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161 Confidence intervals (cont.) moving averages, 126-8 prediction, 444 realized volatility, 133 Constant absolute risk aversion (CARA), 196 Constant correlation, estimates, 108 models, 14-17 Constant relative risk aversion (CRRA), 196, 197 Constant volatility, ATM volatility, 37 Black-Scholes model, 21, 24, 26, 30, 34, 61 geometric Brownian motion (GBM), 117 historic volatility, 120 implied volatility, 22 likelihood functions, 122 models, 12, 13, 18, 21, 22 square root of time rule, 61 unconditional volatility, 12 underlying asset price process, 22 volatility term structures, 61, 99 Convergence, BEKK model, 217 bivariate GARCH, 108, 109 multivariate GARCH, 109, 115 orthogonal GARCH, 217 univariate GARCH, 73, 75, 84, 96, 97-8, 108, 109 volatility term structures, 12, 75, 76, 97-8, 102-3 Copulas, co-dependency, 8 financial risk management, 9 multivariate distributions, 8 Correlation, apparent stability, 16 basic concept, 3-4 co-dependency, 7, 8 co-movements, 5, 6 conditional, 16-17 constant see Constant correlation cross-market, 15-16, 18 exponentially weighted moving average (EWMA), 55, 76, 111 financial markets, 8, 10, 11, 12, 111 GARCH constant, 115 historical see Historic correlation implied, 45 joint density, 5, 15 long-term, 57 ordinary least squares (OLS), 7 orthogonality, 7 portfolio diversification, 186-7 positive/negative, 6, 8 principal component analysis (PCA), 143 quanto correlation, 46-7 scatter plots, 8 statistical nature, 5-9 time horizons, 18 time-varying see Time-varying correlation unconditional see Unconditional correlation Correlation estimates, 11, 15-16, 18, 19, 50 artefact of method, 56 BEKK model, 109, 110 bivariate GARCH, 109, 110, 205 conditional correlation, 16-17 constant correlation, 108 cross market, 15-16 multivariate GARCH, 205 orthogonal GARCH, 109, 110 time-varying, 15, 107 Correlation forecasts, uncertainty, 137-8 underlying correlation, 117 Correlation matrices, eigenvalues, 149, 152, 153, 154, 159 eigenvectors, 149, 152, 153, 154, 159 simple cash portfolios, 262 time-varying, 115 Correlation risk, hedging, 138 Correlation term structures, GARCH models, 109 Correlogam, 333-5 Counter-monotonic dependency, 8 Covariance, capital asset pricing model (CAPM), 109 co-movements, 6-7 stationary, 14-15 unconditional see Unconditional covariance VaR (value-at-risk) models, 237, 250 Covariance forecasts, 50, 109 Covariance matrices, applications, 180-201 capital asset pricing model (CAPM), 115, 238 Cholesky decomposition, 182, 1831 diagonal, 141, 144, 146, 206 exponentially weighted moving average (EWMA), 180, 184, 204, 206-10 large dimensions, 108, 112 linear portfolios, 180-2 meaning, 179 minimum variance portfolios, 187-92 orthogonality, 144, 180, 204-27
portfolio risk, 179 positive semi-definiteness, 116, 179, 180, 181, 206, 211 RiskMetrics data, 201-4 simplifying assumptions, 179 splicing methods, 220-1, 227 stress testing, 184-5 time-varying, 114-16 Covariance VaR models, advantages/limitations, 266-7 aggregation, 266 assumptions, 260-1 cash-flow maps, 263-5 factor models, 262-3 fat tails, 285 normal mixture distributions, 302-3 RiskMetrics data, 260 simple cash portfolios, 261-2 Credit risk capital requirement (CRR), 252, 255 Crude oil, futures, 55, 60, 153, 154, 212, 213 GARCH parameters, 77, 85 Currency markets, cointegration, 366 components GARCH, 79 conditional correlation, 16, 17 conditional volatility, 97 convergence, 102 correlation estimates, 16, 17 DEM, 102, 129-32, 349 excess kurtosis, 82, 301 exchange rates, random walk, 75 GBP, 16, 17, 102, 132-3 high frequency data, 97 I-GARCH, 76 implied correlation, 45-7 JPY, 16, 17, 102, 129-32 market events, 102 mean-reverting volatility, 75 NLG, 349 parameter estimates, 101-2 quanto correlation, 46-7 symmetric smiles, 30 time-varying correlation, 15 USD. 16. 17, 101-2, 129-33 volatility clustering, 65, 97 volatility term structures, 102 Data problems. see also Missing data data errors. 437-9 dummy variables. 440-2 multicollinearity, 144, 1721. 436-7 principal component analysis (PCA), 18, 144, 171-8 DAX 30: 85, 89, 155, 217, 297, 363, 364 Delta, Black-Scholes, 32, 44 calculation, 44, 105 chain rule, 44 constant absolute risk aversion (CARA), 196 delta-gamma approximations, 2734 finite difference approximations, 105 hedging, 43-5, 119, 138, 139 neutrality, 43, 44 price sensitivity, 34, 43 property, 196 sticky delta, 35, 37 Density, non-normal density function, 451-2 normal density function, 449-50 price/volatility, 41, 42, 43 Derivatives portfolios, correlated risk factor movements, 182-3 Monte Carlo simulation, 182 DFP algorithm, 95 Dickey-Fuller (DF) test, 325-8 Dispersion, 4, 119 Dow Jones 30, 231 Durbin-Hausmann test, 328 Durbin-Watson test, 431 Dynamic hedging, 138-0 E-GARCH, 79-80 Efficient frontier, capital allocation, 190 frontier analysis, 186, 191 mean-variance analysis, 201 minimum variance portfolios, 191 optimal portfolios, 199, 200 risk aversion, 197 Eigenvalues and Eigenvectors negative values, 184 orthogonal GARCH, 215, 217 principal component analysis (PCA), 145, 146, 147, 149, 152, 153, 154, 159 Energy markets, crude oil, 55, 60, 77, 85, 153, 154, 212. 213 historic correlation, 54-7 natural gas futures, 54, 55, 111. 326 Engle-Granger methodology, 353, 354-7. 360-1, 368 Equity markets, asymmetric distribution, 31 Australia, 90, 91 Brazil, 90 cointegration, 368-9
Equity markets {com.) convergence, 102 economic doom and gloom, 31 Hong Kong, 90 I-GARCH, 76 implied correlation, 45-7 index returns, beta, 181 jumpy markets, 35, 36, 37-8, 44 large price fall, 31, 99 leverage effect, 31, 68-9, 79 quanto correlation, 46-7 range-bounded markets, 36, 44 Singapore, 90 skew. 30-1, 155 South Africa, 90, 91 Taiwan, 90, 91 time-varying correlation, 15 trending markets, 36, 37, 44 USA see United States volatility clustering, 65 volatility regimes, 34-8 volatility term structures. 101-3 Error, data errors, 437-9 error distributions, 82 forecast error, 123 GARCH error coefficients, 73 mean absolute error, 445 mean square error, 445 RMSE see Root mean square error samples see Sampling error standard see Standard error Error correction model (ECM), cointegration, 347, 355, 362-6 price discovery, 367 spread options, 367 Estima, 80 EVIEWS, 84 Excess kurtosis, currency markets, 82, 301 intra-day returns, 285 non-normal returns, 286-7, 309-11 non-normal symmetric GARCH, 73 normal GARCH models, 82 option pricing, 306 single outliers, 67 volatility clustering, 66, 67 Exponential GARCH see E-GARCH Exponentially weighted moving average (EWMA), accuracy, 119 beta, 111, 167, 239 combined forecasts, 129-32 correlation, 55, 76, 111 covariance matrices, 180, 184, 204, 206-10 efficient frontier, 187 historic correlation, 55, 76 I-GARCH, 60, 76, 111 limitations, 111 look-back periods, 58 orthogonality, 204, 206-10, 212-14 persistence, 59, 207 positive semi-definiteness, 184 reaction, 59, 207 RiskMetrics data, 60, 115, 163, 179, 202, 203 sampling error, 111 smoothing constant. 115, 117, 163, 202, 203, 204, 217 squared returns, 57-8, 207 statistical volatility, 125 variance estimates, 127, 163 volatility estimates, 57-8, 60 volatility forecasts, 60, 119 Factor GARCH models, volatility, 115 Factor models, see also Multi-factor models Bayesian estimation, 242-5 CAPM see Capital asset pricing model cash-flow maps, 262 covariance VaR models, 262-3 decomposing risk, 230-6 factor sensitivities, 229, 23916 instrumental variables, 247 linear portfolios, 181, 187. 229 linear regression, 172 portfolio risk, 237 risk factors, 229 risk measurement, 22948 sources of risk, 232 specification procedures, 246-8 statistical, 235-6 tracking models, 229 Fat tails, see also Leptokurtic distributions covariance VaR models, 285 financial data, 82 GARCH models, 125 QQ plots, 288-90 return distributions, 30, 82, 125 Financial asset prices, random variables, 4 stochastic processes, 1, 3 Financial asset returns, log prices, 4 stochastic volatility, 285 Financial markets, ARCH, 71 autocorrelation, 63 Black Monday, 52-3, 57, 85, 185
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