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162 capital asset pricing model (CAPM), 230 cointegration, 347, 366-9 controlling risk, 250-5 correlation, 8, 10, 11, 12, 111 crises, 250 deterministic trends, 328, 329 historic volatility, 52-4 jumps, 297 lag/persistence coefficients, 73 return distributions, 119 spot-future correlations, 55 trend, testing, 328-9 unconditional correlation, 15 volatility, 9-12, 119, 250, 297 volatility clustering, 17, 64 volatility forecasts, 75, 120 Fixed strike volatility, dynamics, 159-67 gamma, 160, 161, 162 market regimes, 159-67 principal component analysis (PCA), 157, 158, 159-67, 169 sticky models, 38 Forecasts, autoregressive moving average (ARMA), 338-9 correlation see Correlation forecasts covariance see Covariance forecasts hedging performance, 124 high frequency data, 389-407 operational criteria, 124-5 out-of-sample tests, 121 parameters, 117 point see Point forecasts post-sample predictions, 121, 122 statistical see Statistical forecasts statistical criteria, 121-4 trading systems, 180 volatility see Volatility forecasts Foreign exchange see Currency markets Frechet distribution, 293, 294 Frontier analysis, see also Efficient frontier short sales, 186, 191 Futures. cointegration. 367 crude oil, 55, 60, 153, 154, 212, 213 implied volatility, 28 natural gas, 54, 55, 111, 326 orthogonality, 205 prompt futures, 54, 56, 78, 111, 326 spot price, hedging. 27 spot-future correlations. 55 term structures. 153-4 Game theory rationality. 21 Gamma, ATM options, 44 Black-Scholes model, 32 delta sensitivity, underlying price, 43 delta-gamma approximations, 2731 finite difference approximations, 105 GARCH models, academic research, 63-5 aggregation, 82, 83 applications, 97-8 assumptions, 98 asymmetric see A-GARCH asymmetric power GARCH, 81 basic form, 64 bivariate see Bivariate GARCH choosing best, 96-7 combined forecasts, 129-33 components GARCH, 78-9 conditional mean, 69-70, 100 conditional variance, 70, 72, 73, 79, 97, 98, 109 conditional volatility, 81, 96-7 confidence intervals, 128-9 constant (omega), 91 constant correlation, 115 convergence, 73, 75, 84, 96, 97-8 correlation term structures, 109 covariance matrices see Covariance matrices data examination, 67 diffusion limits, 98 error coefficient, 73 estimation problems, 96 exponential GARCH (E-GARCH), 79-80 Factor GARCH, 115 fat tails, 125 first derivatives, 95 forward volatilities, 98 GJR model, 81 gradient vectors, 95 high frequency (HARCH), 82-4 implied volatility, 106, 118 integrated see I-GARCH intra-day data, 82, 84 introduction, 65-70 lag coefficient, 73 Lagrange multiplier (LM) test, 67, 68 leverage effect, 64, 68-9, 79 likelihood function, 91 low-frequency, 82, 83 maximum likelihood, 447 MBRM-GARCH, 64 multivariate see Multivariate GARCH non-linear see N-GARCH non-normal, 73, 82
GARCH models (cont.) option hedging, 1, 98 option pricing, 1, 98 orthogonal see Orthogonal GARCH parameter estimates, 85, 91 parameter estimation algorithms, 92-5 persistence, 92 positive semi-definiteness, 184 quadratic GARCH, 81 reaction, 92 realized volatility, 11 RiskMetrics data, 202 second derivatives, 94 specification and estimation, 84-98 stochastic processes, 98, 105 symmetric see Symmetric GARCH t-GARCH, 82 threshold GARCH, 81 time-varying correlation, 16-17, 108-14 time-varying variance, 70 time-varying volatility models, 14, 95, 115 uncertainty, 128 univariate see Univariate GARCH vanilla, 72, 73, 75. 79, 84, 96 volatility clustering, 96, 97 volatility forecasts, 91, 98, 99-100 volatility term structures, 32, 50, 61, 64, 85, 98-103 GAUSS programs, 82, 92 Gaussian white noise, 316 Generalized autoregressive conditional heteroscedasticity see GARCH models Generalized least squares (GLS), estimators, 433-6 residual analysis, 429 risk factors, 237-8 systems of seemingly unrelated regression equations (SURE), 434-5 Generalized Pareto distributions (GPD), conditional VaR, 260, 295 extreme value distributions, 290 underlying distribution, 295 Geometric Brownian motion (GBM), Black-Scholes model, 10, 21, 30 constant volatility, 117 diffusion process, 104 implied volatility, 10, 23, 28 spread options, 367 underlying price, 30 Ghost effects/features, historic correlation, 55 historic volatility, 53, 128 RiskMetrics data, 204 weighted average, 53, 60, 125 GJR model, 81 Goldfeld-Quandt test, 432, 433 Granger causality, 315-16, 344-6, 347 Granger representation theorem, 361 Granger-Ramanathan procedure. 118, 129 Guidelines for Banking Supervision, 280 Gulf War, 55-6, 76, 78, 85 Gumbel distribution, 293, 294 Hang Seng, 90, 91 HARCH, High Frequency ARCH, 83-4 Hedging, basket options, 56-7 correlation risk, 138 delta, 43-5, 119, 138, 139 dynamically hedged portfolios, 138-0 futures, spot price, 27 implied volatility, 119, 139, 140 options see Option hedging perfect hedge, 21, 106 performance, 124 proxy hedge ratio, 54, 111 rebalancing, 112 standard error, 139, 140 statistical hedge ratios, 109 volatility, 139-40 Hessian matrix, 94, 95 High frequency data, alpha, 83 ARCH, 82, 393 autocorrelation, 391-3 beta, 83 chaotic dynamics, 401-7 conditional mean, 82 data and information sources, 390 data filters, 390-1 data processing, 397-8 error distributions, 82 forecasts, 389107 GARCH models. 824 neural networks, 395101 noise, 17 parametric models, 393-5 persistence, 83 price prediction models, 401-7 sampling error, 17 square root of time rule, 83 Historic correlation, basket options, 56 definition and application, 50-2 equally weighted moving averages, 49, 50 extreme events, 52, 57 ghost effects/features, 55
proxy hedge ratio, 54 sampling error, 51 use of estimates, 57 Historic volatility, computation, 118 constant volatility, 120 definition and application, 50-2 equally weighted moving averages, 49, 50, 52-3 extreme events, 52, 57, 85 ghost effects/features, 53, 128 volatility estimates, 57, 128 I-GARCH, alpha, 77, 78 beta, 77, 78 currency markets, 76 equity markets, 76 exponentially weighted moving average (EWMA), 60, 76, 111 long-term volatility, 90 omega, 77 persistence, 76 unconditional variance, 76 volatility estimates, 76, 77 Ibovespa, 89, 90 Implied correlation, 45-7 Implied trees, 35 Implied volatility, ATM volatility. 30, 34, 35, 279 back out, 26, 34, 106 Black-Scholes model, 12, 22, 23-6, 33, 34, 106 call options, 26-8 constant volatility, 22 features, 30-4 futures options, 28 geometric Brownian motion (GBM), 10, 23, 28 hedging, 119, 139, 140 meaning, 10, 22-30 option maturity, 31 option pricing, 4, 10, 2115 OTM. 30 principal component analysis (PCA), 143 and process volatility, 22 put options, 26-8 scenario analysis. 384-3, 185 skews, 1, 30-1, 32, 33, 68, 155, 158 smile effect, 1, 30, 32, 33, 98, 155 statistical volatility distinguished, 28-30 strikes. 26. 27. 28, 30, 31, 32-3, 155 trading strategy. 125 and underlying. 11, 26. 28. 183 \o!aiilit\ cones. 29 volatility forecasts, 117-18, 124-5 volatility smile surface, 106, 154 volatility surfaces, 32-4 volatility term structures, 31-2, 78 Index tracking models, linear regression, 172 Information ratio (IR), 194, 445 Integration, cointegration see Cointegration GARCH models see I-GARCH high frequency data, 400-1 time series models, 320-2 International equity portfolios, 181, 234 Intra-day data, see High frequency data Investment analysis, capital allocation see Capital allocation cointegration, 369-81 covariance matrices, 179, 185-201 global asset management, 186 mean-variance analysis, 185-6, 197, 198-201 optimal allocations, 186 quadratic programming, 185 trading limits, 186 utility functions, 185, 194-7 ITM (in the money), Black-Scholes model, 136, 297, 306, 309 call options, 24, 26, 28, 30, 31 long-term options, 33 market regimes, 160 straddles, 124 Jarque-Bera (JB) statistic, 287 Johansen methodology, 357, 361, 363, 368 Joint density, correlation, 5, 15 price/volatility, 41, 42, 43 JP Morgan, 163, 179, 201, 202 Jumpy markets, market regimes, 35, 36, 37-8, 44, 156 KCBOT, natural gas, 54, 55, 111 Kurtosis, excess see Excess kurtosis term structures, 303-5 Lagrange multiplier (LM), 67, 68, 336. 428, 432 Leptokurtic distributions, see Fat tails Leverage effect, coefficient, 99, 100 debt/equity ratios, 68 equity markets, 31. 68-9. 79 GARCH models, 64. 68-9. 79
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