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165 Smile effect ( /.) volatility smile surface, 33, 81, 106, 107, 143 Smoothing constant, 115, 117, 163, 202, 203, 204, 217 Spot prices, cointegration, 367 crude oil, 55, 60, 78, 111 hedging, 27 Spot-future correlations, commodity markets, 55 Gulf War, 55-6 Spread options, 367 Square root of time rule, constant volatility, 61 high frequency data, 83 short-term horizons, 97 volatility clustering, 97 Squared returns, ARCH, 83 autocorrelation, 63, 65-6, 68, 97 exponentially weighted moving average (EWMA), 57-8, 207 regression, 123, 124 variance forecasts, 123, 124 Standard deviation, dispersion, 4 volatility forecasts, 121 Standard error, hedging, 139, 140 multicollinearity, 172 realized volatility, 133 regression, 372 variance estimators. 127 volatility estimators, 127 volatility forecasts, 118-19, 126 Standard Industrial Classification, 233 Stationary processes, covariance stationarity, 317-20 jointly covariance stationary, 14-15, 3414 strict stationarity, 317-20 Statistical factor models, 235-6 Statistical inference, analysis of variance (ANOVA), 427-8 confidence intervals, 421-4 /-tests, 426-7 hypothesis testing, 421-4 Lagrange multiplier (LM), 428 likelihood ratio (LR), 428-9 /-tests, 424-6 Wald test, 428 Statistical volatility, confidence limits, 29 exponentially weighted moving average (EWMA), 125 implied volatility distinguished, 28-30 option pricing, 29 volatility forecasts, 118 Sticky models, delta, 35, 37 equity skew, 45 fixed strike/ATM volatility, 38 market regimes, 36-8, 44 strikes, 35, 36 trees, 35, 43 validation, 156 Stochastic processes, asset return series, 12 financial asset prices, 1, 3 GARCH models, 98, 105 option pricing, 105 process volatility, 11, 22 random walk, 351 stationary, 12, 317-20 trends, 328 Stochastic volatility, diffusion limits, 98 financial asset returns, 285 large price changes, 30 option pricing, 104, 105 perfect hedge, 106 risk-neutral probability, 81 Straddles, 118, 124-5, 386 Straights Times, 89, 90 Stress testing, covariance matrices, 184-5 linear portfolios, 185 risk management, 141 VaR (value-at-risk) models, 278, 281-3 Strikes, discounted value, 23, 24 fixed strike deviations, 158, 160, 161 fixed strike volatility, 38, 157, 158, 159-67 implied volatility, 26,27,28, 30, 31, 32-3, 155 local volatility, 34-5 sticky strike model, 35, 36 Student t distributions, 82, 293 Symmetric GARCH, 72-5, 81-2, 95, 99, 123 Systems of seemingly unrelated regression equations (SURE), 434-5 t-GARCH, Student t distributions, 82 Term structures, cointegration, 368 correlation see Correlation term structures futures prices, 1534
kurtosis, 303-5 principal component analysis (PCA), 143, 147-54 volatility see Volatility term structures Threshold GARCH, 81 Time series models, ARMA see Autoregressive moving average autocorrelation, 335-7 autoregression, 329-31 basic properties, 316-29 correlogams, 333-5 detrending financial data, 322-4 Dickey-Fuller (DF) test, 325-8 integrated processes, 320-2 mean-reverting, 317-20 model identification, 333-9 moving averages, 331-2 multivariate see Multivariate time series operators, 316-17 random walk, 320-2 RATS (Regression Analysis of Time Series), 80 stationary processes, 317-20 statistical modelling, 315-46 testing down, 337-8 unit root test, 324-7 univariate, 329-33 Time-varying correlation, bivariate GARCH, 16 conditional see Conditional correlation correlation estimates, 15, 107 GARCH models, 16-17, 108-14 market betas, 109, 238 optimal weights, 112 statistical hedge ratios, 109 Time-varying variance, 13, 70, 95 Time-varying volatility, actual/expected values, 13 conditional volatility, 12, 14 GARCH models, 14, 95, 115 high frequency returns, 17 Tracking models, benchmark tracking, 144, 172 cointegration, 372 factor models. 229 index tracking. 172 Trading limits. risk aversion. 186 short sales. 186 VaR {value-at-risk) models, 257 Trading performance, measurement, 124 Trading strategy implied \olatilit\. 125 P&L. 124. 125 Transaction costs, long-term positions. 29 Treasury bills, 231 Trending markets, market regimes, 36, 37, 44, 156 TSP, 84 Uncertainty, Black-Scholes model, 23 consequences, 134-40 correlation forecasts, 137-8 dynamically hedged portfolios, 13840 marked-to-model value, 136-8 mixture of normal densities, 136 volatility, 18, 119 Unconditional correlation, 15, 17, 49 Unconditional correlation matrices, principal component analysis (PCA), 147 Unconditional covariance, diagonal matrix, 144, 146, 206 principal component analysis (PCA), 162, 206 Unconditional distribution, return process, 12 Unconditional heteroscedasticity, 432-3 Unconditional variance and volatility, I-GARCH, 76 stationary series, 12, 317-20 weighted average, 49-50 Underlying asset prices, ATM volatility, 39 constant volatility, 22 covariance, 183 geometric Brownian motion (GBM), 30 implied volatility, 11, 26, 183 option hedging, 10 scenario analysis, 185 stochastic volatility, 105 volatility term structures, 18 Unit root test, 324-7, 444 United States, alpha, 91 beta, 91 Dow Jones 30: 231 Microsoft, 91-2 Monopolies Commission, 92 persistence, 86, 91 reaction, 91 S&P 500: 86, 89, 129, 155, 202, 231. 234, 371, 403 technology bubble, 91 USD, 16, 17, 101-2, 129-33 zero-coupon bonds, 148 Univariate GARCH, see GARCH models Univariate time series models. 329-33 Utility functions. expected utility. 194-5 exponential utility function. 196-"
Utility functions (cont.) investment analysis, 185, 194-7 logarithmic, 196 marginal utility, 195, 197 mean-variance analysis, 197 power utility function, 196 quadratic, 196 risk, 194-6 risk aversion, 195-6 Value, MtM see Marked-to-market (MtM) value VaR (value-at-risk) models, advantages/limitations, 255-60 alternatives, 257-60 backtesting, 125, 250, 275-7 conditional, 259-60 covariance see Covariance VaR models delta-gamma approximations, 273-4 extreme returns, 185 generalized extreme value (GEV), 293 historical simulation, 267, 268-70 marked-to-market (MtM) value, 135 market risk capital requirement (MRR), 253, 254-5, 274, 276 model validation, 274-8 Monte Carlo simulation, 267, 270-3, 274, 277-8 normality assumption, 185 not sub-additive, 259 P&L, 180, 253, 254, 262-3 portfolio volatility, 185 process volatility, 139 risk evaluation, 249-83 risk management, 236, 259 RiskMetrics data, 102, 2031 scenario analysis, 278-81 sensitivity analysis, 277-8 short risk horizons. 186 simulation, 267-74 stress testing. 278, 281-3 tails prediction, 125 trading limits. 257 traditional measures compared, 256-7 Variance, analysis of variance (ANOVA), 427-8 conditional see Conditional variance conditional heteroscedasticity. 14 confidence intervals, 126 linear portfolios, 180-2, 183 minimum see Minimum variance portfolios P&L. 181-2 quadratic, 181 time-varying see Time-varying variance unconditional see Unconditional variance Variance estimation and forecasting, exponentially weighted moving average (EWMA), 127, 163 h-period, 100 one-step ahead, 99 parameters, 50 regression, 123, 124 squared returns, 123, 124 standard error, 127 weighted average, 126 Vech models, bivariate GARCH, 16, 108, 109, 110 diagonal, 108, 109, 110, 114, 217, 218, 219 parameter estimates, 219 parameterization, 114 Vega, ATM options, 44 Black-Scholes model, 32, 44 volatility sensitivity, 24 Volatility, see also Implied volatility, Process volatility, Statistical volatility, Realized volatility ATM see ATM volatility basic concept, 31 basket options, 56-7 bursts, clusters see Volatility clustering common volatility, 386-7 conditional 12, 14 cones see Volatility cones constant see Constant volatility dispersion, measure, 4, 119 estimates see Volatility estimates financial markets, 9-12, 119, 250, fixed strike see Fixed strike volatility forecasts see Volatility forecasts hedging, 13940 historical see Historic volatility implied see Implied volatility irreducible risk, 9 local see Local volatility long-term, 57. 85 parallel shifts, 38 persistence see Persistence process see Process volatility reaction, 59, 73, 86, 90 realized see Realized volatility sensitivity see Volatility sensitivity smile see Smile effect; Volatility smile surface
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