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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].
157
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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
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19. A. B. Schmidt, ˜˜Modeling the Birth of a Liquid Market,™™ Physica A283,
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20. S. Solomon, ˜˜Importance of Being Discrete: Life Always Wins on the
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Deterministic Agents and Derivation of Limit of GARCH Process,™™
http://xxx.lanl.gov/cond-mat0109139/.
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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

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