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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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