Index
A
Adaptive equilibrium models, 130–132
APT. See Arbitrage Pricing Theory
Arbitrage, 11
convertible, 119
equity market-neutral strategy and
statistical, 119
fixed-income, 119
merger, 119
relative value, 119–120
statistical, 13
trading strategies of, 118–120
Arbitrage Pricing Theory (APT), 116–118
ARCH. See Autoregressive conditional
heteroskedascisity
ARIMA. See Autoregressive moving
integrated average model
ARMA. See Autoregressive moving average
model
Ask, 5
Attractor, 72
quasi-periodic, 78
strange, 69
Autocorrelation function, 47
Autocovariance, 47
Autonomous systems, 75
Autoregressive conditional heteroskedascisity
(ARCH), 52
exponential generalized (EGARCH), 53–54
generalized (GARCH), 52–53, 87
integrated generalized (IGARCH), 53
Autoregressive moving average model
(ARMA), 45–46
integrated (ARIMA), 46
Autoregressive moving integrated average
model (ARIMA), 46
Autoregressive process, 43
B
Basin of attraction, 72
Behavioral finance, 13
Bernoulli trials, 20
Beta, 115
Bid, 5
Bifurcation
global, 82
Hopf, 78
local, 82
point of, 70, 71f
Binomial
cascade, 64–66, 65f
distribution, 21
measure, 64
tree, 98–101, 99f
Black-Scholes equation, 102–104
Black-Scholes Theory (BST), 101–105
Bond, 130–131
Bounded rationality, 14, 133
Box-counting dimension, 61
Brownian motion, 32–35
arithmetic, 34
fractional, 62–63
geometric, 34
C
Capital Asset Pricing Model (CAPM),
114–116, 118
Capital market line, 114
CAPM. See Capital Asset Pricing Model
CARA. See Constant absolute risk aversion
function
Cascade, 64
binomial, 64–66, 65f
canonical, 66
conservative, 65
microcanonical, 65
multifractal, 63–64
multiplicative process of, 64
Cauchy (Lorentzian) distribution, 23, 24f
standard, 23
Central limit theorem, 22
Chaos, 70, 82–85
measuring, 83–85
Chaotic transients, 83
Chapmen-Kolmogorov equation, 30–31
Characteristic function, 25
Chartists, 132, 134–135, 137, 138
Coherent risk measures, 124
Cointegration, 51
Compound stochastic process, 92
Compounded return, 8
continuously, 8
Conditional expectation, 18
Conservative system, 76–77
Constant absolute risk aversion (CARA)
function, 132
Contingent claim. See Derivatives
Continuously compounded return, 8. See also
Log return
Continuous-time random walk, 34
Contract
forward, 93
future, 94
Contrarians, 133
Correlation
coefficient, 20
dimension, 85
Covariance, 20
matrix of, 20
stationarity-, 49
Crises, 83
Cumulative distribution function, 18
D
Damped oscillator, 76, 76f
Data
granularity, 88
snooping, 54
Delta, 103
Delta-neutral portfolios, 104
Derivatives, 93
Deterministic trend v. stochastic trend,
49–50, 50f
Dickey-Fuller method, 45, 51
Dimension
box-counting, 61
correlation, 85
fractal, 60
Discontinuous jumps, 31
Discounted-cash-flow pricing model, 8–9
Discounting, 9
Discrete random walk, 33
Dissipative system, 76
Distribution
binomial, 21
Cauchy (Lorentzian), 23, 24f
extreme value, 23
Frechet, 24
Gumbel, 24
Iibull, 24
Levy, 25–27
lognormal, 22–23
normal (Gaussian), 21–22
Pareto, 24, 26
Poisson, 21
stable, 25
standard Cauchy, 23
standard normal, 22, 24f
standard uniform, 20
uniform, 20
Dividend effects, 8–10, 96
Dogs of the Dow, 14
Doob-Meyer decomposition theorem, 41
Dow-Jones index
returns of, 89
162 Index
Dummy parameters, 51
Dynamic hedging, 104
E
Econometrics, 1
Econophysics, 1–2
Efficient frontier, 114
Efficient market, 12
Efficient Market Hypothesis (EMH), 12–14, 40
random walk, 12–13
semi-strong, 12
strong, 12
weak, 12
Efficient Market Theory, 12
EGARCH. See Exponential generalized
autoregressive conditional
heteroskedascisity
EMH. See Efficient Market Hypothesis
Equilibrium models
adaptive, 130–133
non-, 130, 134–135
Equity hedge, 119
Error function, 22
ETL. See Expected tail loss
Euro, 88
EWMA. See Exponentially weighed
moving average; exponentially weighed
moving average
Exchange rates
foreign, 86
Exogenous variable, 56
Exotic options, 141
Expectation, 18. See also Mean
Expected shortfall, 141
Expected tail loss (ETL), 124, 124f
Expiration date, 94. See also Maturity
Exponential generalized autoregressive
conditional heteroskedascisity
(EGARCH), 53–54
Exponentially weighed moving average
(EWMA), 53
Extreme value distribution, 23
F
Fair game, 40
Fair prices, 12–13
Firm rates, 141
Fisher-Tippett theorem, 23–24
Fixed point, 69–70
Flow, 73–74
Fokker-Planck equation, 30–31
Foreign exchange rates, 141
Forward contract, 93
Fractal. See also Multifractal
box-counting dimension, 61
deterministic, 60–63, 60f
dimension, 60
iterated function systems of, 61
random, 60
stochastic, 60f
technical definitions of, 55–56
Frechet distribution, 24
Fundamental analysis, 12
Fundamentalists, 132, 134–135, 137, 141
Future
contract, 94
value, 9
Future contract, 94
G
Gamma, 103
Gamma-neutral, 104
GARCH. See Generalized autoregressive
conditional heteroskedascisity
Gaussian distribution, 21–22
Generalized autoregressive conditional
heteroskedascisity (GARCH), 52–53,
85
Given future value, 9
Granger causality, 56
Greeks, 103
Gumbel distribution, 24
Index 163
H
Hamiltonian system, 76–77
Hang-Seng index
returns of, 89
Historical simulation, 125
Ho
¨
lder exponent, 63
Homoskedastic process, 51–54
Hopf bifurcation, 78
Hurst exponent, 62
I
IGARCH. See Integrated generalized
autoregressive conditional
heteroskedascisity
Iibull distribution, 24
IID. See Independently and identically
distributed process
Implied volatility, 103
Independent variables, 20
Independently and identically distributed
process (IID), 33
Indicative rates, 141
Initial condition, 30
Integral
stochastic, 36–39
stochastic Ito’s, 38–39
Integrated generalized autoregressive
conditional heteroskedascisity
(IGARCH), 53
Integrated of order, 45
Intermittency, 83
Irrational exuberance, 13
Iterated map, 71
Iteration function, 71
Ito’s integral
stochastic, 38–39
Ito’s lemma, 35–36
J
January Effect, 14
Joint distribution, 19
K
Kolmogorov-Sinai entropy, 84
Kupiec test, 126
Kurtosis, 19
L
Lag operator, 43–44
Langevin equation, 32
Law of One Price, 10
Leptokurtosis, 19
Levy distribution, 25–26
Limit cycle, 77
Limit orders, 6
Log return, 8. See also Continuously
compounded return
Logistic map, 70–72, 73f, 74f
attractor on, 72
basin of attraction on, 72
fixed point on, 71–73
Lognormal distribution, 22–23
Long position, 6
Lorentzian distribution. See Cauchy
(Lorentzian) distribution
Lorenz model, 70–71, 79–82, 80f, 81f, 82f
Lotka-Volterra system, 90
Lyapunov exponent, 82–85
M
Market(s)
bourse, 5
exchange, 5
liquidity, 6, 141–142, 143f
microstructure, 6
orders, 6
over-the-counter, 5
price formation, 5–7
Market microstructure, 136
Market portfolio, 115
Market-neutral strategies, 118
Markov process, 29–32
Martingale, 39–41
164 Index
sub, 40
super, 40
Mathematical Finance, 1
Maturity, 93–94
Maximum likelihood estimate (MLE), 48
‘‘Maxwell’s Demon,’’ 136–137
MBS. See mortgage-backed securities
arbitrage
Mean, 18
reversion, 44
squared error, 48
Mean squared error (MSE), 48
Mean-reverting process, 42
Mean-square limit, 38
Mean-variance efficient portfolio, 108
Median, 18
Microsoft Excel, 4, 25
Mimetic contagion, 134
Minority game, 129–130
MLE. See Maximum likelihood estimate
Mortgage-backed securities (MBS)
arbitrage, 119
Moving average model, 45–47
autoregressive, 45–46
invertible, 46–47
MSE. See Mean squared error
Multifractal, 63–64. See also Fractal
binomial measure, 64
cascade, 63–64
spectrum, 64
Multipliers, 64
Multivariate time series, 54–57
N
Noise
non-white, 38
white, 33, 43
Nonanticipating function, 39
Non-equilibrium price models, 130, 134–136
Non-integrable system, 75
Normal distribution, 21–22
standard, 22, 24f
Notations, 4
O
OLS. See Ordinary least squares
Operational time, 7
Options, 98
American, 94–96
call, 94
European, 94–96
exercise price of, 94
exotic, 141
expiration date of, 94
long call, 95, 97f
long put, 95, 97f
maturity of, 93–94
premium of, 96
put, 94
short call, 95, 97f
short put, 95–96, 97f
strike price of, 94
Orders
limit, 6
market, 6
stop, 6
Ordinary least squares (OLS), 48
Ornstein-Uhlenbeck equation, 42
P
Pair trading, 118
Pareto distribution, 24, 26
Partition function, 67
Partly forcastable prices, 70
Period-doubling, 82
Persistent process, 62
anti-, 63
P/L. See Profits and losses
Poisson distribution, 21
Portfolio
delta-neutral, 106
rebalancing, 106
well-diversified, 117
Portfolio selection, 111–115
Position
Index 165
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