<<

. 5
( 5)




Àc(nþ À nÀ ) if nþ > nÀ
Rþ3 ¼
0, if nþ nÀ

Àc(nÀ À nþ ) if nÀ > nþ
RÀ3 ¼ (12:4:15)
0, if nÀ nþ
We call the parameter c > 0 the ˜˜impatience™™ factor. Here, we neglect
the price variation, so that vþÀ ¼ vÀþ ¼ 0. We also neglect the sto-
chastic rates rÆ . Let us specify
nþ (0) À nÀ (0) ¼ d > 0: (12:4:16)
Then equations (12.4.11)“(12.4.12) have the following form
dnþ =dt ¼ a(nþ þ nÀ ) À bnþ nÀ À c(nþ À nÀ ) (12:4:17)
dnÀ =dt ¼ a(nþ þ nÀ ) À bnþ nÀ (12:4:18)
The equation for the total number of traders n ¼ nþ þ nÀ has the
Riccati form7
dn=dt ¼ 2an À 0:5bn2 þ 0:5bd2 exp (À2ct) À cd exp (Àct) (12:4:19)
Equation (12.4.19) has the asymptotic solution
n0 ¼ 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 ˜˜impatience™™
factor. Obviously, the higher the ˜˜impatience™™ factor, the deeper the
minimum of n(t) will be. At sufficiently high ˜˜impatience™™ 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 Æ (t) ¼ 0 if n Æ (t) < nmin (12:4:21)
Still, further analysis shows that the discrete analog of the system
(12.4.17)“(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 ¼ 0.25, b ¼ 1, nþ(0) ¼ 0.2, and nÀ(0) ¼ 0.1: 1 - c ¼ 1; 2 - c ¼
10; 3 - c ¼ 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)“(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 ˜˜So, how can we make some money with this
stuff?™™ Undoubtedly, ˜˜money-making™™ 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 ˜˜in-house™™ software developed with C/Cþþ) 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) À P(tÀ1)] 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 company™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. Tesfatsion™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 ¼ kT=(6pZR). Then E[r2 ] À r0 2 ¼ 6Dt.
2. The general case of the random walk is discussed in Section 5.1.
3. Here we simplify the notations: m(t) ¼ m, s(y(t), t) ¼ s.
4. The notation y ¼ 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 ˜˜expected™™ news.
Consider a Federal Reserve interest rate cut, which is an economic
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
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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,
5“19 (1998).
5. A. J. McNeil and R. Frey, ˜˜Estimation of Tail-Related Risk for Hetero-
scedastic Financial Time Series: An Extreme Value Approach,™™ Journal
of Empirical Finance 7, 271“300 (2000).
6. J. A. Lopez, ˜˜Regulatory Evaluation of Value-at-risk Models,™™ Journal
of Risk 1, 37“64 (1999).


CHAPTER 12
1. See [1.12].
2. D. Challet, A. Chessa, A. Marsili, and Y. C. Chang, ˜˜From Minority
Games to Real Markets,™™ Quantitative Finance 1, 168“176 (2001).
3. W. B. Arthur, ˜˜Inductive Reasoning and Bounded Rationality,™™ Ameri-
can Economic Review 84, 406“411 (1994).
4. A. Beja and M. B. Goldman, ˜˜On the Dynamic Behavior of Prices in
Disequilibrium,™™ Journal of Finance 35, 235“248 (1980).
5. B. LeBaron, ˜˜A Builder™s Guide to Agent-Based Markets,™™ Quantitative
Finance 1, 254“261 (2001).
6. See [1.11].
7. W. A. Brock and C. H. Hommes, ˜˜Heterogeneous Beliefs and Routes to
Chaos in a Simple Asset Pricing Model,™™ Journal of Economic Dynamics
and Control 22, 1235“1274 (1998).
8. B. LeBaron, W. B. Arthur, and R. Palmer, ˜˜The Time Series Properties
of an Artificial Stock Market,™™ Journal of Economic Dynamics and
Control 23, 1487“1516 (1999).
9. M. Levy, H. Levy, and S. Solomon, ˜˜A Macroscopic Model of the Stock
Market: Cycles, Booms, and Crashes,™™ Economics Letters 45, 103“111
(1994). See also [1.7].
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Heterogeneous Expectations,™™ Quantitative Finance 1, 509“526 (2001).
11. See [7.4].
12. See [8.15].
13. A. B. Schmidt, ˜˜Observable Variables in Agent-Based Modeling of
Financial Markets™™ in [1.11].
14. T. Lux and M. Marchesi, ˜˜Scaling and Criticality in a Stochastic Multi-
Agent Model of Financial Market,™™ Nature 397, 498“500 (1999).
15. B. LeBaron, ˜˜Calibrating an Agent-Based Financial Market to Macro-
economic Time Series,™™ Working Paper, Brandeis University, 2002.
16. F. Wagner, ˜˜Volatility Cluster and Herding,™™ Physica A322, 607“619
(2003).
17. A. B. Schmidt, ˜˜Modeling the Demand-price Relations in a High-
Frequency Foreign Exchange Market,™™ Physica A271, 507“514 (1999).
18. A. B. Schmidt, ˜˜Why Technical Trading May Be Successful: A Lesson
From the Agent-Based Modeling,™™ Physica A303, 185-188 (2002).
19. A. B. Schmidt, ˜˜Modeling the Birth of a Liquid Market,™™ Physica A283,
479“485 (2001).
20. S. Solomon, ˜˜Importance of Being Discrete: Life Always Wins on the
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Answers to Exercises




2.2 (a) $113.56; (b) $68.13.
2.4 Borrow 100000 USD to buy 100000/1.7705 GBP. Then buy (100000/
1.7705)/0.6694 EUR. Exchange the resulting amount to
1.1914[(100000/1.7705)/0.6694] % 100525 USD. Return the loan and
enjoy profits of $525 (minus transaction fees).
3.2 (a) 0.157; (b) 1.645; (c) 1.036
Since aX þ b $ N(am þ b, (as)2 ), it follows that C2 ¼ a2 þ b2 and D ¼
3.4
(a þ b À C) m. Ðt
(t) ¼ X(0)exp( Àmt) þ s exp[ Àm(t À s)]dW (s)
4.3
0
For this process, the AR(2) polynomial (5.1.12) is:1 “ 1.2z þ 0.32z2 ¼ 0.
5.2
Since its roots, z ¼ (1.2 Æ 0.4)/0.64 > 1, are outside the unit circle, the
process is covariance-stationary.
Linear regression for the dividends in 2000 “ 2003 is D ¼ 1.449 þ
5.3
0.044n (where n is number of years since 2000). Hence the dividend
growth is G ¼ 4.4%.p¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬
(a) X* ¼ 0.5 Æ 0:25 À C. Hence there are two fixed points at C <
7.1
0.25, one fixed point at C ¼ 0.25, and none for C > 0.25.
(b) X1* % 0.14645 is attractor with the basin 0 X < X2* where X2* %
0.85355.
(a) 1) c ¼ 2.70, p ¼ 0.26; 2) c ¼ 0.58, p ¼ 2.04.
9.1
(b) The Black-Scholes option prices do not depend on the stock
growth rate (see discussion on the risk-neutral valuation).
9.2 Since the put-call parity is violated, you may sell a call and a T-bill for
$(8 þ 98) ¼ $106. Simultaneously, you buy a share and a put for $(100



159
160 Answers



þ 3.50) ¼ $103.50 to cover your obligations. Then you have profits of
$(106 À 103.50) ¼ $2.50 (minus transaction fees).
(a) E[R] ¼ 0.13, s ¼ 0.159; (b) E[R] ¼ 0.13, s¼ 0.104.
10.1
(a) bA ¼ 1.43;
10.2
(b) For bA¼ 1.43, E[RA] ¼ 0.083 according to eq(10.2.1). However,
the average return for the given sample of returns is 0.103. Hence
CAPM is violated in this case.
w1 ¼ (b21 b32Àb22 b31)/[ b11(b22Àb32) þ b21(b32Àb12) þ b31(b12Àb22)],
10.3
w2 ¼ (b12 b31Àb11b32)/[b22(b11Àb31) þ b12(b31Àb21) þ b32(b21Àb11)].
l1 ¼ [b22(R1ÀRf)Àb12(R2ÀRf)]/(b11b22Àb12b21), l2 ¼ [b11(R2ÀRf)À
10.4
b21(R1ÀRf)]/(b11b22Àb12b21).
10.4
11.1 (a) $136760; (b) $78959.
Index


A integrated (ARIMA), 46
Autoregressive moving integrated average
model (ARIMA), 46
Adaptive equilibrium models, 130“132
Autoregressive process, 43
APT. See Arbitrage Pricing Theory
Arbitrage, 11
convertible, 119
equity market-neutral strategy and
statistical, 119
B
fixed-income, 119
merger, 119
Basin of attraction, 72
relative value, 119“120
Behavioral finance, 13
statistical, 13
Bernoulli trials, 20
trading strategies of, 118“120
Beta, 115
Arbitrage Pricing Theory (APT), 116“118
Bid, 5
ARCH. See Autoregressive conditional
Bifurcation
heteroskedascisity
global, 82
ARIMA. See Autoregressive moving
Hopf, 78
integrated average model
local, 82
ARMA. See Autoregressive moving average
point of, 70, 71f
model
Binomial
Ask, 5
cascade, 64“66, 65f
Attractor, 72
distribution, 21
quasi-periodic, 78
measure, 64
strange, 69
tree, 98“101, 99f
Autocorrelation function, 47
Black-Scholes equation, 102“104
Autocovariance, 47
Black-Scholes Theory (BST), 101“105
Autonomous systems, 75
Bond, 130“131
Autoregressive conditional heteroskedascisity
Bounded rationality, 14, 133
(ARCH), 52
Box-counting dimension, 61
exponential generalized (EGARCH), 53“54
Brownian motion, 32“35
generalized (GARCH), 52“53, 87
arithmetic, 34
integrated generalized (IGARCH), 53
fractional, 62“63
Autoregressive moving average model
geometric, 34
(ARMA), 45“46
162 Index


Crises, 83
C Cumulative distribution function, 18

Capital Asset Pricing Model (CAPM),

D
114“116, 118
Capital market line, 114
CAPM. See Capital Asset Pricing Model
Damped oscillator, 76, 76f
CARA. See Constant absolute risk aversion
Data
function
granularity, 88
Cascade, 64
snooping, 54
binomial, 64“66, 65f
Delta, 103
canonical, 66
Delta-neutral portfolios, 104
conservative, 65
Derivatives, 93
microcanonical, 65
Deterministic trend v. stochastic trend,
multifractal, 63“64
49“50, 50f
multiplicative process of, 64
Dickey-Fuller method, 45, 51
Cauchy (Lorentzian) distribution, 23, 24f
Dimension
standard, 23
box-counting, 61
Central limit theorem, 22
correlation, 85
Chaos, 70, 82“85
fractal, 60
measuring, 83“85
Discontinuous jumps, 31
Chaotic transients, 83
Discounted-cash-flow pricing model, 8“9
Chapmen-Kolmogorov equation, 30“31
Discounting, 9
Characteristic function, 25
Discrete random walk, 33
Chartists, 132, 134“135, 137, 138
Dissipative system, 76
Coherent risk measures, 124
Distribution
Cointegration, 51
binomial, 21
Compound stochastic process, 92
Cauchy (Lorentzian), 23, 24f
Compounded return, 8
extreme value, 23
continuously, 8
Frechet, 24
Conditional expectation, 18
Gumbel, 24
Conservative system, 76“77
Iibull, 24
Constant absolute risk aversion (CARA)
Levy, 25“27
function, 132
lognormal, 22“23
Contingent claim. See Derivatives
normal (Gaussian), 21“22
Continuously compounded return, 8. See also
Pareto, 24, 26
Log return
Poisson, 21
Continuous-time random walk, 34
stable, 25
Contract
standard Cauchy, 23
forward, 93
standard normal, 22, 24f
future, 94
standard uniform, 20
Contrarians, 133
uniform, 20
Correlation
Dividend effects, 8“10, 96
coefficient, 20
Dogs of the Dow, 14
dimension, 85
Doob-Meyer decomposition theorem, 41
Covariance, 20
Dow-Jones index
matrix of, 20
returns of, 89
stationarity-, 49
163
Index



F
Dummy parameters, 51
Dynamic hedging, 104

Fair game, 40

E Fair prices, 12“13
Firm rates, 141
Fisher-Tippett theorem, 23“24
Econometrics, 1 Fixed point, 69“70
Econophysics, 1“2 Flow, 73“74
Efficient frontier, 114 Fokker-Planck equation, 30“31
Efficient market, 12 Foreign exchange rates, 141
Efficient Market Hypothesis (EMH), 12“14, 40 Forward contract, 93
random walk, 12“13 Fractal. See also Multifractal
semi-strong, 12 box-counting dimension, 61
strong, 12 deterministic, 60“63, 60f
weak, 12 dimension, 60
Efficient Market Theory, 12 iterated function systems of, 61
EGARCH. See Exponential generalized random, 60
autoregressive conditional stochastic, 60f
heteroskedascisity technical definitions of, 55“56
EMH. See Efficient Market Hypothesis Frechet distribution, 24
Equilibrium models Fundamental analysis, 12
adaptive, 130“133 Fundamentalists, 132, 134“135, 137, 141
non-, 130, 134“135 Future
Equity hedge, 119 contract, 94
Error function, 22 value, 9
ETL. See Expected tail loss Future contract, 94
Euro, 88
EWMA. See Exponentially weighed
moving average; exponentially weighed
moving average
G
Exchange rates
foreign, 86
Exogenous variable, 56 Gamma, 103
Exotic options, 141 Gamma-neutral, 104
Expectation, 18. See also Mean GARCH. See Generalized autoregressive
Expected shortfall, 141 conditional heteroskedascisity
Expected tail loss (ETL), 124, 124f Gaussian distribution, 21“22
Expiration date, 94. See also Maturity Generalized autoregressive conditional
Exponential generalized autoregressive heteroskedascisity (GARCH), 52“53,
conditional heteroskedascisity 85
(EGARCH), 53“54 Given future value, 9
Exponentially weighed moving average Granger causality, 56
(EWMA), 53 Greeks, 103
Extreme value distribution, 23 Gumbel distribution, 24
164 Index


K
H
Kolmogorov-Sinai entropy, 84
Hamiltonian system, 76“77
Kupiec test, 126
Hang-Seng index
Kurtosis, 19
returns of, 89
Historical simulation, 125
¨
Holder exponent, 63
L
Homoskedastic process, 51“54
Hopf bifurcation, 78
Hurst exponent, 62 Lag operator, 43“44
Langevin equation, 32
I Law of One Price, 10
Leptokurtosis, 19
Levy distribution, 25“26
IGARCH. See Integrated generalized
Limit cycle, 77
autoregressive conditional Limit orders, 6
heteroskedascisity Log return, 8. See also Continuously
Iibull distribution, 24
compounded return
IID. See Independently and identically
Logistic map, 70“72, 73f, 74f
distributed process attractor on, 72
Implied volatility, 103 basin of attraction on, 72
Independent variables, 20
fixed point on, 71“73
Independently and identically distributed Lognormal distribution, 22“23
process (IID), 33 Long position, 6
Indicative rates, 141 Lorentzian distribution. See Cauchy
Initial condition, 30
(Lorentzian) distribution
Integral Lorenz model, 70“71, 79“82, 80f, 81f, 82f
stochastic, 36“39 Lotka-Volterra system, 90
stochastic Ito™s, 38“39 Lyapunov exponent, 82“85
Integrated generalized autoregressive
conditional heteroskedascisity
(IGARCH), 53
M
Integrated of order, 45
Intermittency, 83
Irrational exuberance, 13 Market(s)
Iterated map, 71 bourse, 5
Iteration function, 71 exchange, 5
Ito™s integral liquidity, 6, 141“142, 143f
stochastic, 38“39 microstructure, 6
Ito™s lemma, 35“36 orders, 6
over-the-counter, 5
price formation, 5“7
J Market microstructure, 136
Market portfolio, 115
January Effect, 14 Market-neutral strategies, 118
Joint distribution, 19 Markov process, 29“32
Martingale, 39“41
165
Index


O
sub, 40
super, 40
Mathematical Finance, 1 OLS. See Ordinary least squares
Maturity, 93“94 Operational time, 7
Maximum likelihood estimate (MLE), 48 Options, 98
˜˜Maxwell™s Demon,™™ 136“137 American, 94“96
MBS. See mortgage-backed securities call, 94
arbitrage European, 94“96
Mean, 18 exercise price of, 94
reversion, 44 exotic, 141
squared error, 48 expiration date of, 94
Mean squared error (MSE), 48 long call, 95, 97f
Mean-reverting process, 42 long put, 95, 97f
Mean-square limit, 38 maturity of, 93“94
Mean-variance efficient portfolio, 108 premium of, 96
Median, 18 put, 94
Microsoft Excel, 4, 25 short call, 95, 97f
Mimetic contagion, 134 short put, 95“96, 97f
Minority game, 129“130 strike price of, 94
MLE. See Maximum likelihood estimate Orders
Mortgage-backed securities (MBS) limit, 6
arbitrage, 119 market, 6
Moving average model, 45“47 stop, 6
autoregressive, 45“46 Ordinary least squares (OLS), 48
invertible, 46“47 Ornstein-Uhlenbeck equation, 42
MSE. See Mean squared error
Multifractal, 63“64. See also Fractal
binomial measure, 64
P
cascade, 63“64
spectrum, 64
Multipliers, 64 Pair trading, 118
Multivariate time series, 54“57 Pareto distribution, 24, 26
Partition function, 67
Partly forcastable prices, 70
N Period-doubling, 82
Persistent process, 62
Noise anti-, 63
non-white, 38 P/L. See Profits and losses
white, 33, 43 Poisson distribution, 21
Nonanticipating function, 39 Portfolio
Non-equilibrium price models, 130, 134“136 delta-neutral, 106
Non-integrable system, 75 rebalancing, 106
Normal distribution, 21“22 well-diversified, 117
standard, 22, 24f Portfolio selection, 111“115
Notations, 4 Position
166 Index


R
long, 93
short, 93
Positive excess kurtosis. See Random walk, 12“13, 44
Leptokurtosis continuous-time, 34
Present value, 8“9 with drifts, 45
Present-value pricing model. See Discounted- Rate of return, 139
cash-flow pricing model Rates
Price firm, 141
exercise, 94 foreign exchange, 141
option, 96 indicative, 141
spot, 94 Rational bubble, 9
strike, 94 Rational investors, 12“13
Price-demand relations, 138“139, 138f, Rescaled range (R/S) analysis, 63, 88
139f Return
Pricing model compounded, 8
discounted-cash-flow, 8“9 log, 8
future value, 9 required rate of, 10
given future value, 9 simple, 7
present-value, 8 Return on Equity (ROE), 117
Probability density function, 16 Rho, 104
Process Riemann integral, 36
anti-persistent, 63 Riemann-Stieltjes integral, 36“37
autoregressive, 43 Risk
compound stochastic, 92 cash-flow, 121
homoskedastic, 51“54 coherent, measures, 124
independently and identically distributed credit, 121
(IID), 33 liquidity, 121
Markov, 29“32 market, 121
mean-reverting, 42 operational, 121
multiplicative, 64 Risk-free asset, 130“131. See also Bond
persistent, 62 Risk-neutral valuation, 99
scale-free, 26 Risk-return trade off line, 112
standard Wiener, 31“32, 34“35 Risky asset, 130“131
stationary, 49 ROE. See Return on Equity
stochastic, 29“42 R/S. See Rescaled range analysis
Profits and losses (P/L), 122, 123f, 124f
Put-call parity, 96

S
Q Santa Fe artificial market, 133
Scale-free process, 26
Quasi-periodic attractors, 78 Scaling function, 66“67
Quasi-periodicity, 83 Seasonal effects, 45“46
167
Index


Security market line, 115 deterministic, 49“50, 50f
Self-affine object, 59 stochastic, 49“50
Self-affinity, 59 Truncated Levy flight, 26“27, 88“89
Sharpe ratio, 115
Short
U
position, 93
selling, 6
Simple return, 7
Uniform distribution, 20
Simultaneous equation, 54
standard, 20
Skewness, 19
Unit root, 45
S&P 500 index, 24f, 87
Univariate time series, 43
returns of, 89
Stable distribution, 25
Standard deviation, 18
V
Standard Wiener process, 31“32, 34“35
Stationary process, 49
non-, 49 Value at risk (VaR), 122“124, 123f
Statistical arbitrage, 14 conditional, 141
Stieltjes integral, 37 Van der Pol
Stochastic equation, 77“78
compound, process, 92 oscillator, 78f
differential equation, 35 VAR. See Vector autoregressive model
integral, 36“39 VaR. See Value at risk
Ito™s integral, 38“39 Variance, 18
process, 29“42 matrix, 19
trend, 49“50, 50f Variate, 16
Stochastic trend v. deterministic trend, 49“50, Vector autoregressive model (VAR), 55“56
50f Vega, 104
Stop orders, 6 Volatility, 19
Stratonovich™s integral, 39 implied, 103
Strict stationarity, 49 smile, 104
Submartingale, 40 Volatility smile, 104“105
Super-efficient portfolio, 114
Supermartingale, 40

W
T Weak stationarity, 49
White noise, 33, 43
Technical analysis, 12 non-, 39
Term structure, 104“105 Wiener process
Theta, 103 standard, 31“32, 34“35
Tick, 6
Tick-by-tick data, 6“7

Z
Traders
regular, 139“141
technical, 139“141, 140f
Zipf™s law, 89
Trajectory, 71, 76f, 77“79, 78f, 79f
Trend

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