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problem. In a simple approach, the historical data X(t Ã€ k) are multi-

plied by the factor lk where 0 < l < 1. Another interesting idea is

weighting the historical data with their volatility [4]. Namely, the asset

returns R(t) at time t used in forecasting VaR for time T are scaled

with the volatility ratio

R0 (t) Â¼ R(t)s(T)=s(t) (11:2:1)

where s(t) is the historical forecast of the asset volatility.3 As a result,

the actual return at day t is increased if the volatility forecast at day T

is higher than that of day t, and vice versa. The scaled forecasts R0 (t)

are further used in calculating VaR in the same way as the forecasts

R(t) are used in equal-weight historical simulation. Other more so-

phisticated non-parametric techniques are discussed in [2] and refer-

ences therein.

An obvious advantage of the non-parametric approaches is their

relative conceptual and implementation simplicity. The main disad-

vantage of the non-parametric approaches is their absolute depend-

ence on the historical data: Collecting and filtering empirical data

always comes at a price.

The parametric approach is a plausible alternative to historical

simulation. This approach is based on fitting the P/L probability

distribution to some analytic function. The (log)normal, Student

126 Market Risk Measurement

and extreme value distributions are commonly used in modeling P/L

[2, 5]. The parametric approach is easy to implement since the analytic

expressions can often be used. In particular, the assumption of the

normal distribution reduces calculating VaR to (11.1.2). Also, VaR

for time interval T can be easily expressed via VaR for unit time (e.g.,

via daily VaR (DVaR) providing T is the number of days)

pï¬ƒï¬ƒï¬ƒï¬ƒ

VaR(T) Â¼ DVaR T (11:2:2)

VaR for a portfolio of N assets is calculated using the variance of the

multivariate normal distribution

X

N

2

sN Â¼ (11:2:3)

sij

i, jÂ¼1

If the P/L distribution is normal, ETL can also be calculated analyt-

ically

ETL(a) Â¼ sPSN (Za )=(1 Ã€ a) Ã€ m (11:2:4)

The value za in (11.2.4) is determined with (11.1.3). Obviously, the

parametric approach is as good and accurate as the choice of the

analytic probability distribution.

Calculating VaR has become a part of the regulatory environment

in the financial industry [6]. As a result, several methodologies have

been developed for testing the accuracy of VaR models. The most

widely used method is the Kupiec test. This test is based on the

assumption that if the VaR(a) model is accurate, the number of the

tail losses n in a sample N is determined with the binomial distribu-

tion

N!

(1 Ã€ a)n a(NÃ€n)

PB (n; N, 1 Ã€ a) Â¼ (11:2:5)

n!(N Ã€ n)!

The null hypothesis is that n/N equals 1 Ã€ a, which can be tested with

the relevant likelihood ratio statistic. The Kupiec test has clear mean-

ing but may be inaccurate for not very large data samples. Other

approaches for testing the VaR models are described in [2, 6] and

references therein.

127

Market Risk Measurement

11.3 REFERENCES FOR FURTHER READING

The Jorionâ€™s monograph [1] is a popular reference for VaR-based

risk management. The Dowdâ€™s textbook [2] is a good resource for the

modern risk measurement approaches beyond VaR.

11.4 EXERCISES

1. Consider a portfolio with two assets: asset 1 has current value $1

million and annual volatility 12%; asset 2 has current value $2

million and annual volatility 24%. Assuming that returns are

normally distributed and there are 250 working days per year,

calculate 5-day VaR of this portfolio with 99% confidence level.

Perform calculations for the asset correlation coefficient equal

to (a) 0.5 and (b) Ã€0.5.

2. Verify (11.2.4).

*3. Implement the algorithm of calculating ETL for given P/L

density function. Analyze the algorithm accuracy as a function

of the number of integration points by comparing the calcula-

tion results with the analytic expression for the normal distribu-

tion (11.2.4).

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Chapter 12

Agent-Based Modeling

of Financial Markets

12.1 INTRODUCTION

Agent-based modeling has become a popular methodology in

social sciences, particularly in economics.1 Here we focus on the

agent-based modeling of financial markets [1]. The very idea of

describing markets with models of interacting agents (traders, invest-

ors) does not fit well with the classical financial theory that is based

on the notions of efficient markets and rational investors. However, it

has become obvious that investors are neither perfectly rational nor

have homogeneous expectations of the market trends (see also Section

2.3). Agent-based modeling proves to be a flexible framework for a

realistic description of the investor adaptation and decision-making

process.

The paradigm of agent-based modeling applied to financial markets

implies that trader actions determine price. This concept is similar to

that of statistical physics within which the thermodynamic (macro-

scopic) properties of the medium are described via molecular inter-

actions. A noted expansion of the microscopic modeling methodology

into social systems is the minority game (see [2] and references therein).

Its development was inspired by the famous El Farolâ€™s bar problem [3].

This problem considers a number of patrons N willing to attend a bar

with a number of seats Ns . It is assumed that Ns < N and every patron

prefers to stay at home if he expects that the number of people

129

130 Agent-Based Modeling of Financial Markets

attending the bar will exceed Ns . There is no communication among

patrons and they make decisions using only information on past

attendance and different predictors (e.g., attendance today is the

same as yesterday, or is some average of past attendance).

The minority game is a simple binary choice problem in which

players have to choose between two sides, and those on the minority

side win. Similarly to the El Farolâ€™s bar problem, in the minority

game there is no communication among players and only a given set

of forecasting strategies defines player decisions. The minority game

is an interesting stylized model that may have some financial implica-

tions [2]. But we shall focus further on the models derived specifically

for describing financial markets.

In the known literature, early work on the agent-based modeling of

financial markets can be traced back to 1980 [4]. In this paper, Beja and

Goldman considered two major trading strategies, value investing and

trend following. In particular, they showed that system equilibrium

may become unstable when the number of trend followers grows.

Since then, many agent-based models of financial markets have

been developed (see, e.g., reviews [1, 5], the recent collection [6] and

references therein). We divide these models into two major groups. In

the first group, agents make decisions based on their own predictions

of future prices and adapt their beliefs using different predictor func-

tions of past returns. The principal feature of this group is that price is

derived from the supply-demand equilibrium [7â€“10].2 Therefore, we

call this group the adaptive equilibrium models. In the other group, the

assumption of the equilibrium price is not employed. Instead, price is

assumed to be a dynamic variable determined via its empirical relation

to the excess demand (see, e.g., [11, 12]). We call this group the non-

equilibrium price models. In the following two sections, we discuss two

instructive examples for both groups of models, respectively. Finally,

Section 12.4 describes a non-equilibrium price model that is derived

exclusively in terms of observable variables [13].

12.2 ADAPTIVE EQUILIBRIUM MODELS

In this group of models [7â€“10], agents can invest either in the risk-

free asset (bond) or in the risky asset (e.g., a stock market index). The

risk-free asset is assumed to have an infinite supply and a constant

131

Agent-Based Modeling of Financial Markets

interest rate. Agents attempt to maximize their wealth by using some

risk aversion criterion. Predictions of future return are adapted using

past returns. The solution to the wealth maximization problem yields

the investor demand for the risky asset. This demand in turn deter-

mines the asset price in equilibrium. Let us formalize these assump-

tions using the notations from [10]. The return on the risky asset at

time t is defined as

rt Â¼ (pt Ã€ ptÃ€1 Ã¾ yt )=ptÃ€1 (12:2:1)

where pt and yt are (ex-dividend) price and dividend of one share of

the risky asset, respectively. Wealth dynamics of agent i is given by

Wi, tÃ¾1 Â¼ R(1 Ã€ pi, t )Wi, t Ã¾ pi, t Wi, t (1 Ã¾ rtÃ¾1 )

Â¼ Wi, t [R Ã¾ pi, t (rtÃ¾1 Ã€ r)] (12:2:2)

where r is the interest rate of the risk-free asset, R Â¼ 1 Ã¾ r, and pi, t is

the proportion of wealth of agent i invested in the risky asset at time t.

Every agent is assumed to be a taker of the risky asset at price that is

established in the demand-supply equilibrium. Let us denote Ei, t and

Vi, t the â€˜â€˜beliefsâ€™â€™ of trader i at time t about the conditional expect-

ation of wealth and the conditional variance of wealth, respectively. It

follows from (12.2.2) that

Ei, t [Wi, tÃ¾1 ] Â¼ Wi, t [R Ã¾ pi, t (Ei, t [rtÃ¾1 ] Ã€ r)], (12:2:3)

Vi, t [Wi, tÃ¾1 ] Â¼ p2 t W2 t Vi, t [rtÃ¾1 ] (12:2:4)

i, i,

Also, every agent i believes that return of the risky asset is normally

distributed with mean Ei, t [rtÃ¾1 ] and variance Vi, t [rtÃ¾1 ]. Agents choose

the proportion pi, t of their wealth to invest in the risky asset, which

maximizes the utility function U

max {Ei, t [U(Wi, tÃ¾1 )]} (12:2:5)

pi, t

The utility function chosen in [9, 10] is

U(Wi, t ) Â¼ log (Wi, t ) (12:2:6)

Then demand pi, t that satisfies (12.2.5) equals

Ei, t [rtÃ¾1 ] Ã€ r

pi, t Â¼ (12:2:7)

Vi, t [rtÃ¾1 ]

132 Agent-Based Modeling of Financial Markets

Another utility function used in the adaptive equilibrium models

employs the so-called constant absolute risk aversion (CARA) function

[7, 8]

a

U(Wi, t ) Â¼ Ei, t [Wi, tÃ¾1 ] Ã€ Vi, t [Wi, tÃ¾1 ] (12:2:8)

2

where a is the risk aversion constant. For the constant conditional

variance Vi, t Â¼ s2 , the CARA function yields the demand

Ei, t [rtÃ¾1 ] Ã€ r

pi, t Â¼ (12:2:9)

as2

The number of shares of the risky asset that corresponds to demand

pi, t equals

Ni, t Â¼ pi, t Wi, t =pt (12:2:10)

Since the total number of shares assumed to be fixed

P

Ni, t Â¼ N Â¼ const , the market-clearing price equals

i

1X

pt Â¼ pi, t Wi, t (12:2:11)

Ni

The adaptive equilibrium model described so far does not contradict

the classical asset pricing theory. The new concept in this model is the

heterogeneous beliefs. In its general form [7, 10]

Ei, t [rtÃ¾1 ] Â¼ fi (rtÃ€1 , . . . , rtÃ€Li ), (12:2:12)

Vi, t [rtÃ¾1 ] Â¼ gi (rtÃ€1 , . . . , rtÃ€Li ) (12:2:13)

The deterministic functions fi and gi depend on past returns with lags

up to Li and may vary for different agents.3

While variance is usually assumed to be constant (gi Â¼ s2 ), several

trading strategies fi are discussed in the literature. First, there are

fundamentalists who use analysis of the business fundamentals to

make their forecasts on the risk premium dF

EF, t [rtÃ¾1 ] Â¼ r Ã¾ dF (12:2:14)

In simple models, the risk premium dF > 0 is a constant but it can be a

function of time and/or variance in the general case. Another major

strategy is momentum trading (traders who use it are often called

chartists). Momentum traders use history of past returns to make

their forecasts. Namely, their strategy can be described as

133

Agent-Based Modeling of Financial Markets

X

L

EM, t [rtÃ¾1 ] Â¼ r Ã¾ dM Ã¾ ak rtÃ€k (12:2:15)

kÂ¼1

where dM > 0 is the constant component of the momentum risk

premium and ak > 0 are the weights of past returns rtÃ€k . Finally,

contrarians employ the strategy that is formally similar to the momen-

tum strategy

X

L

EC, t [rtÃ¾1 ] Â¼ r Ã¾ dC Ã¾ bk rtÃ€k (12:2:16)

kÂ¼1

with the principal difference that all bk are negative. This implies that

contrarians expect the market to turn around (e.g., from bull market

to bear market).

An important feature of adaptive equilibrium models is that agents

are able to analyze performance of different strategies and choose the

most efficient one. Since these strategies have limited accuracy, such

adaptability is called bounded rationality.

In the limit of infinite number of agents, Brock and Hommes offer

a discrete analog of the Gibbs probability distribution for the fraction

of traders with the strategy i [7]

X

nit Â¼ exp [b(Fi, tÃ€1 Ã€ Ci )]=Zt , Zt Â¼ exp [b(Fi, tÃ€1 Ã€ Ci )] (12:2:17)

i

In (12.2.17), Ci ! 0 is the cost of the strategy i, the parameter b is

called the intensity of choice, and Fi, t is the fitness function that

characterizes the efficiency of strategy i. The natural choice for the

fitness function is

Fi, t Â¼ gFi, tÃ€1 Ã¾ wi, t , wi, t Â¼ pi, t (Wi, t Ã€ Wi, tÃ€1 )=Wi, tÃ€1 (12:2:18)

where 0 g 1 is the memory parameter that retains part of past

performance in the current strategy.

Adaptive equilibrium models have been studied in several direc-

tions. Some work has focused on analytic analysis of simpler models.

In particular, the system stability and routes to chaos have been

discussed in [7, 10]. In the meantime, extensive computational model-

ing has been performed in [9] and particularly for the so-called Santa

Fe artificial market, in which a significant number of trading strat-

egies were implemented [8].

134 Agent-Based Modeling of Financial Markets

12.3 NON-EQUILIBRIUM PRICE MODELS

The concept of market clearing that is used in determining price of

the risky asset in the adaptive equilibrium models does not accurately

reflect the way real markets work. In fact, the number of shares

involved in trading varies with time, and price is essentially a dynamic

variable. A simple yet reasonable alternative to the price-clearing

paradigm is the equation of price formation that is based on the

empirical relation between price change and excess demand [4].

Different agent decision-making rules may be implemented within

this approach. Here the elaborated model offered by Lux [11] is

described. In this model, two groups of agents, namely chartists and

fundamentalists, are considered. Agents can compare the efficiency of

different trading strategies and switch from one strategy to another.

Therefore, the numbers of chartists, nc (t), and fundamentalists, nf (t),

vary with time while the total number of agents in the market N is

assumed constant. The chartist group in turn is sub-divided into

optimistic (bullish) and pessimistic (bearish) traders with the numbers

nÃ¾ (t) and nÃ€ (t), respectively

nc (t) Ã¾ nf (t) Â¼ N, nÃ¾ (t) Ã¾ nÃ€ (t) Â¼ nc (t) (12:3:1)

Several aspects of trader behavior are considered. First, the chartist

decisions are affected by the peer opinion (so-called mimetic conta-

gion). Secondly, traders change strategy while seeking optimal per-

formance. Finally, traders may exit and enter markets. The bullish

chartist dynamics is formalized in the following way:

dnÃ¾ =dt Â¼ (nÃ€ pÃ¾Ã€ Ã€ nÃ¾ pÃ€Ã¾ )(1 Ã€ nf =N) Ã¾ mimetic contagion

nf nÃ¾ (pÃ¾f Ã€ pfÃ¾ )=N Ã¾ changes of strategy

(b Ã€ a)nÃ¾ market entry and exit (12:3:2)

Here, pab denotes the probability of transition from group b to group

a. Similarly, the bearish chartist dynamics is given by

dnÃ€ =dt Â¼ (nÃ¾ pÃ€Ã¾ Ã€ nÃ€ pÃ¾Ã€ )(1 Ã€ nf =N) Ã¾ mimetic contagion

nf nÃ€ (pÃ€f Ã€ pfÃ€ )=N Ã¾ changes of strategy

(b Ã€ a)nÃ€ market entry and exit (12:3:3)

It is assumed that traders entering the market start with the chartist

strategy. Therefore, constant total number of traders yields the

135

Agent-Based Modeling of Financial Markets

relation b Â¼ aN=nc . Equations (12.3.1)â€“(12.3.3) describe the dynam-

ics of three trader groups (nf , nÃ¾ , nÃ€ ) assuming that all transfer

probabilities pab are determined. The change between the chartist

bullish and bearish mood is given by

pÃ¾Ã€ Â¼ 1=pÃ€Ã¾ Â¼ n1 exp(Ã€U1 ),

U1 Â¼ a1 (nÃ¾ Ã€ nÃ€ )=nc Ã¾ (a2 =n1 )dP=dt (12:3:4)

where n1 , a1 and a2 are parameters and P is price. Conversion of

fundamentalists into bullish chartists and back is described with

pÃ¾f Â¼ 1=pfÃ¾ Â¼ n2 exp(Ã€U21 ),

U21 Â¼ a3 ((r Ã¾ nÃ€1 dP=dt)=P Ã€ R Ã€ sj(Pf Ã€ P)=Pj) (12:3:5)

2

where n2 and a3 are parameters, r is the stock dividend, R is the

average revenue of economy, s is a discounting factor 0 < s < 1, and

Pf is the fundamental price of the risky asset assumed to be an input

parameter. Similarly, conversion of fundamentalists into bearish

chartists and back is given by

pÃ€f Â¼ 1=pfÃ€ Â¼ n2 exp(Ã€U22 ),

U22 Â¼ a3 (R Ã€ (r Ã¾ nÃ€1 dP=dt)=P Ã€ sj(Pf Ã€ P)=Pj) (12:3:6)

2

Price P in (12.3.4)â€“(12.3.6) is a variable that still must be defined.

Hence, an additional equation is needed in order to close the system

(12.3.1)â€“(12.3.6). As it was noted previously, an empirical relation

between the price change and the excess demand constitutes the

specific of the non-equilibrium price models4

dP=dt Â¼ bDex (12:3:7)

In the model [11], the excess demand equals

Dex Â¼ tc (nÃ¾ Ã€ nÃ€ ) Ã¾ gnf (Pf Ã€ P) (12:3:8)

The first and second terms in the right-hand side of (12.3.8) are the

excess demands of the chartists and fundamentalists, respectively;

b, tc and g are parameters.

The system (12.3.1)â€“(12.3.8) has rich dynamic properties deter-

mined by its input parameters. The system solutions include stable

equilibrium, periodic patterns, and chaotic attractors. Interestingly,

the distributions of returns derived from the chaotic trajectories

may have fat tails typical for empirical data. Particularly in [14], the

136 Agent-Based Modeling of Financial Markets

model [11] was modified to describe the arrival of news in the market,

which affects the fundamental price. This process was modeled with

the Gaussian random variable e(t) so that

ln Pf (t) Ã€ ln Pf (t Ã€ 1) Â¼ e(t) (12:3:9)

The modeling results exhibited the power-law scaling and temporal

volatility dependence in the price distributions.

12.4 THE OBSERVABLE VARIABLES MODEL

12.4.1 THE FRAMEWORK

The models discussed so far are capable of reproducing important

features of financial market dynamics. Yet, one may notice a degree

of arbitrariness in this field. The number of different agent types and

the rules of their transition and adaptation vary from one model to

another. Also, little is known about optimal choice of the model

parameters [15, 16]. As a result, many interesting properties, such as

deterministic chaos, may be the model artifacts rather than reflections

of the real world.5

A parsimonious approach to choosing variables in the agent-based

modeling of financial markets was offered in [17]. Namely, it was

suggested to derive agent-based models exclusively in terms of observ-

able variables. Note that the notion of observable data in finance

should be discerned from the notion of publicly available data. While

the transaction prices in regulated markets are publicly available, the

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