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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)

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