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Ã€c(nÃ¾ Ã€ nÃ€ ) if nÃ¾ > nÃ€

RÃ¾3 Â¼

0, if nÃ¾ nÃ€

Ã€c(nÃ€ Ã€ nÃ¾ ) if nÃ€ > nÃ¾

RÃ€3 Â¼ (12:4:15)

0, if nÃ€ nÃ¾

We call the parameter c > 0 the â€˜â€˜impatienceâ€™â€™ factor. Here, we neglect

the price variation, so that vÃ¾Ã€ Â¼ vÃ€Ã¾ Â¼ 0. We also neglect the sto-

chastic rates rÃ† . Let us specify

nÃ¾ (0) Ã€ nÃ€ (0) Â¼ d > 0: (12:4:16)

Then equations (12.4.11)â€“(12.4.12) have the following form

dnÃ¾ =dt Â¼ a(nÃ¾ Ã¾ nÃ€ ) Ã€ bnÃ¾ nÃ€ Ã€ c(nÃ¾ Ã€ nÃ€ ) (12:4:17)

dnÃ€ =dt Â¼ a(nÃ¾ Ã¾ nÃ€ ) Ã€ bnÃ¾ nÃ€ (12:4:18)

The equation for the total number of traders n Â¼ nÃ¾ Ã¾ nÃ€ has the

Riccati form7

dn=dt Â¼ 2an Ã€ 0:5bn2 Ã¾ 0:5bd2 exp (Ã€2ct) Ã€ cd exp (Ã€ct) (12:4:19)

Equation (12.4.19) has the asymptotic solution

n0 Â¼ 4a=b (12:4:20)

An example of evolution of the total number of traders (in units of n0 )

is shown in Figure 12.4 for different values of the â€˜â€˜impatienceâ€™â€™

factor. Obviously, the higher the â€˜â€˜impatienceâ€™â€™ factor, the deeper the

minimum of n(t) will be. At sufficiently high â€˜â€˜impatienceâ€™â€™ factor, the

finite-difference solution to equation (12.4.19) falls to zero. This

means that the market dies out due to trader impatience. However,

the exact solution never reaches zero and always approaches the

asymptotic value (12.4.20) after passing the minimum. This demon-

strates the drawback of the continuous approach. Indeed, a non-zero

number of traders that is lower than unity does not make sense. One

way around this problem is to use a threshold, nmin , such that

n Ã† (t) Â¼ 0 if n Ã† (t) < nmin (12:4:21)

Still, further analysis shows that the discrete analog of the system

(12.4.17)â€“(12.4.18) may be more adequate than the continuous model

[19].8

143

Agent-Based Modeling of Financial Markets

1

n/no

0.9

0.8

0.7

1

0.6

2

0.5

3

0.4

0.3

0.2

0.1

Time

0

0 2 4 6 8 10 12 14 16

Figure 12.4 Dynamics of the number of traders described with equation

(12.4.19) with a Â¼ 0.25, b Â¼ 1, nÃ¾(0) Â¼ 0.2, and nÃ€(0) Â¼ 0.1: 1 - c Â¼ 1; 2 - c Â¼

10; 3 - c Â¼ 20.

12.5 REFERENCES FOR FURTHER READING

Reviews [1, 5] and the recent collection [6] might be a good starting

point for deeper insight into this quickly evolving field.

12.6 EXERCISES

**

1. Discuss the derivation of the GARCH process with the agent-

based model [21].

**

2. Discuss the insider trading model [22]. How would you model

agents having knowledge of upcoming large block trades?

**

3. Discuss the parsimony problem in agent-based modeling of

financial markets (use [16] as the starting point).

**

4. Discuss the agent-based model of business growth [23].

**

5. Verify if the model (12.4.1)â€“(12.4.7) exhibits a price distribu-

tion with fat tails.

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Comments

CHAPTER 1

1. The author calls this part academic primarily because he has difficulty

answering the question â€˜â€˜So, how can we make some money with this

stuff?â€™â€™ Undoubtedly, â€˜â€˜money-makingâ€™â€™ mathematical finance has deep

academic roots.

2. Lots of information on the subject can also be found on the websites

http://www.econophysics.org and http://www.unifr.ch/econophysics.

3. Still, Section 7.1 is a useful precursor for Chapter 12.

4. It should be noted that scientific software packages such as Matlab and

S-Plus (let alone â€˜â€˜in-houseâ€™â€™ software developed with C/CÃ¾Ã¾) are often

used for sophisticated financial data analysis. But Excel, having a wide

array of built-in functions and programming capabilities with Visual

Basic for Applications (VBA) [13], is ubiquitously employed in the finan-

cial industry.

CHAPTER 2

1. In financial literature, return is sometimes defined as [P(t) Ã€ P(tÃ€1)] while

the variable R(t) in (2.2.1) is named rate of return.

2. For the formal definition of IID, see Section 5.1.

3. USD/JPY denotes the price of one USD in units of JPY, etc.

4. Technical analysis is based on the seeking and interpretation of patterns

in past prices [7]. Fundamental analysis is evaluation the companyâ€™s

145

146 Comments

business quality based on its growth expectations, cash flow, and so

on [8].

5. Arbitrage trading strategies are discussed in Section 10.4.

6. An instructive discussion on EMH and rational bubbles is given also on

L. Tesfatsionâ€™s website: http://www.econ.iastate.edu/classes/econ308/tes-

fatsion/emarketh.htm.

CHAPTER 4

1. In the physical literature, the diffusion coefficient is often defined as

D Â¼ kT=(6pZR). Then E[r2 ] Ã€ r0 2 Â¼ 6Dt.

2. The general case of the random walk is discussed in Section 5.1.

3. Here we simplify the notations: m(t) Â¼ m, s(y(t), t) Â¼ s.

4. The notation y Â¼ O(x) means that y and x are of the same asymptotic

order, that is, 0 < lim [y(t)=x(t)] < 1.

t!0

CHAPTER 5

1. See http://econ.la.psu.edu/$hbierens/EASYREG.HTM.

CHAPTER 7

1. Ironically, markets may react unexpectedly even at â€˜â€˜expectedâ€™â€™ news.

Consider a Federal Reserve interest rate cut, which is an economic

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