ñòð. 5 |

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.

This page intentionally left blank

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

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 â€˜â€˜hard-scienceâ€™â€™

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 [1Ã€3].

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.

Ã€1

@F

5. We might choose also one share and Ã€ 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 Poorâ€™s 500 Index is used as proxy for the U.S.

market portfolio.

4. ROE Â¼ E/B where E is earnings; B is the book value that in a nutshell

equals the companyâ€™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. Tesfatsionâ€™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. Brockâ€™s (http://www.ssc.wisc.edu/

$wbrock/),

C. Chiarellaâ€™s (http://www.business.uts.edu.au/finance/staff/carl.html),

J. D. Farmerâ€™s (http://www.santafe.edu/$jdf),

B. LeBaronâ€™s (http://people.brandeis.edu/$blebaron/index.htm),

T. Luxâ€™s (http://www.bwl.uni-kiel.de/vwlinstitute/gwrp/team/lux.htm), and

S. Solomonâ€™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[rtÃ¾1 ] Â¼

fi (rt , . . . , rtÃ€Li ) [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 singlesâ€™ 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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CHAPTER 1

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2. W. H. Green, Econometric Analysis, Prentice Hall, 1998.

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150 References

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CHAPTER 2

1. C. Alexander, Market Models: A Guide to Financial Data Analysis,

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2. M. M. Dacorogna, R. Gencay, U. Muller, R. B. Olsen, and O. V. Pictet,

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6. Z. Bodie and R. C. Merton, Finance, Prentice Hall, 1998.

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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

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14. D. Kahneman and A. Tversky (Eds.), Choices, Values and Frames,

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15. R. H. Thaler (Ed), Advances in Behavioral Finance, Russell Sage Foun-

dation, 1993.

16. D. Kahneman and A. Tversky: â€˜â€˜Prospect Theory: An Analysis of Decision

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17. M. A. H. Dempster and C. M. Jones: â€˜â€˜Can Technical Pattern Trading

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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: â€˜â€˜Foundations of Technical

Analysis: Computational Algorithms, Statistical Inference, and Empir-

ical Implementation,â€™â€™ NBER Working Paper W7613, 2000.

19. B. LeBaron: â€˜â€˜Technical Trading Profitability in Foreign Exchange

Markets in the 1990s,â€™â€™ Working Paper, Brandeis University, 2000.

20. R. Clow: â€˜â€˜Arbitrage Stung by More Efficient Market,â€™â€™ Financial Times

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151

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CHAPTER 3

1. W. Feller, An Introduction to Probability Theory and Its Applications,

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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, â€˜â€˜Fractional Calculus and

Continuous-time Finance,â€™â€™ Physica A284, 376â€“384, (2000).

5. J. Masoliver, M. Montero, and G. H. Weiss, â€˜â€˜A Continuous Time

Random Walk Model for Financial Distributions,â€™â€™ Physical Review

E67, 21112â€“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: â€˜â€˜Data Snooping, Technical

Trading Rule Performance, and the Bootstrap,â€™â€™ Journal of Finance 54,

1647â€“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: â€˜â€˜Limit Lognormal Multifractal Measures,â€™â€™ Physica

A163, 306â€“315 (1990).

CHAPTER 7

1. B. LeBaron, â€˜â€˜Chaos and Nonlinear Forecastability in Economics and

Finance,â€™â€™ Philosophical Transactions of the Royal Society of London

348A, 397â€“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, â€˜â€˜The Socio-economic Dynamics of Speculative Markets: Inter-

acting Agents, Chaos, and the Fat Tails of Return Distributions,â€™â€™

Journal of Economic Behavior and Organization 33,143â€“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, â€˜â€˜Zipfâ€™s Law in Income

Distributions of Companies,â€™â€™ Physica A269, 125â€“131 (1999).

4. R. Axtell, â€˜â€˜Zipf Distribution of U.S. Firm Sizes,â€™â€™ Science, 293, 1818â€“

1820 (2001).

5. C. A. O. Goodhart and M. Oâ€™Hara, â€˜â€˜High Frequency Data in Financial

Markets: Issues and Applications,â€™â€™ Journal of Empirical Finance 4,

73â€“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, â€˜â€˜Autocorrela-

tion as a Source of Truncated Levy Flights in Foreign Exchange Rates,â€™â€™

Physica A323, 601â€“625 (2003).

11. P. Gopikrishnan, V. Plerou, L. A. N. Amaral, M. Meyer, and E. H.

Stanley, â€˜â€˜Scaling of the Distribution of Fluctuations of Financial

Market Indices,â€™â€™ Physical Review E60, 5305â€“5316 (1999).

12. V. Plerou, P. Gopikrishnan, L. A. N. Amaral, M. Meyer, and E. H.

Stanley, â€˜â€˜Scaling of the Distribution of Price Fluctuations of Individual

Companies,â€™â€™ Phys. Rev. E60, 6519â€“6529 (1999).

13. X. Gabaix, P. Gopikrishnan, V. Plerou, and H. E. Stanley, â€˜â€˜A Theory

of Power-law Distributions in Financial Market Fluctuations, Nature,â€™â€™

423, 267â€“270 (2003).

14. O. Biham, O. Malcai, M. Levy, and S. Solomon, â€˜â€˜Generic Emergence of

Power-Law Distributions and Levy-Stable Intermittent Fluctuations in

Discrete Logistic Systems,â€™â€™ Phys. Rev. E58, 1352â€“1358 (1998).

15. J. D. Farmer, â€˜â€˜Market Force, Ecology, and Evolution,â€™â€™ Working

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16. See [1.10].

17. See [1.9].

18. B. LeBaron, â€˜â€˜Stochastic Volatility as a Simple Generator of Apparent

Financial Power Laws and Long Memory,â€™â€™ Quantitative Finance 1, 621â€“

631 (2001).

19. T. Lux, â€˜â€˜Power Laws and Long Memory,â€™â€™ Quantitative Finance 1, 560â€“

562 (2001).

154 References

20. F. Schmitt, D. Schertzer, and S. Lovejoy, â€˜â€˜Multifractal Fluctuations in

Finance,â€™â€™ International Journal of Theoretical and Applied Finance 3,

361â€“364 (2000).

21. N. Vandewalle and M. Ausloos, â€˜â€˜Multi-Affine Analysis of Typical

Currency Exchange Rates,â€™â€™ Eur. Phys. J. B4, 257â€“261 (1998).

22. B. Mandelbrot, A. Fisher, and L. Calvet, â€˜â€˜A Multifractal Model of

Asset Returns,â€™â€™ Cowless Foundation Discussion Paper 1164, 1997.

23. T. Lux, â€˜â€˜Turbulence in Financial Markets: The Surprising Explanatory

Power of Simple Cascade Models,â€™â€™ Quantitative Finance 1, 632â€“640

(2001).

24. L. Calvet and A. Fisher, â€˜â€˜Multifractality in Asset Returns: Theory and

Evidence,â€™â€™ Review of Economics and Statistics 84, 381â€“406 (2002).

25. L. Calvet and A. Fisher, â€˜â€˜Regime-Switching and the Estimation of

Multifractal Processes,â€™â€™ Working Paper, Harvard University, 2003.

26. T. Lux, â€˜â€˜The Multifractal Model of Asset Returns: Its Estimation via

GMM and Its Use for Volatility Forecasting,â€™â€™ Working Paper, Univer-

sity of Kiel, 2003.

27. J. D. Farmer and F. Lillo, â€˜â€˜On the Origin of Power-Law Tails in Price

Fluctuations,â€™â€™ Quantitative Finance 4, C7â€“C10 (2004).

28. V. Plerou, P. Gopikrishnan, X. Gabaix, and H. E. Stanley, â€˜â€˜On the

Origin of Power-Law Fluctuations in Stock Prices,â€™â€™ Quantitative

Finance 4, C11â€“C15 (2004).

29. P. Weber and B. Rosenow, â€˜â€˜Large Stock Price Changes: Volume or

Liquidity?â€™â€™ http://xxx.lanl.gov/cond-mat 0401132.

30. T. Di Matteo, T. Aste, and M. Dacorogna, â€˜â€˜Long-Term Memories of

Developed and Emerging Markets: Using the Scaling Analysis to Char-

acterize Their Stage of Development,â€™â€™ 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

Engineerâ€™s Approach, World Scientific, 2001.

4. See [4.2].

5. F. Black and M. Scholes, â€˜â€˜The Pricing of Options and Corporate

Liabilities,â€™â€™ Journal of Political Economy 81, 637â€“659 (1973).

6. See [2.5].

155

References

7. J. P. Bouchaud, â€˜â€˜Welcome to a Non-Black-Scholes World,â€™â€™ Quantita-

tive Finance 1, 482â€“483 (2001).

8. L. Borland, â€˜â€˜A Theory of Non-Gaussian Option Pricing,â€™â€™ Quantitative

Finance 2:415â€“431, 2002.

9. A. B. Schmidt, â€˜â€˜True Invariant of an Arbitrage Free Portfolio,â€™â€™ Physica

320A, 535â€“538 (2003).

10. A. Krakovsky, â€˜â€˜Pricing Liquidity into Derivatives,â€™â€™ Risk 12, 65 (1999).

11. U. Cetin, R. A. Jarrow, and P. Protter: â€˜â€˜Liquidity Risk and Arbitrage

Pricing Theory,â€™â€™ Working Paper, Cornell University, 2002.

12. J. Perella, J. M. Porra, M. Montero, and J. Masoliver, â€˜â€˜Black-Sholes

Option Pricing Within Ito and Stratonovich Conventions.â€™â€™ 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, â€˜â€˜Large Returns, Conditional Cor-

relation and Portfolio Diversification: A-Value-at-Risk Approach,â€™â€™

Quantitative Finance 1, 542â€“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, â€˜â€˜An Anatomy of Trading Strategies,â€™â€™

Review of Financial Studies 11, 489â€“519 (1998).

10. E. G. Galev, W. N. Goetzmann, and K. G. Rouwenhorst, â€˜â€˜Pairs

Trading: Performance of a Relative Value Arbitrage Rule,â€™â€™ NBER

Working Paper W7032, 1999.

11. W. Fung and D. A. Hsieh, â€˜â€˜The Risk in Hedge Fund Strategies: Theory

and Evidence From Trend Followers,â€™â€™ The Review of Financial Studies

14, 313â€“341 (2001).

12. M. Mitchell and T. Pulvino, â€˜â€˜Characteristics of Risk and Return in Risk

Arbitrage,â€™â€™ Journal of Finance 56, 2135â€“2176 (2001).

13. S. Hogan, R. Jarrow, and M. Warachka, â€˜â€˜Statistical Arbitrage and

Market Efficiency,â€™â€™ 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, â€˜â€˜Coherent Measures

of Risk,â€™â€™ Mathematical Finance 9, 203â€“228 (1999).

4. J. Hull and A. White, â€˜â€˜Incorporating Volatility Updating into the

Historical Simulation Method for Value-at-Risk,â€™â€™ Journal of Risk 1,

5â€“19 (1998).

5. A. J. McNeil and R. Frey, â€˜â€˜Estimation of Tail-Related Risk for Hetero-

scedastic Financial Time Series: An Extreme Value Approach,â€™â€™ Journal

of Empirical Finance 7, 271â€“300 (2000).

6. J. A. Lopez, â€˜â€˜Regulatory Evaluation of Value-at-risk Models,â€™â€™ Journal

of Risk 1, 37â€“64 (1999).

CHAPTER 12

1. See [1.12].

2. D. Challet, A. Chessa, A. Marsili, and Y. C. Chang, â€˜â€˜From Minority

Games to Real Markets,â€™â€™ Quantitative Finance 1, 168â€“176 (2001).

3. W. B. Arthur, â€˜â€˜Inductive Reasoning and Bounded Rationality,â€™â€™ Ameri-

can Economic Review 84, 406â€“411 (1994).

4. A. Beja and M. B. Goldman, â€˜â€˜On the Dynamic Behavior of Prices in

Disequilibrium,â€™â€™ Journal of Finance 35, 235â€“248 (1980).

5. B. LeBaron, â€˜â€˜A Builderâ€™s Guide to Agent-Based Markets,â€™â€™ Quantitative

Finance 1, 254â€“261 (2001).

6. See [1.11].

7. W. A. Brock and C. H. Hommes, â€˜â€˜Heterogeneous Beliefs and Routes to

Chaos in a Simple Asset Pricing Model,â€™â€™ Journal of Economic Dynamics

and Control 22, 1235â€“1274 (1998).

8. B. LeBaron, W. B. Arthur, and R. Palmer, â€˜â€˜The Time Series Properties

of an Artificial Stock Market,â€™â€™ Journal of Economic Dynamics and

Control 23, 1487â€“1516 (1999).

9. M. Levy, H. Levy, and S. Solomon, â€˜â€˜A Macroscopic Model of the Stock

Market: Cycles, Booms, and Crashes,â€™â€™ Economics Letters 45, 103â€“111

(1994). See also [1.7].

157

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10. C. Chiarella and X. He, â€˜â€˜Asset Pricing and Wealth Dynamics Under

Heterogeneous Expectations,â€™â€™ Quantitative Finance 1, 509â€“526 (2001).

11. See [7.4].

12. See [8.15].

13. A. B. Schmidt, â€˜â€˜Observable Variables in Agent-Based Modeling of

Financial Marketsâ€™â€™ in [1.11].

14. T. Lux and M. Marchesi, â€˜â€˜Scaling and Criticality in a Stochastic Multi-

Agent Model of Financial Market,â€™â€™ Nature 397, 498â€“500 (1999).

15. B. LeBaron, â€˜â€˜Calibrating an Agent-Based Financial Market to Macro-

economic Time Series,â€™â€™ Working Paper, Brandeis University, 2002.

16. F. Wagner, â€˜â€˜Volatility Cluster and Herding,â€™â€™ Physica A322, 607â€“619

(2003).

17. A. B. Schmidt, â€˜â€˜Modeling the Demand-price Relations in a High-

Frequency Foreign Exchange Market,â€™â€™ Physica A271, 507â€“514 (1999).

18. A. B. Schmidt, â€˜â€˜Why Technical Trading May Be Successful: A Lesson

From the Agent-Based Modeling,â€™â€™ Physica A303, 185-188 (2002).

19. A. B. Schmidt, â€˜â€˜Modeling the Birth of a Liquid Market,â€™â€™ Physica A283,

479â€“485 (2001).

20. S. Solomon, â€˜â€˜Importance of Being Discrete: Life Always Wins on the

Surface,â€™â€™ Proceedings of National Academy of Sciences 97, 10322â€“10324

(2000).

21. A. H. Sato and H. Takayasu, â€˜â€˜Artificial Market Model Based on

Deterministic Agents and Derivation of Limit of GARCH Process,â€™â€™

http://xxx.lanl.gov/cond-mat0109139/.

22. E. Scalas, S. Cincotti, C. Dose, and M. Raberto, â€˜â€˜Fraudulent Agents in

an Artificial Financial Market,â€™â€™ http://xxx.lanl.gov/cond-mat0310036.

23. D. Delli Gatti, C. Di Guilmi, E. Gaffeo, G. Giulioni, M. Gallegati, and

A. Palestrini, â€˜â€˜A New Approach to Business Fluctuations: Heteroge-

neous Interacting Agents, Scaling Laws and Financial Fragility,â€™â€™ http://

xxx.lanl.gov/cond-mat0312096.

This page intentionally left blank

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 Ã¾ b $ N(am Ã¾ b, (as)2 ), it follows that C2 Â¼ a2 Ã¾ b2 and D Â¼

3.4

(a Ã¾ b Ã€ C) m. Ãt

(t) Â¼ X(0)exp( Ã€mt) Ã¾ s exp[ Ã€m(t Ã€ s)]dW (s)

4.3

0

For this process, the AR(2) polynomial (5.1.12) is:1 â€“ 1.2z Ã¾ 0.32z2 Â¼ 0.

5.2

Since its roots, z Â¼ (1.2 Ã† 0.4)/0.64 > 1, are outside the unit circle, the

process is covariance-stationary.

Linear regression for the dividends in 2000 â€“ 2003 is D Â¼ 1.449 Ã¾

5.3

0.044n (where n is number of years since 2000). Hence the dividend

growth is G Â¼ 4.4%.pï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒï¬ƒ

(a) X* Â¼ 0.5 Ã† 0:25 Ã€ C. Hence there are two fixed points at C <

7.1

0.25, one fixed point at C Â¼ 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 Â¼ 2.70, p Â¼ 0.26; 2) c Â¼ 0.58, p Â¼ 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 Ã¾ 98) Â¼ $106. Simultaneously, you buy a share and a put for $(100

159

160 Answers

Ã¾ 3.50) Â¼ $103.50 to cover your obligations. Then you have profits of

$(106 Ã€ 103.50) Â¼ $2.50 (minus transaction fees).

(a) E[R] Â¼ 0.13, s Â¼ 0.159; (b) E[R] Â¼ 0.13, sÂ¼ 0.104.

10.1

(a) bA Â¼ 1.43;

10.2

(b) For bAÂ¼ 1.43, E[RA] Â¼ 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 Â¼ (b21 b32Ã€b22 b31)/[ b11(b22Ã€b32) Ã¾ b21(b32Ã€b12) Ã¾ b31(b12Ã€b22)],

10.3

w2 Â¼ (b12 b31Ã€b11b32)/[b22(b11Ã€b31) Ã¾ b12(b31Ã€b21) Ã¾ b32(b21Ã€b11)].

l1 Â¼ [b22(R1Ã€Rf)Ã€b12(R2Ã€Rf)]/(b11b22Ã€b12b21), l2 Â¼ [b11(R2Ã€Rf)Ã€

10.4

b21(R1Ã€Rf)]/(b11b22Ã€b12b21).

10.4

11.1 (a) $136760; (b) $78959.

Index

A integrated (ARIMA), 46

Autoregressive moving integrated average

model (ARIMA), 46

Adaptive equilibrium models, 130â€“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, 119â€“120

Behavioral finance, 13

statistical, 13

Bernoulli trials, 20

trading strategies of, 118â€“120

Beta, 115

Arbitrage Pricing Theory (APT), 116â€“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, 64â€“66, 65f

Attractor, 72

distribution, 21

quasi-periodic, 78

measure, 64

strange, 69

tree, 98â€“101, 99f

Autocorrelation function, 47

Black-Scholes equation, 102â€“104

Autocovariance, 47

Black-Scholes Theory (BST), 101â€“105

Autonomous systems, 75

Bond, 130â€“131

Autoregressive conditional heteroskedascisity

Bounded rationality, 14, 133

(ARCH), 52

Box-counting dimension, 61

exponential generalized (EGARCH), 53â€“54

Brownian motion, 32â€“35

generalized (GARCH), 52â€“53, 87

arithmetic, 34

integrated generalized (IGARCH), 53

fractional, 62â€“63

Autoregressive moving average model

geometric, 34

(ARMA), 45â€“46

162 Index

Crises, 83

C Cumulative distribution function, 18

Capital Asset Pricing Model (CAPM),

D

114â€“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, 64â€“66, 65f

Delta, 103

canonical, 66

Delta-neutral portfolios, 104

conservative, 65

Derivatives, 93

microcanonical, 65

Deterministic trend v. stochastic trend,

multifractal, 63â€“64

49â€“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, 82â€“85

fractal, 60

measuring, 83â€“85

Discontinuous jumps, 31

Chaotic transients, 83

Discounted-cash-flow pricing model, 8â€“9

Chapmen-Kolmogorov equation, 30â€“31

Discounting, 9

Characteristic function, 25

Discrete random walk, 33

Chartists, 132, 134â€“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, 76â€“77

Iibull, 24

Constant absolute risk aversion (CARA)

Levy, 25â€“27

function, 132

lognormal, 22â€“23

Contingent claim. See Derivatives

normal (Gaussian), 21â€“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, 8â€“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, 12â€“13

Firm rates, 141

Fisher-Tippett theorem, 23â€“24

Econometrics, 1 Fixed point, 69â€“70

Econophysics, 1â€“2 Flow, 73â€“74

Efficient frontier, 114 Fokker-Planck equation, 30â€“31

Efficient market, 12 Foreign exchange rates, 141

Efficient Market Hypothesis (EMH), 12â€“14, 40 Forward contract, 93

random walk, 12â€“13 Fractal. See also Multifractal

semi-strong, 12 box-counting dimension, 61

strong, 12 deterministic, 60â€“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, 55â€“56

EMH. See Efficient Market Hypothesis Frechet distribution, 24

Equilibrium models Fundamental analysis, 12

adaptive, 130â€“133 Fundamentalists, 132, 134â€“135, 137, 141

non-, 130, 134â€“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, 21â€“22

Expiration date, 94. See also Maturity Generalized autoregressive conditional

Exponential generalized autoregressive heteroskedascisity (GARCH), 52â€“53,

conditional heteroskedascisity 85

(EGARCH), 53â€“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, 76â€“77

Kupiec test, 126

Hang-Seng index

Kurtosis, 19

returns of, 89

Historical simulation, 125

Â¨

Holder exponent, 63

L

Homoskedastic process, 51â€“54

Hopf bifurcation, 78

Hurst exponent, 62 Lag operator, 43â€“44

Langevin equation, 32

I Law of One Price, 10

Leptokurtosis, 19

Levy distribution, 25â€“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, 70â€“72, 73f, 74f

distributed process attractor on, 72

Implied volatility, 103 basin of attraction on, 72

Independent variables, 20

fixed point on, 71â€“73

Independently and identically distributed Lognormal distribution, 22â€“23

process (IID), 33 Long position, 6

Indicative rates, 141 Lorentzian distribution. See Cauchy

Initial condition, 30

(Lorentzian) distribution

Integral Lorenz model, 70â€“71, 79â€“82, 80f, 81f, 82f

stochastic, 36â€“39 Lotka-Volterra system, 90

stochastic Itoâ€™s, 38â€“39 Lyapunov exponent, 82â€“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

Itoâ€™s integral liquidity, 6, 141â€“142, 143f

stochastic, 38â€“39 microstructure, 6

Itoâ€™s lemma, 35â€“36 orders, 6

over-the-counter, 5

price formation, 5â€“7

J Market microstructure, 136

Market portfolio, 115

January Effect, 14 Market-neutral strategies, 118

Joint distribution, 19 Markov process, 29â€“32

Martingale, 39â€“41

165

Index

O

sub, 40

super, 40

Mathematical Finance, 1 OLS. See Ordinary least squares

Maturity, 93â€“94 Operational time, 7

Maximum likelihood estimate (MLE), 48 Options, 98

â€˜â€˜Maxwellâ€™s Demon,â€™â€™ 136â€“137 American, 94â€“96

MBS. See mortgage-backed securities call, 94

arbitrage European, 94â€“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, 93â€“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, 95â€“96, 97f

Minority game, 129â€“130 strike price of, 94

MLE. See Maximum likelihood estimate Orders

Mortgage-backed securities (MBS) limit, 6

arbitrage, 119 market, 6

Moving average model, 45â€“47 stop, 6

autoregressive, 45â€“46 Ordinary least squares (OLS), 48

invertible, 46â€“47 Ornstein-Uhlenbeck equation, 42

MSE. See Mean squared error

Multifractal, 63â€“64. See also Fractal

binomial measure, 64

P

cascade, 63â€“64

spectrum, 64

Multipliers, 64 Pair trading, 118

Multivariate time series, 54â€“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, 134â€“136 delta-neutral, 106

Non-integrable system, 75 rebalancing, 106

Normal distribution, 21â€“22 well-diversified, 117

standard, 22, 24f Portfolio selection, 111â€“115

Notations, 4 Position

166 Index

R

long, 93

short, 93

Positive excess kurtosis. See Random walk, 12â€“13, 44

Leptokurtosis continuous-time, 34

Present value, 8â€“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, 12â€“13

Price-demand relations, 138â€“139, 138f, Rescaled range (R/S) analysis, 63, 88

139f Return

Pricing model compounded, 8

discounted-cash-flow, 8â€“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, 36â€“37

autoregressive, 43 Risk

compound stochastic, 92 cash-flow, 121

homoskedastic, 51â€“54 coherent, measures, 124

independently and identically distributed credit, 121

(IID), 33 liquidity, 121

Markov, 29â€“32 market, 121

mean-reverting, 42 operational, 121

multiplicative, 64 Risk-free asset, 130â€“131. See also Bond

persistent, 62 Risk-neutral valuation, 99

scale-free, 26 Risk-return trade off line, 112

standard Wiener, 31â€“32, 34â€“35 Risky asset, 130â€“131

stationary, 49 ROE. See Return on Equity

stochastic, 29â€“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, 66â€“67

Quasi-periodicity, 83 Seasonal effects, 45â€“46

167

Index

Security market line, 115 deterministic, 49â€“50, 50f

Self-affine object, 59 stochastic, 49â€“50

Self-affinity, 59 Truncated Levy flight, 26â€“27, 88â€“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, 31â€“32, 34â€“35

Stationary process, 49

non-, 49 Value at risk (VaR), 122â€“124, 123f

Statistical arbitrage, 14 conditional, 141

Stieltjes integral, 37 Van der Pol

Stochastic equation, 77â€“78

compound, process, 92 oscillator, 78f

differential equation, 35 VAR. See Vector autoregressive model

integral, 36â€“39 VaR. See Value at risk

Itoâ€™s integral, 38â€“39 Variance, 18

process, 29â€“42 matrix, 19

trend, 49â€“50, 50f Variate, 16

Stochastic trend v. deterministic trend, 49â€“50, Vector autoregressive model (VAR), 55â€“56

50f Vega, 104

Stop orders, 6 Volatility, 19

Stratonovichâ€™s integral, 39 implied, 103

Strict stationarity, 49 smile, 104

Submartingale, 40 Volatility smile, 104â€“105

Super-efficient portfolio, 114

Supermartingale, 40

W

T Weak stationarity, 49

White noise, 33, 43

Technical analysis, 12 non-, 39

Term structure, 104â€“105 Wiener process

Theta, 103 standard, 31â€“32, 34â€“35

Tick, 6

Tick-by-tick data, 6â€“7

Z

Traders

regular, 139â€“141

technical, 139â€“141, 140f

Zipfâ€™s law, 89

Trajectory, 71, 76f, 77â€“79, 78f, 79f

Trend

ñòð. 5 |