. 4
( 5)


Gamma is small, rebalancing can be performed less frequently.
Adding options to the portfolio can change its Gamma. In particular,
delta-neutral portfolio with Gamma G can be made gamma-neutral if
it is supplemented with n ¼ ÀG=GF options having Gamma GF .
Theta characterizes the time decay of the portfolio price. In add-
ition, two other Greeks, Vega and Rho, are used to measure the
portfolio sensitivity to its volatility and risk-free rate, respectively
@P @P
y¼ ,r¼ (9:4:13)
@s @r
Several assumptions that are made in BST can be easily relaxed. In
particular, dividends can be accounted. Also, r and s can be treated as
time-dependent parameters. BST has been expanded in several ways
(see [1“3, 7, 8] and references therein). One of the main directions
addresses so-called volatility smile. The problem is that if all charac-
teristics of the European option besides the strike price are fixed, its
implied volatility derived from the Black-Scholes expression is con-
stant. However, real market price volatilities do depend on the strike
price, which manifests in ˜˜smile-like™™ graphs. Several approaches
have been developed to address this problem. One of them is introdu-
cing the time dependencies into the interest rates or/and volatilities
Option Pricing

(so-called term structure). In a different approach, the lognormal
stock price distribution is substituted with another statistical distri-
bution. Also, the jump-diffusion stochastic processes are sometimes
used instead of the geometric Brownian motion.
Other directions for expanding BST address the market imperfec-
tions, such as transaction costs and finite liquidity. Finally, the option
price in the current option pricing theory depends on time and price
of the underlying asset. This seemingly trivial assumption was ques-
tioned in [9]. Namely, it was shown that the option price might
depend also on the number of shares of the underlying asset in the
arbitrage-free portfolio. Discussion of this paradox is given in the
Appendix section of this chapter.

Hull™s book is the classical reference for the first reading on finan-
cial derivatives [1]. A good introduction to mathematics behind the
option theory can be found in [4]. Detailed presentation of the option
theory, including exotic options and extensions to BST, is given in
[2, 3].

As we discussed in Section 9.4, the option price F(S, t) in BST is a
function of the stock price and time. The arbitrage-free portfolio in
BST consists of one share and of a number of options (M0 ) that hedge
this share [5]. BST can also be derived with the arbitrage-free port-
folio consisting of one option and of a number of shares MÀ1 (see,
e.g., [1]). However, if the portfolio with an arbitrary number of shares
N is considered, and N is treated as an independent variable, that is,
F ¼ F(S, t, N) (9:6:1)
then a non-zero derivative, @F=@N, can be recovered within the
arbitrage-free paradigm [9]. Since options are traded independently
from their underlying assets, the relation (9.6.1) may look senseless to
the practitioner. How could this dependence ever come to mind?
106 Option Pricing

Recall the notion of liquidity discussed in Section 2.1. If a market
order exceeds supply of an asset at current ˜˜best™™ price, then the
order is executed within a price range rather than at a single price. In
this case within continuous presentation,
S ¼ S(t, N) (9:6:2)
and the expense of buying N shares at time t equals

S(t, x)dx (9:6:3)

The liquidity effect in pricing derivatives has been addressed in [10,
11] without proposing (9.6.1). Yet, simply for mathematical general-
ity, one could assume that (9.6.1) may hold if (9.6.2) is valid. Surpris-
ingly, the dependence (9.6.1) holds even for infinite liquidity. Indeed,
consider the arbitrage-free portfolio P with an arbitrary number of
shares N at price S and M options at price F:
P(S, t, N) ¼ NS(t) þ MF(S, t, N) (9:6:4)
Let us assume that N is an independent variable and M is a parameter
to be defined from the arbitrage-free condition, similar to M0 in BST.
As in BST, the asset price S ¼ S(t) is described with the geometric
Brownian process
dS ¼ mSdt þ sSdW: (9:6:5)
In (9.6.5), m and s are the price drift and volatility, and W is the
standard Wiener process. According to the Ito™s Lemma,
s2 2 @ 2 F
@F @F @F
dF ¼ dt þ dS þ S dt þ dN (9:6:6)
@t @S @N
It follows from (9.6.4) that the portfolio dynamic is
dP ¼ MdF þ NdS þ SdN (9:6:7)
Substituting equation (9.6.6) into equation (9.6.7) yields
@F s2 2 @ 2 F
@F @F
dP ¼ [M þ N]dS þ [M þ S]dN þ M þS dt
@S @N @t
Option Pricing

As within BST, the arbitrage-free portfolio grows with the risk-free
interest rate, r
dP ¼ rPdt (9:6:9)
Then the combination of equation (9.6.8) and equation (9.6.9)
@F s2 2
2@ F
þ N]dS þ [M þ MS À rMF À rNS]dtþ
@S @t (9:6:10)
þ S]dN ¼ 0
Since equation (9.6.10) must be valid for arbitrary values of dS, dt
and dN, it can be split into three equations
M (9:6:11)
@F s2 2 @ 2 F
þS À rF À rNS ¼ 0
M (9:6:12)
M (9:6:13)
Let us present F(S, t, N) in the form
F(S, t, N) ¼ F0 (S, t)Z(N) (9:6:14)
where Z(N) satisfies the condition
Z(1) ¼ 1 (9:6:15)
Then it follows from equation (9.6.11) that
M ¼ ÀN= Z (9:6:16)
This transforms equation (9.6.15) and equation (9.6.16), respectively,
@F0 s2 2 @ 2 F0
þ rS þS À rF0 ¼ 0 (9:6:17)
@S 2
@t @S
dZ @F0
¼ (S=F0 ) (Z=N), (9:6:18)
dN @S
108 Option Pricing

Equation (9.6.17) is the classical Black-Scholes equation (cf. with
(9.4.6)) while equations (9.6.16) and (9.6.18) define the values of M
and Z(N). Solution to equation (9.6.18) that satisfies the condition
(9.6.15) is
Z(N) ¼ Na (9:6:19)
¼ ÀMÀ1 . Equation (9.6.13) and equa-
where a ¼ (S=F0 )D, D ¼ 0
tion (9.6.16) yield
M ¼ ÀN1Àa =D ¼ N1Àa M0 (9:6:20)
Hence, the option price in the arbitrage-free portfolio with N shares
F(S, t, N) ¼ F0 (S, t)Na (9:6:21)
It coincides with the BST solution F0 (S, t) only if N ¼ 1, that is when
the portfolio has one share. However, the total expense of hedging N
shares in the arbitrage-free portfolio
Q ¼ MF ¼ À(N=D)F0 ¼ NM0 F0 (9:6:22)
is the same as within BST. Therefore, Q is the true invariant of the
arbitrage-free portfolio.
Invariance of the hedging expense is easy to understand using the
dimensionality analysis. Indeed, the arbitrage-free condition (9.6.9) is
given in units of the portfolio and therefore can only be used for
defining part of the portfolio. Namely, the arbitrage-free condition
can be used for defining the hedging expense Q ¼ MF but not for
defining both factors M and F. Similarly, the law of energy conser-
vation can be used for defining the kinetic energy of a body,
K ¼ 0:5mV2 . Yet, this law alone cannot be used for calculating the
body™s mass, m, and velocity, V. Note, however, that if a body has
unit mass (m ¼ 1), then the energy conservation law effectively yields
the body™s velocity. Similarly, the arbitrage-free portfolio with one
share does not reveal dependence of the option price on the number of
shares in the portfolio.
Option Pricing

1. (a) Calculate the Black-Scholes prices of the European call and
put options with six-month maturity if the current stock
price is $20 and grows with average rate of m ¼ 10%, vola-
tility is 20%, and risk-free interest rate is 5%. The strike price
is: (1) $18; (2) $22.
(b) How will the results above change if m ¼ 5%?
2. Is there an arbitrage opportunity with the following assets: the
price of the XYZ stock with no dividends is $100; the European
put options at $98 with six-month maturity are sold for $3.50;
the European call options at $98 with the same maturity are sold
for $8; T-bills with the same maturity are sold for $98. Hint:
Check the put-call parity.
**3. Compare the Ito™s and Stratonovich™s approaches for derivation
of the Black-Scholes equation (consult [12]).
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Chapter 10

Portfolio Management

This chapter begins with the basic ideas of portfolio selection.
Namely, in Section 10.1, the combination of two risky assets and
the combination of a risky asset and a risk-free asset are considered.
Then two major portfolio management theories are discussed: the
capital asset pricing model (Section 10.2) and the arbitrage pricing
theory (Section 10.3). Finally, several investment strategies based on
exploring market arbitrage opportunities are introduced in Section

Optimal investing is an important real-life problem that has been
translated into elegant mathematical theories. In general, opportun-
ities for investing include different assets: equities (stocks), bonds,
foreign currency, real estate, antique, and others. Here portfolios
that contain only financial assets are considered.
There is no single strategy for portfolio selection, because there is
always a trade-off between expected return on portfolio and risk of
portfolio losses. Risk-free assets such as the U.S. Treasury bills guar-
antee some return, but it is generally believed that stocks provide
higher returns in the long run. The trouble is that the notion of ˜˜long
run™™ is doomed to bear an element of uncertainty. Alas, a decade of

112 Portfolio Management

market growth may end up with a market crash that evaporates a
significant part of the equity wealth of an entire generation. Hence,
risk aversion (that is often well correlated with investor age) is an
important factor in investment strategy.
Portfolio selection has two major steps [1]. First, it is the selection
of a combination of risky and risk-free assets and, secondly, it is the
selection of risky assets. Let us start with the first step.
For simplicity, consider a combination of one risky asset and one
risk-free asset. If the portion of the risky asset in the portfolio is
a(a 1), then the expected rate of return equals
E[R] ¼ aE[Rr ] þ (1 À a)Rf ¼ Rf þ a(E[Rr ] À Rf ) (10:1:1)
where Rf and Rr are rates of returns of the risk-free and risky assets,
respectively. In the classical portfolio management theory, risk is
characterized with the portfolio standard deviation, s.1 Since no
risk is associated with the risk-free asset, the portfolio risk in our
case equals
s ¼ asr (10:1:2)
Substituting a from (10.1.2) into (10.1.1) yields
E[R] ¼ Rf þ s(E[Rr ] À Rf )=sr (10:1:3)
The dependence of the expected return on the standard deviation is
called the risk-return trade-off line. The slope of the straight line
s ¼ (E[Rr ] À Rf )=sr (10:1:4)
is the measure of return in excess of the risk-free return per unit of
risk. Obviously, investing in a risky asset makes sense only if s > 0,
that is, E[Rr ] > Rf . The risk-return trade-off line defines the mean-
variance efficient portfolio, that is, the portfolio with the highest
expected return at a given risk level.
On the second step of portfolio selection, let us consider the port-
folio consisting of two risky assets with returns R1 and R2 and with
standard deviations s1 and s2 , respectively. If the proportion of the
risky asset 1 in the portfolio is g(g 1), then the portfolio rate of
return equals
E[R] ¼ gE[R1 ] þ (1 À g)E[R2 ] (10:1:5)
Portfolio Management

and the portfolio standard deviation is

s2 ¼ g2 s1 2 þ (1 À g)2 s2 2 þ 2g(1 À g)s12 (10:1:6)
In (10.1.6), s12 is the covariance between the returns of asset 1 and
asset 2. For simplicity, it is assumed further that the asset returns are
uncorrelated, that is, s12 ¼ 0. The value of g that yields minimal risk
for this portfolio equals

gm ¼ s2 2 =(s1 2 þ s2 2 ), (10:1:7)
This value yields the minimal portfolio risk sm

sm 2 ¼ s1 2 s2 2 =(s1 2 þ s2 2 ) (10:1:8)
Consider an example with E[R1 ] ¼ 0:1, E[R2 ] ¼ 0:2, s1 ¼ 0:15,
s2 ¼ 0:3. If g ¼ 0:8, then s % 0:134 < s1 and E[R] ¼ 0:12 > E[R1 ].
Hence, adding the more risky asset 2 to asset 1 decreases the portfolio
risk and increases the portfolio return. This somewhat surprising
outcome demonstrates the advantage of portfolio diversification.
Finally, let us combine the risk-free asset with a portfolio that
contains two risky assets. The optimal combination of the risky
asset portfolio and the risk-free asset can be found at the tangency
point between the straight risk-return trade-off line with the intercept
E[R] ¼ Rf and the risk-return trade-off curve for the risky asset
portfolio (see Figure 10.1). For the portfolio with two risky uncorrel-
ated assets, the proportion g at the tangency point T equals

gT ¼ (E[R1 ] À Rf )s2 2 ={(E[R1 ] À Rf )s2 2 þ (E[R2 ] À Rf )s1 2 }
Substituting gT from (10.1.9) into (10.1.5) and (10.1.6) yields the
coordinates of the tangency point (i.e., E[RT ] and sT ). A similar
approach can be used in the general case with an arbitrary number
of risky assets. The return E[RT ] for a given portfolio with risk sT is
˜˜as good as it gets.™™ Is it possible to have returns higher than E[RT ]
while investing in the same portfolio? In other words, is it possible to
reach say point P on the risk-return trade-off line depicted in Figure.
10.1? Yes, if you borrow money at rate Rf and invest it in the portfolio
with g ¼ gT . Obviously, the investment risk is then higher than that of
sT .
114 Portfolio Management





0 0.04 0.08 0.12 0.16
Figure 10.1 The return-risk trade-off lines: portfolio with the risk-free
asset and a risky asset (dashed line); portfolio with two risky assets (solid
line); Rf ¼ 0:05, s1 ¼ 0:12, s2 ¼ 0:15, E[R1 ] ¼ 0:08, E[R2 ] ¼ 0:14.

The Capital Asset Pricing Model (CAPM) is based on the portfolio
selection approach outlined in the previous section. Let us consider
the entire universe of risky assets with all possible returns and risks.
The set of optimal portfolios in this universe (i.e., portfolios with
maximal returns for given risks) forms what is called a efficient
frontier. The straight line that is tangent to the efficient frontier and
has intercept Rf is called the capital market line.2 The tangency point
between the capital market line and the efficient frontier corresponds
to the so-called super-efficient portfolio.
In CAPM, it is assumed that all investors have homogenous expect-
ations of returns, risks, and correlations among the risky assets. It is
also assumed that investors behave rationally, meaning they all hold
optimal mean-variance efficient portfolios. This implies that all invest-
ors have risky assets in their portfolio in the same proportions as the
entire market. Hence, CAPM promotes passive investing in the index
Portfolio Management

mutual funds. Within CAPM, the optimal investing strategy is simply
choosing a portfolio on the capital market line with acceptable risk
level. Therefore, the difference among rational investors is determined
only by their risk aversion, which is characterized with the proportion
of their wealth allocated to the risk-free assets. Within the CAPM
assumptions, it can be shown that the super-efficient portfolio consists
of all risky assets weighed with their market values. Such a portfolio is
called a market portfolio.3
CAPM defines the return of a risky asset i with the security market
E[Ri ] ¼ Rf þ bi (E[RM ] À Rf ) (10:2:1)
where RM is the market portfolio return and parameter beta bi equals
bi ¼ Cov[Ri , RM ]=Var[RM ] (10:2:2)
Beta defines sensitivity of the risky asset i to the market dynamics.
Namely, bi > 1 means that the asset is more volatile than the entire
market while bi < 1 implies that the asset has a lower sensitivity to the
market movements. The excess return of asset i per unit of risk (so-
called Sharpe ratio) is another criterion widely used for estimation of
investment performance
Si ¼ (E[Ri ] À Rf )=si (10:2:3)
CAPM, being the equilibrium model, has no time dependence. How-
ever, econometric analysis based on this model can be conducted
providing that the statistical nature of returns is known [2]. It is
often assumed that returns are independently and identically distrib-
uted. Then the OLS method can be used for estimating bi in the
regression equation for the excess return Zi ¼ Ri À Rf
Zi (t) ¼ ai þ bi ZM (t) þ ei (t) (10:2:4)
It is usually assumed that ei (t) is a normal process and the S&P 500
Index is the benchmark for the market portfolio return RM (t). More
details on the CAPM validation and the general results for the mean-
variance efficient portfolios can be found in [2, 3].
As indicated above, CAPM is based on the belief that investing in
risky assets yields average returns higher than the risk-free return.
Hence, the rationale for investing in risky assets becomes question-
able in bear markets. Another problem is that the asset diversification
116 Portfolio Management

advocated by CAPM is helpful if returns of different assets are
uncorrelated. Unfortunately, correlations between asset returns may
grow in bear markets [4]. Besides the failure to describe prolonged
bear markets, another disadvantage of CAPM is its high sensitivity to
proxy for the market portfolio. The latter drawback implies that
CAPM is accurate only conditionally, within a given time period,
where the state variables that determine economy are fixed [2]. Then
it seems natural to extend CAPM to a multifactor model.

The CAPM equation (10.2.1) implies that return on risky assets is
determined only by a single non-diversifiable risk, namely by the risk
associated with the entire market. The Arbitrage Pricing Theory
(APT) offers a generic extension of CAPM into the multifactor
APT is based on two postulates. First, the return for an asset i
(i ¼ 1, . . . , N) at every time period is a weighed sum of the risk factor
contributions fj (t) (j ¼ 1, . . . , K, K < N) plus an asset-specific com-
ponent ei (t)
Ri (t) ¼ ai þ bi1 f1 þ bi2 f2 þ . . . þ biK fK þ ei (t) (10:3:1)
In (10.3.1), bij are the factor weights (betas). It is assumed that the
expectations of all factor values and for the asset-specific innovations
are zero
E[f1 (t)] ¼ E[f2 (t)] ¼ . . . ¼ E[fK (t)] ¼ E[ei (t)] ¼ 0 (10:3:2)
Also, the time distributions of the risk factors and asset-specific
innovations are independent
Cov[fj (t), fj (t0 )] ¼ 0, Cov[ei (t), ei (t0 )] ¼ 0, t 6¼ t (10:3:3)
and uncorrelated
Cov[fj (t), ei (t)] ¼ 0 (10:3:4)
Within APT, the correlations between the risk factors and the asset-
specific innovations may exist, that is Cov[fj (t), fk (t)] and
Cov[ei (t), ej (t)] may differ from zero.
Portfolio Management

The second postulate of APT requires that there are no arbitrage
opportunities. This implies, in particular, that any portfolio in which
all factor contributions are canceled out must have return equal to
that of the risk-free asset (see Exercise 3). These two postulates lead to
the APT theorem (see, e.g., [5]). In its simple form, it states that there
exist such K þ 1 constants l0 , l1 , . . . lK (not all of them equal zero)
E[Ri (t)] ¼ l0 þ bi1 l1 þ . . . þ biK lK (10:3:5)
While l0 has the sense of the risk-free asset return, the numbers lj are
named the risk premiums for the j-th risk factors.
Let us define a well-diversified portfolio as a portfolio that consists
wi ¼ 1, so that wi < W=N
of N assets with the weights wi where
and W % 1 is a constant. Hence, the specific of a well-diversified
portfolio is that it is not overweighed by any of its asset components.
APT turns out to be more accurate for well-diversified portfolios
than for individual stocks. The general APT states that if the return of
a well-diversified portfolio equals
R(t) ¼ a þ b1 f1 þ b2 f2 þ . . . þ bK fK þ e(t) (10:3:6)
a¼ wi a i , b i ¼ wk bik (10:3:7)

then the expected portfolio return is
E[R(t)] ¼ l0 þ b1 l1 þ . . . þ bK lK (10:3:8)
In addition, the returns of the assets that constitute the portfolio
satisfy the simple APT (10.3.5).
APT does not specify the risk factors. Yet, the essential sources of
risk are well described in the literature [6]. They include both macro-
economic factors including inflation risk, interest rate, and corporate
factors, for example, Return on Equity (ROE).4 Development of
statistically reliable multifactor portfolio models poses significant
challenges [2]. Yet, multifactor models are widely used in active
portfolio management.
118 Portfolio Management

Both CAPM and APT consider only one time period and treat the
risk-free interest rate as an exogenous parameter. However, in real
life, investors make investing and consumption decisions that are in
effect for long periods of time. An interesting direction in the port-
folio theory (that is beyond the scope of this book) describes invest-
ment and consumption processes within a single framework. The risk-
free interest rate is then determined by the consumption growth and
by investor risk aversion. The most prominent theories in this direc-
tion are the intertemporal CAPM (ICAPM) and the consumption
CAPM (CCAPM) [2, 3, 7].

The simple investment strategy means ˜˜buy and hold™™ securities
of ˜˜good™™ companies until their performance worsens, then sell them
and buy better assets. A more sophisticated approach is sensitive to
changing economic environment and an investor™s risk tolerance,
which implies periodic rebalancing of the investor portfolio between
cash, fixed income, and equities. Proponents of the conservative
investment strategy believe that this is everything an investor should
do while investing for the ˜˜long run.™™ Yet, many investors are not
satisfied with the long-term expectations: they want to make money
at all times (and who could blame them?). Several concepts being
intensively explored by a number of financial institutions, particu-
larly by the hedge funds, are called market-neutral strategies.
In a nutshell, market-neutral strategy implies hedging the risk of
financial losses by combining long and short positions in the port-
folio. For example, consider two companies within the same industry,
A and B, one of which (A) yields consistently higher returns. The
strategy named pair trading involves simultaneously buying shares
A and short selling shares B. Obviously, if the entire sector rises,
this strategy does not bring as much money as simply buying
shares A. However, if the entire market falls, presumably shares B
will have higher losses than shares A. Then the profits from short
selling shares B would more than compensate for the losses from
buying shares A.
Portfolio Management

Specifics of the hedging strategies are not widely advertised for
obvious reasons: the more investors target the same market ineffi-
ciency, the faster it is wiped out. Several directions in the market-
neutral investing are described in the literature [8].
Convertible arbitrage. Convertible bonds are bonds that can be
converted into shares of the same company. Convertible bonds
often decline less in a falling market than shares of the same company
do. Hence, the idea of the convertible arbitrage is buying convertible
bonds and short selling the underlying stocks.
Fixed-income arbitrage. This strategy implies taking long and short
positions in different fixed-income securities. By watching the correl-
ations between different securities, one can buy those securities
that seem to become underpriced and sell short those that look
Mortgage-backed securities (MBS) arbitrage. MBS is actually a
form of fixed income with a prepayment option. Yet, there are so
many different MBS that this makes them a separate business.
Merger arbitrage. This form of arbitrage involves buying shares of a
company that is being bought and short selling the shares of the buying
company. The rationale behind this strategy is that companies are
usually acquired at a premium, which sends down the stock prices of
acquiring companies.
Equity hedge. This strategy is not exactly the market-neutral one, as
the ratio between long and short equity positions may vary depending
on the market conditions. Sometimes one of the positions is the stock
index future while the other positions are the stocks that constitute
this index (so-called index arbitrage). Pair trading also fits this
Equity market-neutral strategy and statistical arbitrage. Nicholas
discerns these two strategies by the level of constraints (availability of
resources) imposed upon the portfolio manager [8]. The common
feature of these strategies is that (in contrast to the equity hedge),
they require complete offsetting of the long positions by the short
positions. Statistical arbitrage implies fewer constraints in the devel-
opment of quantitative models and hence a lower amount of the
portfolio manager™s discretion in constructing a portfolio.
Relative value arbitrage. This is a synthetic approach that may
embrace several hedging strategies and different securities including
120 Portfolio Management

equities, bonds, options, and foreign currencies. Looking for the arbi-
trage opportunities ˜˜across the board™™ is technically more challenging
but potentially rewarding.
Some academic research on efficiency of the arbitrage trading
strategies can be found in [9“12] and references therein. Note that
the research methodology in this field is itself a non-trivial problem

A good introduction into the finance theory, including CAPM, is
given in [1]. For a description of the portfolio theory and investment
science with an increasing level of technical detail, see [5, 14].

1. Consider a portfolio with two assets having the following
returns and standard deviations: E[R1 ] ¼ 0:15, E[R2 ] ¼ 0:1,
s1 ¼ 0:2, s2 ¼ 0:15. The proportion of asset 1 in the portfolio
g ¼ 0:5. Calculate the portfolio return and standard deviation.
The correlation coefficient between assets is (a) 0.5; (b) À0.5.
2. Consider returns of stock A and the market portfolio M in three
A À7% 12% 26%
M À5% 9% 18%
Assuming the risk-free rate is 5%, (a) calculate b of stock A; and
(b) verify if CAPM describes pricing of stock A.
3. Providing the stock returns follow the two-factor APT:
Ri (t) ¼ ai þ bi1 f1 þ bi2 f2 þ ei (t), construct a portfolio with
three stocks (i.e., define w1 , w2 , and w3 ¼ 1 À w1 À w2 ) that
yields return equal to that of the risk-free asset.
4. Providing the stock returns follow the two-factor simple APT,
derive the values of the risk premiums. Assume the expected
returns of two stocks and the risk-free rate are equal to R1 , R2 ,
and Rf , respectively.
Chapter 11

Market Risk Measurement

The widely used risk measure, value at risk (VaR), is discussed in
Section 11.1. Furthermore, the notion of the coherent risk measure is
introduced and one such popular measure, namely expected tail losses
(ETL), is described. In Section 11.2, various approaches to calculating
risk measures are discussed.

There are several possible causes of financial losses. First, there is
market risk that results from unexpected changes in the market prices,
interest rates, or foreign exchange rates. Other types of risk relevant
to financial risk management include liquidity risk, credit risk, and
operational risk [1]. The liquidity risk closely related to market risk is
determined by a finite number of assets available at a given price (see
discussion in Section 2.1). Another form of liquidity risk (so-called
cash-flow risk) refers to the inability to pay off a debt in time. Credit
risk arises when one of the counterparts involved in a financial
transaction does not fulfill its obligation. Finally, operational risk is
a generic notion for unforeseen human and technical problems, such
as fraud, accidents, and so on. Here we shall focus exclusively on
measurement of the market risk.
In Chapter 10, we discussed risk measures such as the asset return
variance and the CAPM beta. Several risk factors used in APT were

122 Market Risk Measurement

also mentioned. At present, arguably the most widely used risk meas-
ure is value at risk (VaR) [1]. In short, VaR refers to the maximum
amount of an asset that is likely to be lost over a given period at a
specific confidence level. This implies that the probability density
function for profits and losses (P=L)1 is known. In the simplest case,
this distribution is normal
PN (x) ¼ p¬¬¬¬¬¬ exp [À(x À m)2 =2s2 ] (11:1:1)
where m and s are the mean and standard deviation, respectively.
Then for the chosen confidence level a,
VAR(a) ¼ Àsza À m (11:1:2)
The value of za can be determined from the cumulative distribution
function for the standard normal distribution (3.2.10)
p¬¬¬¬¬¬ exp [Àz2 =2]dz ¼ 1 À a
Pr(Z < za ) ¼ (11:1:3)

Since za < 0 at a > 50%, the definition (11.1.2) implies that positive
values of VaR point to losses. In general, VaR(a) grows with the
confidence level a. Sufficiently high values of the mean
P=L (m > Àsza ) for given a move VaR(a) into the negative region,
which implies profits rather than losses. Examples of za for typical
values of a ¼ 95% and a ¼ 99% are given in Figure 11.1. Note that
the return variance s corresponds to za ¼ À1 and yields a % 84%.
The advantages of VaR are well known. VaR is a simple and
universal measure that can be used for determining risks of different
financial assets and entire portfolios. Still, VaR has some drawbacks
[2]. First, accuracy of VaR is determined by the model assumptions
and is rather sensitive to implementation. Also, VaR provides an
estimate for losses within a given confidence interval a but says
nothing about possible outcomes outside this interval. A somewhat
paradoxical feature of VaR is that it can discourage investment
diversification. Indeed, adding volatile assets to a portfolio may
move VaR above the chosen risk threshold. Another problem with
VaR is that it can violate the sub-additivity rule for portfolio risk.
According to this rule, the risk measure r must satisfy the condition
Market Risk Measurement



VaR at 84%
z = ’1

VaR at 95%
z = ’1.64

VaR at 99%
z = ’2.33


’5 ’4 ’3 ’2 ’1 0 1 2 3 4 5
Figure 11.1 VaR for the standard normal probability distribution of P/L.

r(A þ B) r(A) þ r(B) (11:1:4)
which means the risk of owning the sum of two assets must not be
higher than the sum of the individual risks of these assets. The
condition (11.1.4) immediately yields an upper estimate of combined
risk. Violation of the sub-additivity rule may lead to several problems.
In particular, it may provoke investors to establish separate accounts
for every asset they have. Unfortunately, VaR satisfies (11.1.4) only if
the probability density function for P/L is normal (or, more generally,
elliptical) [3].
The generic criterions for the risk measures that satisfy the require-
ments of the modern risk management are formulated in [3]. Besides
the sub-additivity rule (11.1.4), they include the following conditions.
r(lA) ¼ lr(A), l > 0 (homogeneity) (11:1:5)
r(A) r(B), if A B (monotonicity) (11:1:6)
r(A þ C) ¼ r(A) À C (translation invariance) (11:1:7)
In (11.1.7), C represents a risk-free amount. Adding this amount to
a risky portfolio should decrease the total risk, since this amount is
124 Market Risk Measurement

not subjected to potential losses. The risk measures that satisfy the
conditions (11.1.4)“(11.1.7) are called coherent risk measures. It can
be shown that any coherent risk measure represents the maximum of
the expected loss on a set of ˜˜generalized scenarios™™ where every such
scenario is determined with its value of loss and probability of occur-
rence [3]. This result yields the coherent risk measure called expected
tail loss (ETL):2
ETL ¼ E[LjL > VaR] (11:1:8)
While VaR is an estimate of loss within a given confidence level,
ETL is an estimate of loss within the remaining tail. For a given
probability distribution of P/L and a given a, ETL is always higher
than VaR (cf. Figures 11.1 and 11.2).
ETL has several important advantages over VaR [2]. In short, ETL
provides an estimate for an average ˜˜worst case scenario™™ while VaR
only gives a possible loss within a chosen confidence interval. ETL
has all the benefits of the coherent risk measure and does not discour-
age risk diversification. Finally, ETL turns out to be a more conveni-
ent measure for solving the portfolio optimization problem.



ETL at 84%
z = ’1.52

ETL at 95%
z = ’2.06

ETL at 99%
z = ’2.66


’5 ’4 ’3 ’2 ’1 0 1 2 3 4 5
Figure 11.2 ETL for the standard normal probability distribution of P/L.
Market Risk Measurement

Two main approaches are used for calculating VaR and ETL [2].
First, there is historical simulation, a non-parametric approach that
employs historical data. Consider a sample of 100 P/L values as a
simple example for calculating VaR and ETL. Let us choose the
confidence level of 95%. Then VaR is the sixth smallest number in
the sample while ETL is the average of the five smallest numbers
within the sample. In the general case of N observations, VaR at the
confidence level a is the [N(1 À a) þ 1] lowest observation and ETL is
the average of N(1 À a) smallest observations.
The well-known problem with the historical simulation is handling
of old data. First, ˜˜too old™™ data may lose their relevance. Therefore,
moving data windows (i.e., fixed number of observations prior to
every new period) are often used. Another subject of concern is
outliers. Different data weighting schemes are used to address this
problem. In a simple approach, the historical data X(t À 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) ¼ 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)
VaR(T) ¼ DVaR T (11:2:2)

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

sN ¼ (11:2:3)
i, j¼1

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

ETL(a) ¼ sPSN (Za )=(1 À a) À 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-
(1 À a)n a(NÀn)
PB (n; N, 1 À a) ¼ (11:2:5)
n!(N À n)!
The null hypothesis is that n/N equals 1 À 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.
Market Risk Measurement

The Jorion™s monograph [1] is a popular reference for VaR-based
risk management. The Dowd™s textbook [2] is a good resource for the
modern risk measurement approaches beyond VaR.

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) À0.5.
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

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
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 Farol™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

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 Farol™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 [7“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].

In this group of models [7“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
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 ¼ (pt À ptÀ1 þ yt )=ptÀ1 (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, tþ1 ¼ R(1 À pi, t )Wi, t þ pi, t Wi, t (1 þ rtþ1 )
¼ Wi, t [R þ pi, t (rtþ1 À r)] (12:2:2)
where r is the interest rate of the risk-free asset, R ¼ 1 þ 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 ˜˜beliefs™™ 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, tþ1 ] ¼ Wi, t [R þ pi, t (Ei, t [rtþ1 ] À r)], (12:2:3)
Vi, t [Wi, tþ1 ] ¼ p2 t W2 t Vi, t [rtþ1 ] (12:2:4)
i, i,

Also, every agent i believes that return of the risky asset is normally
distributed with mean Ei, t [rtþ1 ] and variance Vi, t [rtþ1 ]. 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, tþ1 )]} (12:2:5)
pi, t

The utility function chosen in [9, 10] is
U(Wi, t ) ¼ log (Wi, t ) (12:2:6)
Then demand pi, t that satisfies (12.2.5) equals
Ei, t [rtþ1 ] À r
pi, t ¼ (12:2:7)
Vi, t [rtþ1 ]
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]
U(Wi, t ) ¼ Ei, t [Wi, tþ1 ] À Vi, t [Wi, tþ1 ] (12:2:8)
where a is the risk aversion constant. For the constant conditional
variance Vi, t ¼ s2 , the CARA function yields the demand
Ei, t [rtþ1 ] À r
pi, t ¼ (12:2:9)
The number of shares of the risky asset that corresponds to demand
pi, t equals
Ni, t ¼ pi, t Wi, t =pt (12:2:10)
Since the total number of shares assumed to be fixed
Ni, t ¼ N ¼ const , the market-clearing price equals
pt ¼ pi, t Wi, t (12:2:11)

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 [rtþ1 ] ¼ fi (rtÀ1 , . . . , rtÀLi ), (12:2:12)
Vi, t [rtþ1 ] ¼ gi (rtÀ1 , . . . , rtÀLi ) (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 ¼ 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 [rtþ1 ] ¼ r þ 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
Agent-Based Modeling of Financial Markets

EM, t [rtþ1 ] ¼ r þ dM þ ak rtÀk (12:2:15)

where dM > 0 is the constant component of the momentum risk
premium and ak > 0 are the weights of past returns rtÀk . Finally,
contrarians employ the strategy that is formally similar to the momen-
tum strategy
EC, t [rtþ1 ] ¼ r þ dC þ bk rtÀk (12:2:16)

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]
nit ¼ exp [b(Fi, tÀ1 À Ci )]=Zt , Zt ¼ exp [b(Fi, tÀ1 À Ci )] (12:2:17)

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 ¼ gFi, tÀ1 þ wi, t , wi, t ¼ pi, t (Wi, t À Wi, tÀ1 )=Wi, tÀ1 (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

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
nþ (t) and nÀ (t), respectively
nc (t) þ nf (t) ¼ N, nþ (t) þ nÀ (t) ¼ 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:
dnþ =dt ¼ (nÀ pþÀ À nþ pÀþ )(1 À nf =N) þ mimetic contagion
nf nþ (pþf À pfþ )=N þ changes of strategy
(b À a)nþ 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
dnÀ =dt ¼ (nþ pÀþ À nÀ pþÀ )(1 À nf =N) þ mimetic contagion
nf nÀ (pÀf À pfÀ )=N þ changes of strategy
(b À a)nÀ 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
Agent-Based Modeling of Financial Markets

relation b ¼ aN=nc . Equations (12.3.1)“(12.3.3) describe the dynam-
ics of three trader groups (nf , nþ , nÀ ) assuming that all transfer
probabilities pab are determined. The change between the chartist
bullish and bearish mood is given by
pþÀ ¼ 1=pÀþ ¼ n1 exp(ÀU1 ),
U1 ¼ a1 (nþ À nÀ )=nc þ (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
pþf ¼ 1=pfþ ¼ n2 exp(ÀU21 ),
U21 ¼ a3 ((r þ nÀ1 dP=dt)=P À R À sj(Pf À P)=Pj) (12:3:5)

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
pÀf ¼ 1=pfÀ ¼ n2 exp(ÀU22 ),
U22 ¼ a3 (R À (r þ nÀ1 dP=dt)=P À sj(Pf À P)=Pj) (12:3:6)

Price P in (12.3.4)“(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)“(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 ¼ bDex (12:3:7)
In the model [11], the excess demand equals
Dex ¼ tc (nþ À nÀ ) þ gnf (Pf À 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)“(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) À ln Pf (t À 1) ¼ e(t) (12:3:9)
The modeling results exhibited the power-law scaling and temporal
volatility dependence in the price distributions.


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
market microstructure is not (see Section 2.1). Still, every event in the
financial markets that affects the market microstructure (such as
quote submission, quote cancellation, transactions, etc.) is recorded
and stored for business and legal purposes. This information allows
one to reconstruct the market microstructure at every moment. We
define observable variables in finance as those that can be retrieved or
calculated from the records of market events. Whether these records
are publicly available at present is a secondary issue. More import-
antly, these data exist and can therefore potentially be used for
calibrating and testing the theoretical models.
The numbers of agents of different types generally are not observ-
able. Indeed, consider a market analog of ˜˜Maxwell™s Demon™™ who is
Agent-Based Modeling of Financial Markets

able to instantly parse all market events. The Demon cannot discern
˜˜chartists™™ and ˜˜fundamentalists™™ in typical situations, such as when
the current price, being lower than the fundamental price, is growing.
In this case, all traders buy rather than sell. Similarly, when the
current price, being higher than the fundamental price, is falling, all
traders sell rather than buy.
Only price, the total number of buyers, and the total number of
sellers are always observable. Whether a trader becomes a buyer or
seller can be defined by mixing different behavior patterns in the
trader decision-making rule. Let us describe a simple non-equilibrium
price model derived along these lines [17]. We discern ˜˜buyers™™ (þ)
and ˜˜sellers™™ (À). Total number of traders is N
Nþ (t) þ NÀ (t) ¼ N (12:4:1)
The scaled numbers of buyers, nþ (t) ¼ Nþ (t)=N, and sellers, nÀ (t)
¼ NÀ (t)=N, are described with equations
dnþ =dt ¼ vþÀ nÀ À vÀþ nþ (12:4:2)
dnÀ =dt ¼ vÀþ nþ À vþÀ nÀ (12:4:3)
The factors vþÀ and vÀþ characterize the probabilities for transfer
from seller to buyer and back, respectively
vþÀ ¼ 1=vÀþ ¼ n exp (U), U ¼ apÀ1 dp=dt þ b(1 À p) (12:4:4)
Price p(t) is given in units of its fundamental value. The first term in
the utility function, U, characterizes the ˜˜chartist™™ behavior while the
second term describes the ˜˜fundamentalist™™ pattern. The factor n has
the sense of the frequency of transitions between seller and buyer
behavior. Since nþ (t) ¼ 1 À nÀ (t), the system (12.4.1)“(12.4.3) is re-
duced to the equation
dnþ =dt ¼ vþÀ (1 À nþ ) À vÀþ nþ (12:4:5)
The price formation equation is assumed to have the following
dp=dt ¼ gDex (12:4:6)
where the excess demand, Dex , is proportional to the excess number of
Dex ¼ d(nþ À nÀ ) ¼ d(2nþ À 1) (12:4:7)
138 Agent-Based Modeling of Financial Markets

The model described above is defined with two observable vari-
ables, nþ (t) and p(t). In equilibrium, its solution is nþ ¼ 0:5 and
p ¼ 1. The necessary stability condition for this model is
1 (12:4:8)
The typical stable solution for this model (relaxation of the initially
perturbed values of nþ and p) is given in Figure 12.1. Lower values of
a and g suppress oscillations and facilitate relaxation of the initial
perturbations. Thus, the rise of the ˜˜chartist™™ component in the utility
function increases the price volatility. Numerical solutions with the
values of a and g that slightly violate the condition (12.4.8) can lead to
the limit cycle providing that the initial conditions are very close to
the equilibrium values (see Figure 12.2). Otherwise, violation of the
condition (12.4.8) leads to system instability, which can be interpreted
as a market crash.
The basic model (12.4.1)“(12.4.7) can be extended in several
ways. First, the condition of the constant number of traders (12.4.1)

1.2 0.4

1.15 0.3

1.1 0.2

1.05 0.1


1 0



Time Dex
0 5 10 15 20 25 30 35 40 45
Figure 12.1 Dynamics of excess demand (Dex) and price for the model
(12.4.5)“(12.4.7) with a ¼ b ¼ g ¼ 1, nþ(0) ¼ 0.4 and p(0) ¼ 1.05.
Agent-Based Modeling of Financial Markets

1.6 1

1.2 0.4



0 5 10 15 20 25 30 35 40 45 50 55 60
Figure 12.2 Dynamics of excess demand (Dex) and price for the model
(12.4.5)“(12.4.7) with a ¼ 1.05, b ¼ g ¼ 1, nþ(0) ¼ 0.4 and p(0) ¼ 1.05.

can be dropped. The system has three variables (nþ , nÀ , p) and
therefore may potentially describe deterministic chaos (see Chapter
7). Also, one can randomize the model by adding noise to the utility
function (12.4.4) or to the price formation equation (12.4.6). Interest-
ingly, the latter option may lead to a negative correlation between
price and excess demand, which is not possible for the deterministic
equation (12.4.6) [17].


A simple extension of the basic model (12.4.1)“(12.4.7) provides
some explanation as to why technical trading may sometimes be
successful [18]. Consider a system with a constant number of traders
N that consists of ˜˜regular™™ traders NR and ˜˜technical™™ traders
NT : NT þ NR ¼ N ¼ const. The ˜˜regular™™ traders are divided into
buyers, Nþ (t), and sellers, NÀ (t): Nþ þ NÀ ¼ NR ¼ const. The rela-
tive numbers of ˜˜regular™™ traders, nþ (t) ¼ Nþ (t)=N and
nÀ (t) ¼ NÀ (t)=N, are described with the equations (12.4.2)“(12.4.4).
The price formation in equation (12.4.6) is also retained. However,
140 Agent-Based Modeling of Financial Markets

the excess demand, in contrast to (12.4.7), incorporates the ˜˜tech-
nical™™ traders
Dex ¼ d(nþ À nÀ þ FnT ) (12:4:9)
In (12.4.9), nT ¼ NT =N and function F is defined by the technical
trader strategy. We have chosen a simple technical rule ˜˜buying on
dips “ selling on tops,™™ that is, buying at the moment when the price
starts rising, and selling at the moment when price starts falling
< 1, p(k) > p(k À 1) and p(k À 1) < p(k À 2)
F(k) ¼ À1, p(k) < p(k À 1) and p(k À 1) > p(k À 2) (12:4:10)
0, otherwise
Figure 12.3 shows that inclusion of the ˜˜technical™™ traders in the
model strengthens the price oscillations. This result can be easily
interpreted. If ˜˜technical™™ traders decide that price is going to fall,
they sell and thus decrease demand. As a result, price does fall and
the ˜˜chartist™™ mood of ˜˜regular™™ traders forces them to sell. This
suppresses price further until the ˜˜fundamentalist™™ motivation of







nT = 0
nT = 0.005
1 5 9 13 17 21 25 29 33 37 41 45 49
Figure 12.3 Price dynamics for the technical strategy (12.4.10) for
a ¼ g ¼ d ¼ n ¼ 1 and b ¼ 4 with initial conditions nþ(0) ¼ 0.4 and p(0)
¼ 1.05.
Agent-Based Modeling of Financial Markets

˜˜regular™™ traders becomes overwhelming. The opposite effect occurs
if ˜˜technical™™ traders decide that it is time to buy: they increase
demand and price starts to grow until it notably exceeds its funda-
mental value. Hence, if the ˜˜technical™™ traders are powerful enough in
terms of trading volumes, their concerted action can sharply change
demand upon ˜˜technical™™ signal. This provokes the ˜˜regular™™ traders
to amplify a new trend, which moves price in the direction favorable
to the ˜˜technical™™ strategy.


Market liquidity implies the presence of traders on both the bid/ask
sides of the market. In emergent markets (e.g., new electronic
auctions), this may be a matter of concern. To address this problem,
the basic model (12.4.1)“(12.4.7) was expanded in the following way
dnþ =dt ¼ vþÀ nÀ À vÀþ nþ þ SRþi þ rþ (12:4:11)
dnÀ =dt ¼ vÀþ nþ À vþÀ nÀ þ SRÀi þ rÀ (12:4:12)
The functions RÆi (i ¼ 1, 2, . . . , M) and rÆ are the deterministic and
stochastic rates of entering and exiting the market, respectively. Let us
consider three deterministic effects that define the total number of
traders.6 First, we assume that some traders stop trading immediately
after completing a trade as they have limited resources and/or need
some time for making new decisions
Rþ1 ¼ RÀ1 ¼ Àbnþ nÀ , b > 0 (12:4:13)
Also, we assume that some traders currently present in the market will
enter the market again and will possibly bring in some ˜˜newcomers.™™
Therefore, the inflow of traders is proportional to the number of
traders present in the market
Rþ2 ¼ RÀ2 ¼ a(nþ þ nÀ ), a > 0 (12:4:14)
Lastly, we account for ˜˜unsatisfied™™ traders leaving the market.
Namely, we assume that those traders who are not able to find the
trading counterparts within a reasonable time exit the market
142 Agent-Based Modeling of Financial Markets


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