
the original model in [4] the market is clearing through Walrasian mechanism which
allows, to some extent, treat the model analytically. We will compare this setting with
the order-driven market.
3.1 Walrasian Auction
Under Walrasian setting there exist a unique price at period t. It can be found as the
solution of the following equation
X
©¡
¢
ª
nh,t Eh,t [pt+1 + yt+1 ] − Rpt /aσ 2 = zs,t ,
(2)
h
where zs,t is the supply of the risky asset, while a and σ 2 stand for the risk-aversion
coefficient and expected variance, respectively, which are shared by all traders. In
some simple case, like if H = 2 and the market is populated by fundamentalists and
technical traders, one can define a low-dimensional dynamical system using equations (1) and (2). The fundamental equilibrium of this system is the point where
price is equal to pf . The local stability analysis reveals that this point loses its stability with β increasing.
3.2 Order-driven Market
In the order-driven market a period of time does not correspond to a single trade
any longer. Instead, there is one trading session over period t and price pt is the
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Mikhail Anufriev and Valentyn Panchenko
1
1020
1
1020
0.75
1000
0.75
1000
980
0.5
960
980
0.5
960
0.25
940
0.25
940
160
180
200
220
240
time
260
280
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300
160
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time
260
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Fig. 1. Close price and the share of fundamentalists in the order-driven market. Price is measured in ticks on the left vertical axis, share is measured on the the right vertical axis. Left
Panel: β = 2, g = 1.5; Right Panel: β = 2, g = 1.2.
closing price of the session. Each agent can place only one buy or sell order during
the session. The sequence in which agents place their orders is determined randomly.
During the session the market operates according to the following mechanism.
There is an electronic book containing unsatisfied agents’ buy and sell orders placed
during current trading session. When a new buy or sell order arrives to the market, it
is checked against the counter-side of the book. The order is partially or completely
executed if it finds a match, i.e. a counter-side order at requested or better price,
starting from the best available price. An unsatisfied order or its part is placed in the
book. At the end of the session all unsatisfied orders are removed from the book.
There are two types of the orders: limit and market order. Each limit order consists of a price and the amount of shares an agent wants to buy or sell at this price. The
price of a limit order is randomly generated in the range ±5% from the last transaction price. Such price formation provides liquidity, similarly to what is done in [11].
Analogously to [2], the amount of shares requested is computed as the corresponding point on the demand function. The expectations Eh,t [pt+1 + yt+1 ] entering the
demand function, about prevailing market price during the next session are formed
before opening the market in a way described in Section 2. In such a way we keep
the behavioral assumptions as close as possible to the analytical counterpart of [4].
This is done to facilitate a comparison between two different market architectures.
The market order consists only of an amount of shares and is satisfied at the best
available price in the order book. The requested amount is determined in a similar
way as the amount of the limit order.
4 Simulation Results
In Fig. 1 we present the outcomes of some typical simulations from our agent-based
model. The results are given solely for illustrative purposes and are to be expanded
in the future. Each panel depicts time series of (i) close price (solid line) with mea-
Heterogeneous Beliefs under Different Market Architectures
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surements on the left vertical axis and (ii) share of the fundamentalists (dashed line)
with measurements on the right vertical axis.
Qualitatively different dynamics of prices is observed due to the change in the
parameter of extrapolation, g, trend followers use to form their expectations. For
g = 1.5 (left panel) we observe order-based analog of steady cycle of model in [4],
while for g = 1.2 (right panel) the dynamics converges to steady state with small
deviations around it. Notice that the noise is endogenously introduced by the market
architecture. Relatively large intensity of choice β = 3 guarantees frequent switches
between the different strategies. In the periods with high share of fundamentalists
the price approaches the fundamental price and deviates from it when the share of
fundamentalists is small.
4.1 On the impact of architecture on the ecology
4.2 On the impact of architecture and ecology on the time series properties
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