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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