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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.


CHAPTER 8
1. See, for example, [1] and references therein. Note that the GARCH
models generally assume that the unconditional innovations are
normal.
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™™
147
Comments



background may often be dissatisfied with rigorousness of empirical
research.
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].


CHAPTER 9
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
asset.
4. Here and further, the transaction fees are neglected.
 À1
@F
5. We might choose also one share and À options.
@S

CHAPTER 10
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.


CHAPTER 11
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).
148 Comments



CHAPTER 12
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.
ac.jp/cs/AI/wehia04.
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/
$wbrock/),
C. Chiarella™s (http://www.business.uts.edu.au/finance/staff/carl.html),
J. D. Farmer™s (http://www.santafe.edu/$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].
References




CHAPTER 1
1. J. Y. Campbell, A. W. Lo, and A. C. MacKinlay, The Econometrics of
Financial Markets, Princeton University Press, 1997.
2. W. H. Green, Econometric Analysis, Prentice Hall, 1998.
3. S. R. Pliska, Introduction to Mathematical Finance: Discrete Time
Models, Blackwell, 1997.
4. S. M. Ross, Elementary Introduction to Mathematical Finance: Options
and Other Topics, Cambridge University Press, 2002.
5. R. N. Mantegna and H. E. Stanley, An Introduction in Econophysics: Cor-
relations and Complexity in Finance, Cambridge University Press, 2000.
6. J. P. Bouchaud and M. Potters, Theory of Financial Risks: From Statis-
tical Physics to Risk Management, Cambridge University Press, 2000.
7. M. Levy, H. Levy, and S. Solomon, The Microscopic Simulation of
Financial Markets: From Investor Behavior to Market Phenomena, Aca-
demic Press, 2000.
8. K. Ilinski, Physics of Finance: Gauge Modeling in Non-Equilibrium
Pricing, Wiley, 2001.
9. J. Voit, Statistical Mechanics of Financial Markets, Springer, 2003.
10. D. Sornette, Why Stock Markets Crash: Critical Events in Complex
Financial Systems, Princeton University Press, 2003.
11. S. Da Silva (Ed), The Physics of the Open Economy, Nova Science, 2005.
12. B. LeBaron, ˜˜Agent-Based Computational Finance: Suggested Read-
ings and Early Research,™™ Journal of Economic Dynamics and Control
24, 679“702 (2000).



149
150 References



13. M. Jackson and M. Staunton, Advanced Modeling in Finance Using
Excel and VBA, Wiley, 2001.


CHAPTER 2
1. C. Alexander, Market Models: A Guide to Financial Data Analysis,
Wiley, 2001.
2. M. M. Dacorogna, R. Gencay, U. Muller, R. B. Olsen, and O. V. Pictet,
An Introduction to High-Frequency Finance, Academic Press, 2001.
3. See [1.1].
4. T. Lux and D. Sornette: ˜˜On Rational Bubbles and Fat Tails,™™ Journal
of Money, Credit, and Banking 34, 589-610 (2002).
5. R. C. Merton, Continuous Time Finance, Blackwell, 1990.
6. Z. Bodie and R. C. Merton, Finance, Prentice Hall, 1998.
7. R. Edwards and J. Magee, Technical Analysis of Stock Trends, 8th Ed.,
AMACOM, 2001.
8. S. Cottle, R. F. Murray, and F. E. Block, Security Analysis, McGraw-
Hill, 1988.
9. B. G. Malkiel, A Random Walk Down Wall Street, Norton, 2003.
10. R. J. Shiller, Irrational Exuberance, Princeton University Press, 2000.
11. E. Peters, Chaos and Order in Capital Markets, Wiley, 1996.
12. A. W. Lo and A. C. MacKinlay, A Non-Random Walk Down Wall
Street, Princeton University Press, 1999.
13. See [1.9].
14. D. Kahneman and A. Tversky (Eds.), Choices, Values and Frames,
Cambridge University Press, 2000.
15. R. H. Thaler (Ed), Advances in Behavioral Finance, Russell Sage Foun-
dation, 1993.
16. D. Kahneman and A. Tversky: ˜˜Prospect Theory: An Analysis of Decision
Under Risk,™™ Econometrica 47, 263-291 (1979). See also [14], pp. 17“43.
17. M. A. H. Dempster and C. M. Jones: ˜˜Can Technical Pattern Trading
Be Profitably Automated? 1. The Channel; 2. The Head and Shoulders,
Working Papers, The Judge Institute of Management Studies, Univer-
sity of Cambridge, November and December, 1999.
18. A. W. Lo, H. Mamaysky, and J. Wang: ˜˜Foundations of Technical
Analysis: Computational Algorithms, Statistical Inference, and Empir-
ical Implementation,™™ NBER Working Paper W7613, 2000.
19. B. LeBaron: ˜˜Technical Trading Profitability in Foreign Exchange
Markets in the 1990s,™™ Working Paper, Brandeis University, 2000.
20. R. Clow: ˜˜Arbitrage Stung by More Efficient Market,™™ Financial Times
April 21, 2002.
151
References



CHAPTER 3
1. W. Feller, An Introduction to Probability Theory and Its Applications,
Wiley, 1968.
2. See [1.5].
3. See [1.6].
4. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Wetterling,
Numerical Recipes: Art of Scientific Programming, Cambridge Univer-
sity Press, 1992.
5. P. Embrechts, C. Klupperberg, and T. Mikosch, Modeling External
Events for Insurance and Finance, Springer, 1997.
6. J. P. Nolan, Stable Distributions, Springer-Verlag, 2002.
7. B. B. Mandelbrot, Fractals and Scaling in Finance, Springer-Verlag,
1997.
8. See [1.9].


CHAPTER 4
1. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemis-
try, and the Natural Sciences, Springer-Verlag, 1997.
2. S. N. Neftci, An Introduction to the Mathematics of Financial Derivatives,
2nd Ed., Academic Press, 1996.
3. See [1.1].
4. E. Scalas, R. Gorenflo, and F. Mainardi, ˜˜Fractional Calculus and
Continuous-time Finance,™™ Physica A284, 376“384, (2000).
5. J. Masoliver, M. Montero, and G. H. Weiss, ˜˜A Continuous Time
Random Walk Model for Financial Distributions,™™ Physical Review
E67, 21112“21121 (2003).
6. W. Horsthemke and R. Lefevr, Noise-Induced Transitions. Theory and
Applications in Physics, Chemistry, and Biology, Springer-Verlag, 1984.
7. B. Oksendal: Stochastic Differential Equations, An Introduction with
Applications, Springer-Verlag, 2000.
8. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,
Springer-Verlag, 1997.


CHAPTER 5
1. P. H. Franses, Time Series Models for Business and Economic Forecast-
ing, Cambridge University Press, 1998.
2. J. D. Hamilton, Time Series Analysis, Princeton University Press, 1994.
152 References



3. See [2.1].
4. See [1.1].
5. R. Sullivan, A. Timmermann, and H. White: ˜˜Data Snooping, Technical
Trading Rule Performance, and the Bootstrap,™™ Journal of Finance 54,
1647“1692 (1999).
6. See [1.2].
7. See [2.2].


CHAPTER 6
1. See [2.4].
2. H. O. Peitgen, H. Jurgens, and D. Saupe, Chaos and Fractals: New
Frontiers in Science, Springer-Verlag, 1992.
3. See [2.11].
4. See [1.1].
5. C. J. G. Evertsz and B. B. Mandelbrot, Multifractal Measures, in [2].
6. B. B. Mandelbrot: ˜˜Limit Lognormal Multifractal Measures,™™ Physica
A163, 306“315 (1990).


CHAPTER 7
1. B. LeBaron, ˜˜Chaos and Nonlinear Forecastability in Economics and
Finance,™™ Philosophical Transactions of the Royal Society of London
348A, 397“404 (1994).
2. W. A. Brock, D. Hsieh, and B. LeBaron, Nonlinear Dynamics, Chaos,
and Instability: Statistical Theory and Economic Evidence, MIT Press,
1991.
3. See [2.11].
4. T. Lux, ˜˜The Socio-economic Dynamics of Speculative Markets: Inter-
acting Agents, Chaos, and the Fat Tails of Return Distributions,™™
Journal of Economic Behavior and Organization 33,143“165 (1998).
5. R. C. Hilborn, Chaos and Nonlinear Dynamics: An Introduction for
Scientists and Engineers, Oxford University Press, 2000.
6. See [2.5].
7. P. Berge, Y. Pomenau, and C. Vidal, Order Within Chaos: Towards a
Deterministic Approach to Turbulence, Wiley, 1986.
8. J. Gleick, Chaos: Making New Science, Penguin, 1988.
9. D. Ruelle, Chance and Chaos, Princeton University Press, 1991.
10. See [6.2].
153
References



CHAPTER 8
1. See [2.2].
2. See [1.5].
3. K. Okuyama, M. Takayasu, and H. Tajkayasu, ˜˜Zipf™s Law in Income
Distributions of Companies,™™ Physica A269, 125“131 (1999).
4. R. Axtell, ˜˜Zipf Distribution of U.S. Firm Sizes,™™ Science, 293, 1818“
1820 (2001).
5. C. A. O. Goodhart and M. O™Hara, ˜˜High Frequency Data in Financial
Markets: Issues and Applications,™™ Journal of Empirical Finance 4,
73“114 (1997).
6. See [3.7].
7. See [2.11].
8. See [1.1].
9. See [1.6].
10. A. Figueiredo, I. Gleria, R. Matsushita, and S. Da Silva, ˜˜Autocorrela-
tion as a Source of Truncated Levy Flights in Foreign Exchange Rates,™™
Physica A323, 601“625 (2003).
11. P. Gopikrishnan, V. Plerou, L. A. N. Amaral, M. Meyer, and E. H.
Stanley, ˜˜Scaling of the Distribution of Fluctuations of Financial
Market Indices,™™ Physical Review E60, 5305“5316 (1999).
12. V. Plerou, P. Gopikrishnan, L. A. N. Amaral, M. Meyer, and E. H.
Stanley, ˜˜Scaling of the Distribution of Price Fluctuations of Individual
Companies,™™ Phys. Rev. E60, 6519“6529 (1999).
13. X. Gabaix, P. Gopikrishnan, V. Plerou, and H. E. Stanley, ˜˜A Theory
of Power-law Distributions in Financial Market Fluctuations, Nature,™™
423, 267“270 (2003).
14. O. Biham, O. Malcai, M. Levy, and S. Solomon, ˜˜Generic Emergence of
Power-Law Distributions and Levy-Stable Intermittent Fluctuations in
Discrete Logistic Systems,™™ Phys. Rev. E58, 1352“1358 (1998).
15. J. D. Farmer, ˜˜Market Force, Ecology, and Evolution,™™ Working
Paper, Santa Fe Institute, 1998.
16. See [1.10].
17. See [1.9].
18. B. LeBaron, ˜˜Stochastic Volatility as a Simple Generator of Apparent
Financial Power Laws and Long Memory,™™ Quantitative Finance 1, 621“
631 (2001).
19. T. Lux, ˜˜Power Laws and Long Memory,™™ Quantitative Finance 1, 560“
562 (2001).
154 References



20. F. Schmitt, D. Schertzer, and S. Lovejoy, ˜˜Multifractal Fluctuations in
Finance,™™ International Journal of Theoretical and Applied Finance 3,
361“364 (2000).
21. N. Vandewalle and M. Ausloos, ˜˜Multi-Affine Analysis of Typical
Currency Exchange Rates,™™ Eur. Phys. J. B4, 257“261 (1998).
22. B. Mandelbrot, A. Fisher, and L. Calvet, ˜˜A Multifractal Model of
Asset Returns,™™ Cowless Foundation Discussion Paper 1164, 1997.
23. T. Lux, ˜˜Turbulence in Financial Markets: The Surprising Explanatory
Power of Simple Cascade Models,™™ Quantitative Finance 1, 632“640
(2001).
24. L. Calvet and A. Fisher, ˜˜Multifractality in Asset Returns: Theory and
Evidence,™™ Review of Economics and Statistics 84, 381“406 (2002).
25. L. Calvet and A. Fisher, ˜˜Regime-Switching and the Estimation of
Multifractal Processes,™™ Working Paper, Harvard University, 2003.
26. T. Lux, ˜˜The Multifractal Model of Asset Returns: Its Estimation via
GMM and Its Use for Volatility Forecasting,™™ Working Paper, Univer-
sity of Kiel, 2003.
27. J. D. Farmer and F. Lillo, ˜˜On the Origin of Power-Law Tails in Price
Fluctuations,™™ Quantitative Finance 4, C7“C10 (2004).
28. V. Plerou, P. Gopikrishnan, X. Gabaix, and H. E. Stanley, ˜˜On the
Origin of Power-Law Fluctuations in Stock Prices,™™ Quantitative
Finance 4, C11“C15 (2004).
29. P. Weber and B. Rosenow, ˜˜Large Stock Price Changes: Volume or
Liquidity?™™ http://xxx.lanl.gov/cond-mat 0401132.
30. T. Di Matteo, T. Aste, and M. Dacorogna, ˜˜Long-Term Memories of
Developed and Emerging Markets: Using the Scaling Analysis to Char-
acterize Their Stage of Development,™™ http://xxx.lanl.gov/cond-mat
0403681.


CHAPTER 9
1. J. C. Hull, Options, Futures, and Other Derivatives, 3rd Ed., Prentice
Hall, 1997.
2. P. Wilmott, Derivatives: The Theory and Practice of Financial Engineer-
ing, Wiley, 1998.
3. A. Lipton, Mathematical Methods for Foreign Exchange, A Financial
Engineer™s Approach, World Scientific, 2001.
4. See [4.2].
5. F. Black and M. Scholes, ˜˜The Pricing of Options and Corporate
Liabilities,™™ Journal of Political Economy 81, 637“659 (1973).
6. See [2.5].
155
References



7. J. P. Bouchaud, ˜˜Welcome to a Non-Black-Scholes World,™™ Quantita-
tive Finance 1, 482“483 (2001).
8. L. Borland, ˜˜A Theory of Non-Gaussian Option Pricing,™™ Quantitative
Finance 2:415“431, 2002.
9. A. B. Schmidt, ˜˜True Invariant of an Arbitrage Free Portfolio,™™ Physica
320A, 535“538 (2003).
10. A. Krakovsky, ˜˜Pricing Liquidity into Derivatives,™™ Risk 12, 65 (1999).
11. U. Cetin, R. A. Jarrow, and P. Protter: ˜˜Liquidity Risk and Arbitrage
Pricing Theory,™™ Working Paper, Cornell University, 2002.
12. J. Perella, J. M. Porra, M. Montero, and J. Masoliver, ˜˜Black-Sholes
Option Pricing Within Ito and Stratonovich Conventions.™™ Physica
A278, 260-274 (2000).


CHAPTER 10
1. See [2.6].
2. See [1.1].
3. See [2.5].
4. P. Silvapulle and C. W. J. Granger, ˜˜Large Returns, Conditional Cor-
relation and Portfolio Diversification: A-Value-at-Risk Approach,™™
Quantitative Finance 1, 542“551 (2001).
5. D. G. Luenberger, Investment Science, Oxford University Press, 1998.
6. R. C. Grinold and R. N. Kahn, Active Portfolio Management, McGraw-
Hill, 2000.
7. R. Korn, Optimal Portfolios: Stochastic Models for Optimal Investment
and Risk Management in Continuous Time, World Scientific, 1999.
8. J. G. Nicholas, Market-Neutral Investing: Long/Short Hedge Fund Strat-
egies, Bloomberg Press, 2000.
9. J. Conrad and K. Gautam, ˜˜An Anatomy of Trading Strategies,™™
Review of Financial Studies 11, 489“519 (1998).
10. E. G. Galev, W. N. Goetzmann, and K. G. Rouwenhorst, ˜˜Pairs
Trading: Performance of a Relative Value Arbitrage Rule,™™ NBER
Working Paper W7032, 1999.
11. W. Fung and D. A. Hsieh, ˜˜The Risk in Hedge Fund Strategies: Theory
and Evidence From Trend Followers,™™ The Review of Financial Studies
14, 313“341 (2001).
12. M. Mitchell and T. Pulvino, ˜˜Characteristics of Risk and Return in Risk
Arbitrage,™™ Journal of Finance 56, 2135“2176 (2001).
13. S. Hogan, R. Jarrow, and M. Warachka, ˜˜Statistical Arbitrage and
Market Efficiency,™™ Working Paper, Wharton-SMU Research Center,
2003.
156 References



14. E. J. Elton, W. Goetzmann, M. J. Gruber, and S. Brown, Modern
Portfolio: Theory and Investment Analysis, Wiley, 2002.


CHAPTER 11
1. P. Jorion, Value at Risk: The New Benchmark for Managing Financial
Risk, McGraw-Hill, 2000.
2. K. Dowd, An Introduction to Market Risk Measurement, Wiley, 2002.
3. P. Artzner, F. Delbaen, J. M. Eber, and D. Heath, ˜˜Coherent Measures
of Risk,™™ Mathematical Finance 9, 203“228 (1999).
4. J. Hull and A. White, ˜˜Incorporating Volatility Updating into the
Historical Simulation Method for Value-at-Risk,™™ Journal of Risk 1,

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