ñòð. 4 |

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 Â¼ Ã€G=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

yÂ¼ ,rÂ¼ (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 [1â€“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 â€˜â€˜smile-likeâ€™â€™ 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

105

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.

9.5 REFERENCES FOR FURTHER READING

Hullâ€™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].

9.6 APPENDIX: THE INVARIANT

OF THE ARBITRAGE-FREE PORTFOLIO

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 MÃ€1 (see,

0

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 Â¼ 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 â€˜â€˜bestâ€™â€™ price, then the

order is executed within a price range rather than at a single price. In

this case within continuous presentation,

S Â¼ S(t, N) (9:6:2)

and the expense of buying N shares at time t equals

Ã°

N

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

0

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) Â¼ NS(t) Ã¾ 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 Â¼ S(t) is described with the geometric

Brownian process

dS Â¼ mSdt Ã¾ 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 Itoâ€™s Lemma,

s2 2 @ 2 F

@F @F @F

dF Â¼ dt Ã¾ dS Ã¾ S dt Ã¾ dN (9:6:6)

@S2

2

@t @S @N

It follows from (9.6.4) that the portfolio dynamic is

dP Â¼ MdF Ã¾ NdS Ã¾ SdN (9:6:7)

Substituting equation (9.6.6) into equation (9.6.7) yields

@F s2 2 @ 2 F

@F @F

dP Â¼ [M Ã¾ N]dS Ã¾ [M Ã¾ S]dN Ã¾ M Ã¾S dt

@S2

2

@S @N @t

(9:6:8)

107

Option Pricing

As within BST, the arbitrage-free portfolio grows with the risk-free

interest rate, r

dP Â¼ rPdt (9:6:9)

Then the combination of equation (9.6.8) and equation (9.6.9)

yields

@F s2 2

2@ F

@F

Ã¾ N]dS Ã¾ [M Ã¾ MS Ã€ rMF Ã€ rNS]dtÃ¾

[M

@S2

2

@S @t (9:6:10)

@F

Ã¾ S]dN Â¼ 0

[M

@N

Since equation (9.6.10) must be valid for arbitrary values of dS, dt

and dN, it can be split into three equations

@F

Ã¾NÂ¼0

M (9:6:11)

@S

@F s2 2 @ 2 F

Ã¾S Ã€ rF Ã€ rNS Â¼ 0

M (9:6:12)

@S2

2

@t

@F

Ã¾SÂ¼0

M (9:6:13)

@N

Let us present F(S, t, N) in the form

F(S, t, N) Â¼ F0 (S, t)Z(N) (9:6:14)

where Z(N) satisfies the condition

Z(1) Â¼ 1 (9:6:15)

Then it follows from equation (9.6.11) that

@F0

M Â¼ Ã€N= Z (9:6:16)

:

@S

This transforms equation (9.6.15) and equation (9.6.16), respectively,

to

@F0 s2 2 @ 2 F0

@F0

Ã¾ rS Ã¾S Ã€ rF0 Â¼ 0 (9:6:17)

@S 2

2

@t @S

dZ @F0

Â¼ (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) Â¼ Na (9:6:19)

@F0

Â¼ Ã€MÃ€1 . Equation (9.6.13) and equa-

where a Â¼ (S=F0 )D, D Â¼ 0

@S

tion (9.6.16) yield

M Â¼ Ã€N1Ã€a =D Â¼ N1Ã€a M0 (9:6:20)

Hence, the option price in the arbitrage-free portfolio with N shares

equals

F(S, t, N) Â¼ F0 (S, t)Na (9:6:21)

It coincides with the BST solution F0 (S, t) only if N Â¼ 1, that is when

the portfolio has one share. However, the total expense of hedging N

shares in the arbitrage-free portfolio

Q Â¼ MF Â¼ Ã€(N=D)F0 Â¼ 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 Â¼ 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 Â¼ 0:5mV2 . Yet, this law alone cannot be used for calculating the

bodyâ€™s mass, m, and velocity, V. Note, however, that if a body has

unit mass (m Â¼ 1), then the energy conservation law effectively yields

the bodyâ€™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.

109

Option Pricing

9.7 EXERCISES

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 Â¼ 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 Â¼ 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 Itoâ€™s and Stratonovichâ€™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

10.4.

10.1 PORTFOLIO SELECTION

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 â€˜â€˜long

runâ€™â€™ is doomed to bear an element of uncertainty. Alas, a decade of

111

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] Â¼ aE[Rr ] Ã¾ (1 Ã€ a)Rf Â¼ Rf Ã¾ a(E[Rr ] Ã€ 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 Â¼ asr (10:1:2)

Substituting a from (10.1.2) into (10.1.1) yields

E[R] Â¼ Rf Ã¾ s(E[Rr ] Ã€ 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

(10.1.3)

s Â¼ (E[Rr ] Ã€ 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] Â¼ gE[R1 ] Ã¾ (1 Ã€ g)E[R2 ] (10:1:5)

113

Portfolio Management

and the portfolio standard deviation is

s2 Â¼ g2 s1 2 Ã¾ (1 Ã€ g)2 s2 2 Ã¾ 2g(1 Ã€ 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 Â¼ 0. The value of g that yields minimal risk

for this portfolio equals

gm Â¼ s2 2 =(s1 2 Ã¾ s2 2 ), (10:1:7)

This value yields the minimal portfolio risk sm

sm 2 Â¼ s1 2 s2 2 =(s1 2 Ã¾ s2 2 ) (10:1:8)

Consider an example with E[R1 ] Â¼ 0:1, E[R2 ] Â¼ 0:2, s1 Â¼ 0:15,

s2 Â¼ 0:3. If g Â¼ 0:8, then s % 0:134 < s1 and E[R] Â¼ 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] Â¼ 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 Â¼ (E[R1 ] Ã€ Rf )s2 2 ={(E[R1 ] Ã€ Rf )s2 2 Ã¾ (E[R2 ] Ã€ Rf )s1 2 }

(10:1:9)

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

â€˜â€˜as good as it gets.â€™â€™ 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 Â¼ gT . Obviously, the investment risk is then higher than that of

sT .

114 Portfolio Management

0.16

E[R]

P

T

0.12

0.08

RF

0.04

sigma

0

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 Â¼ 0:05, s1 Â¼ 0:12, s2 Â¼ 0:15, E[R1 ] Â¼ 0:08, E[R2 ] Â¼ 0:14.

10.2 CAPITAL ASSET PRICING MODEL (CAPM)

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

115

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

line

E[Ri ] Â¼ Rf Ã¾ bi (E[RM ] Ã€ Rf ) (10:2:1)

where RM is the market portfolio return and parameter beta bi equals

bi Â¼ 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 Â¼ (E[Ri ] Ã€ 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 Â¼ Ri Ã€ Rf

Zi (t) Â¼ ai Ã¾ bi ZM (t) Ã¾ 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.

10.3 ARBITRAGE PRICING THEORY (APT)

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

paradigm.

APT is based on two postulates. First, the return for an asset i

(i Â¼ 1, . . . , N) at every time period is a weighed sum of the risk factor

contributions fj (t) (j Â¼ 1, . . . , K, K < N) plus an asset-specific com-

ponent ei (t)

Ri (t) Â¼ ai Ã¾ bi1 f1 Ã¾ bi2 f2 Ã¾ . . . Ã¾ biK fK Ã¾ 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)] Â¼ E[f2 (t)] Â¼ . . . Â¼ E[fK (t)] Â¼ E[ei (t)] Â¼ 0 (10:3:2)

Also, the time distributions of the risk factors and asset-specific

innovations are independent

0

Cov[fj (t), fj (t0 )] Â¼ 0, Cov[ei (t), ei (t0 )] Â¼ 0, t 6Â¼ t (10:3:3)

and uncorrelated

Cov[fj (t), ei (t)] Â¼ 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.

117

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 Ã¾ 1 constants l0 , l1 , . . . lK (not all of them equal zero)

that

E[Ri (t)] Â¼ l0 Ã¾ bi1 l1 Ã¾ . . . Ã¾ 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

P

N

wi Â¼ 1, so that wi < W=N

of N assets with the weights wi where

iÂ¼1

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) Â¼ a Ã¾ b1 f1 Ã¾ b2 f2 Ã¾ . . . Ã¾ bK fK Ã¾ e(t) (10:3:6)

where

X X

N N

aÂ¼ wi a i , b i Â¼ wk bik (10:3:7)

kÂ¼1

iÂ¼1

then the expected portfolio return is

E[R(t)] Â¼ l0 Ã¾ b1 l1 Ã¾ . . . Ã¾ 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].

10.4 ARBITRAGE TRADING STRATEGIES

The simple investment strategy means â€˜â€˜buy and holdâ€™â€™ securities

of â€˜â€˜goodâ€™â€™ companies until their performance worsens, then sell them

and buy better assets. A more sophisticated approach is sensitive to

changing economic environment and an investorâ€™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 â€˜â€˜long run.â€™â€™ 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.

119

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

overpriced.

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

strategy.

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 managerâ€™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 â€˜â€˜across the boardâ€™â€™ is technically more challenging

but potentially rewarding.

Some academic research on efficiency of the arbitrage trading

strategies can be found in [9â€“12] and references therein. Note that

the research methodology in this field is itself a non-trivial problem

[13].

10.5 REFERENCES FOR FURTHER READING

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].

10.6 EXERCISES

1. Consider a portfolio with two assets having the following

returns and standard deviations: E[R1 ] Â¼ 0:15, E[R2 ] Â¼ 0:1,

s1 Â¼ 0:2, s2 Â¼ 0:15. The proportion of asset 1 in the portfolio

g Â¼ 0:5. Calculate the portfolio return and standard deviation.

The correlation coefficient between assets is (a) 0.5; (b) Ã€0.5.

2. Consider returns of stock A and the market portfolio M in three

years:

A Ã€7% 12% 26%

M Ã€5% 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) Â¼ ai Ã¾ bi1 f1 Ã¾ bi2 f2 Ã¾ ei (t), construct a portfolio with

three stocks (i.e., define w1 , w2 , and w3 Â¼ 1 Ã€ w1 Ã€ 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.

11.1 RISK MEASURES

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

121

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

1

PN (x) Â¼ pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp [Ã€(x Ã€ m)2 =2s2 ] (11:1:1)

2ps

where m and s are the mean and standard deviation, respectively.

Then for the chosen confidence level a,

VAR(a) Â¼ Ã€sza Ã€ m (11:1:2)

The value of za can be determined from the cumulative distribution

function for the standard normal distribution (3.2.10)

Ã°

za

1

pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ exp [Ã€z2 =2]dz Â¼ 1 Ã€ a

Pr(Z < za ) Â¼ (11:1:3)

2p

Ã€1

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 > Ã€sza ) for given a move VaR(a) into the negative region,

which implies profits rather than losses. Examples of za for typical

values of a Â¼ 95% and a Â¼ 99% are given in Figure 11.1. Note that

the return variance s corresponds to za Â¼ Ã€1 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

123

Market Risk Measurement

0.45

0.4

0.35

VaR at 84%

z = âˆ’1

0.3

0.25

VaR at 95%

z = âˆ’1.64

0.2

0.15

VaR at 99%

z = âˆ’2.33

0.1

0.05

0

âˆ’5 âˆ’4 âˆ’3 âˆ’2 âˆ’1 0 1 2 3 4 5

Figure 11.1 VaR for the standard normal probability distribution of P/L.

r(A Ã¾ B) r(A) Ã¾ 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) Â¼ lr(A), l > 0 (homogeneity) (11:1:5)

r(A) r(B), if A B (monotonicity) (11:1:6)

r(A Ã¾ C) Â¼ r(A) Ã€ 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)â€“(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 â€˜â€˜generalized scenariosâ€™â€™ 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 Â¼ 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 â€˜â€˜worst case scenarioâ€™â€™ 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.

0.45

0.4

0.35

ETL at 84%

z = âˆ’1.52

0.3

0.25

ETL at 95%

z = âˆ’2.06

0.2

0.15

ETL at 99%

z = âˆ’2.66

0.1

0.05

0

âˆ’5 âˆ’4 âˆ’3 âˆ’2 âˆ’1 0 1 2 3 4 5

Figure 11.2 ETL for the standard normal probability distribution of P/L.

125

Market Risk Measurement

11.2 CALCULATING RISK

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 Ã€ a) Ã¾ 1] lowest observation and ETL is

the average of N(1 Ã€ a) smallest observations.

The well-known problem with the historical simulation is handling

of old data. First, â€˜â€˜too oldâ€™â€™ 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 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

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 â€˜â€˜Maxwellâ€™s Demonâ€™â€™ who is

137

Agent-Based Modeling of Financial Markets

able to instantly parse all market events. The Demon cannot discern

â€˜â€˜chartistsâ€™â€™ and â€˜â€˜fundamentalistsâ€™â€™ 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 â€˜â€˜buyersâ€™â€™ (Ã¾)

and â€˜â€˜sellersâ€™â€™ (Ã€). Total number of traders is N

NÃ¾ (t) Ã¾ NÃ€ (t) Â¼ N (12:4:1)

The scaled numbers of buyers, nÃ¾ (t) Â¼ NÃ¾ (t)=N, and sellers, nÃ€ (t)

Â¼ NÃ€ (t)=N, are described with equations

dnÃ¾ =dt Â¼ vÃ¾Ã€ nÃ€ Ã€ vÃ€Ã¾ nÃ¾ (12:4:2)

dnÃ€ =dt Â¼ vÃ€Ã¾ nÃ¾ Ã€ vÃ¾Ã€ nÃ€ (12:4:3)

The factors vÃ¾Ã€ and vÃ€Ã¾ characterize the probabilities for transfer

from seller to buyer and back, respectively

vÃ¾Ã€ Â¼ 1=vÃ€Ã¾ Â¼ n exp (U), U Â¼ apÃ€1 dp=dt Ã¾ b(1 Ã€ 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 â€˜â€˜chartistâ€™â€™ behavior while the

second term describes the â€˜â€˜fundamentalistâ€™â€™ pattern. The factor n has

the sense of the frequency of transitions between seller and buyer

behavior. Since nÃ¾ (t) Â¼ 1 Ã€ nÃ€ (t), the system (12.4.1)â€“(12.4.3) is re-

duced to the equation

dnÃ¾ =dt Â¼ vÃ¾Ã€ (1 Ã€ nÃ¾ ) Ã€ vÃ€Ã¾ nÃ¾ (12:4:5)

The price formation equation is assumed to have the following

form

dp=dt Â¼ gDex (12:4:6)

where the excess demand, Dex , is proportional to the excess number of

buyers

Dex Â¼ d(nÃ¾ Ã€ nÃ€ ) Â¼ d(2nÃ¾ Ã€ 1) (12:4:7)

138 Agent-Based Modeling of Financial Markets

12.4.2 PRICE-DEMAND RELATIONS

The model described above is defined with two observable vari-

ables, nÃ¾ (t) and p(t). In equilibrium, its solution is nÃ¾ Â¼ 0:5 and

p Â¼ 1. The necessary stability condition for this model is

1 (12:4:8)

adgn

The typical stable solution for this model (relaxation of the initially

perturbed values of nÃ¾ 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 â€˜â€˜chartistâ€™â€™ 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)â€“(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

Price

Dex

1 0

âˆ’0.1

0.95

âˆ’0.2

0.9

âˆ’0.3

0.85

Price

Time Dex

âˆ’0.4

0.8

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)â€“(12.4.7) with a Â¼ b Â¼ g Â¼ 1, nÃ¾(0) Â¼ 0.4 and p(0) Â¼ 1.05.

139

Agent-Based Modeling of Financial Markets

1.6 1

0.8

1.4

0.6

1.2 0.4

0.2

1

Price

Dex

0

0.8

âˆ’0.2

âˆ’0.4

0.6

âˆ’0.6

0.4

âˆ’0.8

Price

Time

Dex

âˆ’1

0.2

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)â€“(12.4.7) with a Â¼ 1.05, b Â¼ g Â¼ 1, nÃ¾(0) Â¼ 0.4 and p(0) Â¼ 1.05.

can be dropped. The system has three variables (nÃ¾ , nÃ€ , 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].

12.4.3 WHY TECHNICAL TRADING SUCCESSFUL

MAY BE

A simple extension of the basic model (12.4.1)â€“(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 â€˜â€˜regularâ€™â€™ traders NR and â€˜â€˜technicalâ€™â€™ traders

NT : NT Ã¾ NR Â¼ N Â¼ const. The â€˜â€˜regularâ€™â€™ traders are divided into

buyers, NÃ¾ (t), and sellers, NÃ€ (t): NÃ¾ Ã¾ NÃ€ Â¼ NR Â¼ const. The rela-

tive numbers of â€˜â€˜regularâ€™â€™ traders, nÃ¾ (t) Â¼ NÃ¾ (t)=N and

nÃ€ (t) Â¼ NÃ€ (t)=N, are described with the equations (12.4.2)â€“(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 â€˜â€˜tech-

nicalâ€™â€™ traders

Dex Â¼ d(nÃ¾ Ã€ nÃ€ Ã¾ FnT ) (12:4:9)

In (12.4.9), nT Â¼ NT =N and function F is defined by the technical

trader strategy. We have chosen a simple technical rule â€˜â€˜buying on

dips â€“ selling on tops,â€™â€™ that is, buying at the moment when the price

starts rising, and selling at the moment when price starts falling

8

< 1, p(k) > p(k Ã€ 1) and p(k Ã€ 1) < p(k Ã€ 2)

F(k) Â¼ Ã€1, p(k) < p(k Ã€ 1) and p(k Ã€ 1) > p(k Ã€ 2) (12:4:10)

:

0, otherwise

Figure 12.3 shows that inclusion of the â€˜â€˜technicalâ€™â€™ traders in the

model strengthens the price oscillations. This result can be easily

interpreted. If â€˜â€˜technicalâ€™â€™ traders decide that price is going to fall,

they sell and thus decrease demand. As a result, price does fall and

the â€˜â€˜chartistâ€™â€™ mood of â€˜â€˜regularâ€™â€™ traders forces them to sell. This

suppresses price further until the â€˜â€˜fundamentalistâ€™â€™ motivation of

1.08

1.06

1.04

1.02

Price

1

0.98

0.96

nT = 0

nT = 0.005

0.94

Time

0.92

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 Â¼ g Â¼ d Â¼ n Â¼ 1 and b Â¼ 4 with initial conditions nÃ¾(0) Â¼ 0.4 and p(0)

Â¼ 1.05.

141

Agent-Based Modeling of Financial Markets

â€˜â€˜regularâ€™â€™ traders becomes overwhelming. The opposite effect occurs

if â€˜â€˜technicalâ€™â€™ 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 â€˜â€˜technicalâ€™â€™ traders are powerful enough in

terms of trading volumes, their concerted action can sharply change

demand upon â€˜â€˜technicalâ€™â€™ signal. This provokes the â€˜â€˜regularâ€™â€™ traders

to amplify a new trend, which moves price in the direction favorable

to the â€˜â€˜technicalâ€™â€™ strategy.

12.4.4 THE BIRTH LIQUID MARKET

OF A

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)â€“(12.4.7) was expanded in the following way

[19]

dnÃ¾ =dt Â¼ vÃ¾Ã€ nÃ€ Ã€ vÃ€Ã¾ nÃ¾ Ã¾ SRÃ¾i Ã¾ rÃ¾ (12:4:11)

dnÃ€ =dt Â¼ vÃ€Ã¾ nÃ¾ Ã€ vÃ¾Ã€ nÃ€ Ã¾ SRÃ€i Ã¾ rÃ€ (12:4:12)

The functions RÃ†i (i Â¼ 1, 2, . . . , M) and rÃ† 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

RÃ¾1 Â¼ RÃ€1 Â¼ Ã€bnÃ¾ nÃ€ , 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 â€˜â€˜newcomers.â€™â€™

Therefore, the inflow of traders is proportional to the number of

traders present in the market

RÃ¾2 Â¼ RÃ€2 Â¼ a(nÃ¾ Ã¾ nÃ€ ), a > 0 (12:4:14)

Lastly, we account for â€˜â€˜unsatisfiedâ€™â€™ 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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