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stimulus. One may expect market rally after its announcement. However,
prices might have already grown in anticipation of this event. Then
investors may start immediate profit taking, which leads to falling prices.
2. In the case with g < 0, the system has an energy source and the trajectory
is an unbounded outward spiral.

1. See, for example, [1] and references therein. Note that the GARCH
models generally assume that the unconditional innovations are
2. While several important findings have been reported after publishing
[2], I think this conclusion still holds. On a philosophical note, statistical
data analysis in general is hardly capable of attaining perfection of
mathematical proof. Therefore, scholars with the ˜˜hard-science™™

background may often be dissatisfied with rigorousness of empirical
3. There has been some interesting research on the distribution of the
company sizes [3, 4].
4. The foreign exchange data available to academic research are overwhelm-
ingly bank quotes (indicative rates) rather than the real inter-bank trans-
action rates (so-called firm rates) [5].

1. In financial literature, derivatives are also called contingent claims.
2. The names of the American and European options refer to the exercising
rule and are not related to geography. Several other types of options with
complicated payoff rules (so-called exotic options) have been introduced
in recent years [1À3].
3. The U.S. Treasury bills are often used as a benchmark for the risk-free
4. Here and further, the transaction fees are neglected.
5. We might choose also one share and À options.

1. See Chapter 11.
2. Qualitative graphical presentation of the efficient frontier and the capital
market line is similar to the trade-off curve and the trade-off straight line,
respectively, depicted in Figure 10.1.
3. Usually, Standard and Poor™s 500 Index is used as proxy for the U.S.
market portfolio.
4. ROE ¼ E/B where E is earnings; B is the book value that in a nutshell
equals the company™s assets minus its debt.

1. In risk management, the self-explanatory notion of P/L is used rather
than return.
2. In the current literature, the following synonyms of ETL are sometimes
used: expected shortfall and conditional VaR [2].
3. EWMA or GARCH are usually used for the historical volatility forecasts
(see Section 4.3).

1. Lots of useful information on agent-based computational economics are
present on L. Tesfatsion™s website: http://www.econ.iastate.edu/tesfatsi/
ace.htm. Recent developments in this field can also be found in the
materials of the regularly held Workshops on Economics and Heteroge-
neous Interacting Agents (WEHIA), see, for example, http://www.nda.
2. I have listed the references to several important models. Early research
and recent working papers on the agent-based modeling of financial
markets can be found on W. A. Brock™s (http://www.ssc.wisc.edu/
C. Chiarella™s (http://www.business.uts.edu.au/finance/staff/carl.html),
J. D. Farmer™s (http://www.santafe.edu/$jdf),
B. LeBaron™s (http://people.brandeis.edu/$blebaron/index.htm),
T. Lux™s (http://www.bwl.uni-kiel.de/vwlinstitute/gwrp/team/lux.htm), and
S. Solomon™s (http://shum.huji.ac.il/$sorin/) websites.
3. In a more consistent yet computationally demanding formulation, the
function fi depends also on current return rt , that is, Ei , t[rtþ1 ] ¼
fi (rt , . . . , rtÀLi ) [8, 9].
4. Log price in the left-hand side of equation (12.3.7) may be a better choice
in order to avoid possible negative price values [12].
5. See also Section 7.1.
6. This model has some similarity with the mating dynamics model where
only agents of opposite sex interact and deactivate each other, at least
temporarily. In particular, this model could be used for describing at-
tendance of the singles™ clubs.
7. Equation (12.4.19) can be transformed into the Schrodinger equation
with the Morse-type potential [19].
8. Another interesting example of qualitative difference between the con-
tinuous and discrete evolutions of the same system is given in [20].

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