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163

Leverage effect (cont.)

skews, 31, 68 Likelihood functions,

boundaries, 96

constant volatility, 122

definition, 91

gradient, 94

gradient algorithm, 96

local optimum, 96

log-likelihood, 95

maximum see Maximum likelihood

minimum data, 91

out-of-sample, 121

post-sample predictive tests, 122

prediction, 445

second derivatives, 94

time-varying variance, 95 Likelihood ratio (LR), 428-9 Linear portfolios,

covariance matrices, 180-2

factor models, 181, 187, 229

linear regression, 172

optimization, 179

risk factors, 181

stress testing, 185

variance, 180-2, 183 Linear regression models,

CAPM see Capital asset pricing model

combined forecasts, 133

conditional variance equation, 64

estimation, 230

explanatory variables/regressors, 172, 409-10

homsoscedastic returns, 14

multivariate models, 412-14

ordinary least squares (OLS), 414-19

simple linear model, 410-12

slope parameters, 7 Local volatility,

moneyness, 37

sticky tree model, 38

strikes, 34-5

trending markets, 37 Lorenz butterfly, 401 Lyapunov exponent, 403, 404

Marked-to-market (MtM) value, adjustment for uncertain volatility, 136-8

ATM options, 135, 136-7 long-term volatility, 57 option portfolios, 119 portfolio valuation, 134-5 VaR (value-at-risk) models, 135

Market events, Asian crisis, 90, 185 Black Monday, 52-3, 57, 85, 185 currency markets, 102 equity crash (1987), 90, 250 equity crash (1998), 166, 169-70, 171 extreme events, 31, 52, 57, 80, 85, 99,

117, 185, 203, 204 ghost effects see Ghost effects/features Gulf War, 55-6, 76, 78, 85 intensity of reaction, 102 news announcements, 65-6, 79 Russian debt crisis, 35, 251

Market price, model price, 21, 22

Market regimes,

ATM volatility, 36-8, 39, 43 fixed strike volatility, 159-67 jumpy markets, 35, 36, 37-8, 44, 156 range-bounded markets, 36, 44, 117, 156 sticky models, 36-8, 44 trending markets, 36, 37, 44, 156

Market risk capital requirements (MMR). 251, 252, 253, 254-5. 274, 276, 279, 280

internal models, 252-5 scenario analysis, 253 Markets,

commodities see Commodity markets energy see Energy markets financial see Financial markets foreign exchange see Currency markets shares see Equity markets Maximum likelihood, see also Likelihood functions GARCH models, 447 maximum likelihood estimators (MLEs),

447, 448, 449-52 methods, 447-52

non-normal density function, 451-2

normal density function, 449-50 MBRM-GARCH, 64 Mean-reversion,

commodity markets, 75

currency markets, 75

time series models, 317-20

univariate GARCH, 111

volatility clustering, 31

volatility forecasts, 75, 79

volatility term structures, 61, 98, 109, 111

Mean-variance analysis, asset allocation, 230 efficient frontier, 201 investment analysis. 185-6, 197, 198-201 portfolio theory, 186 utility functions, 197



MFIT statistical package, 82, 84 Minimum variance portfolios,

constrained minimum variance, 191

covariance matrices, 187-92

efficient frontier, 191

global minimum variance, 188, 189, 191

risk-free assets, 192 Missing data,

missing observations, 439-40

new assets, 144

principal component analysis (PCA), 18, 144, 171, 174-8, 439 Model price, market price, 21, 22 Moneyness,

constant volatility, 24

definitions, 23, 24, 25, 44

local volatility, 37 Monte Carlo simulation,

covariance matrices, use of, 185

for derivatives portfolios, 104, 105, 182

spread options, 367

VaR (value-at-risk) models, 267, 270-3, 274, 277-8 Morgan Stanley indices, 217, 369, 371 Moving averages,

ARMA see Autoregressive moving average

confidence intervals, 126-8

equally weighted see Weighted average

exponentially weighted (EWMA), 49, 57-60

historic correlation, 49

historic volatility, 49

long-term, 18

sampling error, 63

short/long, 51

time series models, 331-2

unconditional correlation, 49

unconditional volatility, 49

volatility forecasts, 126-8 Multi-factor models,

see also Factor models

arbitrage pricing theory (APT), 181, 233. 247

explanatory variables, 172

fundamental factors, 233-5

ordinary least squares (OLS), 237

regression analysis, 144 Multicollinearity. 144, 172-4, 436-7 Multivariate distributions, copulas, 8 Multivariate GARCH.

Baba, Engle, Kraft and Kroner see BEKK model

see also Bivariate GARCH

academic research, 64

asymmetry, 114

computational problems, 107, 115 conditional mean, 111 convergence, 109, 115 correlation estimates and forecasts, 205 covariance matrices, 114, 115, 210 cross equation restrictions, 114 large-dimensions, 108, 112 parameterization, 112-14, 217 portfolio risk, 112 vech model, 113, 218 volatility estimates and forecasts, 217 Multivariate time series, covariance stationarity, 341-4 Granger causality, 315-16, 344-6 time series models, 340-6 vector autoregressions, 340-1

N-GARCH,

local risk-neutral valuation, 106

parameter estimates, 107

volatility smile surface, 81, 106, 107 Neural networks, 395-6, 447 New assets, missing data points, 144 News announcements,

see also Market events

asymmetric volatility, 79

volatility clustering, 65-6 Nikkei 225: 15, 86, 89, 129, 155, 199, 200,

403 Noise,

Gaussian white noise, 316 high frequency data, 17 moving averages, 49, 63 orthogonal GARCH, 216 Non-normal returns,

distributions, testing, 286-90 excess kurtosis, 286-7, 309-11 extreme value distributions, 290-6 generalized extreme value (GEV), 290, 293

hyperbolic distributions, 296-7 non-normal distributions, 290-301 normal mixture distributions, 297-311 peaks over threshold (POT), 290-2, 294, 295

QQ plots, 288-90

skews, 286-7 Normal mixture distributions,

applications, 301-11

covariance VaR models, 302-3

method of moments, 300 Normal mixture distributions (com.)

non-normal returns, 297-311

option pricing, 305-11 NYMEX, 54, 56. 111. 153. 154. 326. 355



Omega, GARCH, 77, 86, 111 Option hedging,

see also delta, gamma, vega

GARCH models, 98

perfect hedge assumed, 21

underlying assets, 10 Option pricing, 24,

see also Call options; Put options

Black-Scholes model, 1, 10, 21, 104, 106, 305-6

calibrating model, 34, 43

excess kurtosis, 306

GARCH models, 98

implied volatility, 4, 10, 2115

Monte Carlo simulation, 104, 105

normal mixture distributions, 305-11

observed market price, 21

over/under priced, 29

path dependent, 100

smile effect, 106-7, 136

statistical volatility, 29 Option value, discounted expectation, 104, 138

Ordinary least squares (OLS), alpha, 167

autocorrelation, 431, 432 beta, 111, 167, 231, 236, 238 capital asset pricing model (CAPM), 111 constrained allocations, 371 correlation, 7

covariance matrices, 237, 419-20 Engle-Granger methodology, 354, 356 estimators, 414-19, 433-6 high/low risk stock, 109 linear regression models, 414-19 multi-factor models, 237 non-stochastic regressors, 417-18 normal distribution, 419 orthogonality, 172 portfolio risk, 237 regression, 167, 173, 174 residual analysis, 429 sampling error, 236 stochastic regressors, 418-19 Orthogonal GARCH, accuracy, 205 calibration, 211-20 conditional correlation, 211 convergence, 217

correlation estimates and forecasts, 109, 110

covariance matrices, 116, 180, 210-20 eigenvalues and eigenvectors, 215, 217 estimation time periods, 220 illiquid maturities, 214 noise, 216

positive semi-definiteness, 211

volatility term structures, 212 Orthogonality,

covariance matrices, 144, 180, 204-27

exponentially weighted moving average (EWMA), 204, 206-10, 212-14

ordinary least squares (OLS), 172

principal component analysis (PCA), 141, 146, 204, 207-9 OTC options, 135, 138 OTM (out of the money),

Black-Scholes model, 136, 297, 306, 309

call options, 24, 30

equity options, 104

implied volatility, 30

long-term options, 33

market regimes, 160

put options, 26, 28, 30, 31

risk horizon, 43

short-term options, 33

volatility forecasts, 119

P&L,

prediction, 446

simple cash portfolios, 261-2

trading strategy, 124, 125

VaR (value-at-risk) models, 180, 253, 254, 262-3

variance, 181-2 Parameterization,

BEKK model, 114

bivariate GARCH, 108, 109, 111

multivariate GARCH, 112-14, 217

vech models, 113, 114

volatility surfaces, 167-70 Persistence, in volatility

see also Beta

exponentially weighted moving average

(EWMA), 59, 207 GARCH models, 92 high frequency data, 83 I-GARCH, 76 Japan, 86

RiskMetrics data, 76, 202

United States, 86, 91 Phillips-Perron test, 328 Point forecasts,

evaluation, accuracy, 119-25

meaning, 118

option valuation, 136

volatility, 118, 126 Portfolio diversification, correlation, 186-7 Portfolio risk,

see also Risk

covariance matrices, 179

determinants, 1



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