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164 factor models, 237 irreducible risk, 9-10 minimum-risk, 112, 179 multivariate GARCH, 112 ordinary least squares (OLS), 237 and returns, 190, 197, 199 stress covariance matrices, 185 Portfolio theory, mean-variance analysis, 186 Portfolios, beta, 231-2 derivatives see Derivatives portfolios dynamically hedged, 138-40 efficient portfolios in practice, 198-201 linear see Linear portfolios marked-to-market (MtM) value, 119, 134-5 minimum variance see Minimum variance portfolios rebalancing, 200-1 simple cash portfolios, 261-2 volatility, 9-10 Positive semi-defmiteness, covariance matrices, 116, 179, 180, 181, 206, 211 exponentially weighted moving average (EWMA), 184 GARCH models, 184 orthogonal GARCH, 211 rounding error, 184 Prediction, backtesting, 444-5 confidence intervals, 444 interval predictions, 133-4, 444 likelihood, 445 long-term, 52 mean absolute error, 445 mean square error, 445 out-of-sample correlation coefficient, 445 P&L, 446 point predictions, 443 post-sample predictions, 121, 122 price prediction models, 401-7 root mean square error (RMSE), 445 statistical operation evaluation methods, 445-7 tails prediction, 125 Prices. see also Option pricing, Financial asset prices. Market prices. Spot prices, Underhing prices Principal component analysis (PCA), advantages. 143 collinear variables. 143. 144. 171 -4 conditional covariance. 162-3 correlation, 143, 220 covariance matrices, 144, 204, 205-6 data problems overcome, 18, 144, 171-8 dimensions, reduction, 141. 143-4. 153 eigenvalues, 145, 146, 147, 152, 153. 154, 159 eigenvectors, 145, 152, 153, 154, 159 fixed strike volatility, 157, 158, 159-67. 169 mathematical background, 145-6 missing data, 18, 144, 171, 174-8, 439 orthogonality, 141, 146, 204, 207-9 principal components representation, 146 purpose, 141 risk factors, 204 scenario analysis, 39, 143, 144, 154-5, 282 skews, 154-71 smile effect, 154-71 term structures, 143, 147-54 unconditional correlation matrices, 147 unconditional covariance, 162, 206 variance, 146 yield curves, 147-53 Process volatility, Black-Scholes model, 11, 23 F-test, 123 price process volatility, 22, 118 realized volatility, distinguished, 11, 121, 123 VaR (value-at-risk) models, 139 volatility forecasts, 117, 118, 129 volatility surface, 34 Put options, ATM options, 30 Black-Scholes model, 23-4 implied volatility, 26-8 ITM, 30 OTM, 26, 28, 30, 31 volatility term structures, 31, 32 QQ plots, 288-90 Quadratic GARCH, 81 Quadratic programming, investment analysis, 185 Quadratic variance, 181 Quadratic volatility surfaces, 38. 45. 144. Random walk, components GARCH. 78-9 exchange rates, 75 I-GARCH, 90
Random walk {com.) stochastic processes, 351 time series models, 320-2 tracking error, 350 Range-bounded markets, market regimes, 36, 44, 117, 156 RATS (Regression Analysis of Time Series), 80 Reaction, see also Alpha exponentially weighted moving average (EWMA), 59, 207 GARCH models, 92 RiskMetrics data, 202 volatility, 59, 73, 86, 90 Realized volatility, combined forecasts, 132-3 confidence intervals, 133 process volatility, distinguished, 11, 121, 123 standard error, 133 Recursion, gradient vectors, 95 Regression, autoregression, 329-31, 340-1 benchmark tracking models, 144 equation, 115 errors-in-variables, 123 linear see Linear regression models multi-factor models, 144 squared returns, 123, 124 standard error, 372 systems of seemingly unrelated egression equations (SURE), 434-5 variance forecasts, 123, 124 Regret, downside risk, 259 Residual analysis, autocorrelation, 429-32 ordinary least squares (OLS), 429 Return distributions, dispersion, 119 fat-tailed, 30, 82, 125 specification, 122 Risk, see also Portfolio risk attitudes, modelling, 194-8 certainty equivalence, 194, 195 correlation risk, 138 decomposing risk, 230-6 downside risk, 258-9 financial markets, 250-5 indifference, 194, 197-8 irreducible risk, 9-10 market risk capital requirement (MRR), 251, 252, 253, 254-5, 274, 276, 279, 280 return, relationship, 189-93 risk premium, 104, 106 risk-free rate of return, 21, 23, 104 risk-loving, 195, 198 sources of risk, 232, 256 transitive preferences, 194 utility functions, 194-6 Risk aversion, assumption, 198 coefficients, 195 constant absolute risk aversion (CARA), 196 constant relative risk aversion (CRRA), 196, 197 efficient frontier, 197 indifference, 197-8 trading limits, 186 utility function, 195-6 Risk factors, arbitrage pricing theory (APT), 233 capital asset pricing model (CAPM), 232 covariance matrices, 179 derivatives portfolios, 182-3 factor models, 229 generalized least squares, 237-8 linear portfolios, 181 Monte Carlo simulation, 185 principal component analysis (PCA), 204 RiskMetrics data, 202 variance, 183 Risk horizons, volatility forecasts, 11, 43, 57, 60, 117 Risk management, back office, 180 beta, 109, 236 constant parameter assumptions, 237-8 copulas, 9 covariance matrices, 179, 180-5 risk factors, 143 scenario analysis, 34, 3813, 141 stress covariance matrices, 185 stress testing, 141 VaR (value-at-risk) models, 236, 259 Risk measurement, cash-flow maps, 256 classical techniques, 236-9 coherent risk measures, 259 constant parameter assumptions. 237-8 factor models, 22948 index stripping, 238-9 present value of a basis point move (PVBP), 256 risk managers/asset managers, 236-7 time-varying parameter assumptions, traditional measures, 256-7
Risk-adjusted performance measures (RAPM), capital allocation, 1931 information ratio (IR), 194 Sharpe ratio (SR), 194 Risk-adjusted returns, capital allocation, 186, 187 traders, performance, 186 Risk-free assets, minimum variance portfolios, 192 risk/return, 198 zero variance, 231 Risk-neutrality, assumption, 106 certainty equivalence, 195 hypothesis, 32 local, 81, 106 probability, 81, 106 valuation, 104, 106 RiskMetrics data, covariance matrices, 201-4 covariance VaR models, 260 exponentially weighted moving average (EWMA), 60, 115, 163, 179, 202, 203 ghost features, 204 limitations, 202-3 methodology, 179-80, 201 persistence, 76, 202 reaction, 202 risk factors, 202 smoothing constant, 202, 203, 204 VaR (value-at-risk) models, 102, 2034 weighted average, 179, 201 Root mean square error (RMSE), distance metric, 123 prediction, 445 volatility forecasts, 122-3 Rounding error, positive semi-definiteness, 184 Russian debt crisis, 35, 251 S&P 500: 86, 89, 129, 155, 202, 231, 234, 371. 403 S-PLUS. 84 Sampling error, constant volatility, 52 exponentially weighted moving average (EWMA). Ill high frequency data. 17 historic correlation. 51 moving averages. 63 ordinar> least squares (OLS). 236 unconditional correlation. 15. 17 weighted average. 49 Scatter plots. 5. 8~ 15. 39. 40 Scenario analysis, implied volatility, 3813, 185 long-term volatility, 92 market risk capital requirements (MRR), 253, 279 principal component analysis (PCA), 39, 143. 144, 154-5, 282 probabilistic, 280-1 risk management, 34, 38-43, 141 skews, 39, 159 smile effect, 39 VaR (value-at-risk) models, 278-81 volatility term structures, 92 yield curves, 143 Schmidt-Phillips test, 328 Sharpe ratio (SR), 194 Short sales, frontier analysis, 186, 191, 198 Simulation, delta, 105 gamma, 105 historical simulation, 267, 268-70 Monte Carlo see Monte Carlo simulation random numbers, 105 VaR (value-at-risk) models, 267-74 Single outliers, alpha, 96 beta, 96 excess kurtosis, 67 Skews, deviations, 160-1 equity markets, 155 implied volatility, 1, 30-1, 32, 33, 68, 155, 158 jumpy markets, 37-8, 156 leverage effect, 31, 68 modelling, 154-71 negative, 30-1 parallel shifts, 45 principal component analysis (PCA), 154-71 range-bounded markets, 36, 44, 156 scenario analysis, 39, 159 trending markets, 37, 156 volatility clustering, 66, 67 Smile effect, implied volatility, 1, 30, 32, 33. 98. 155 modelling, 154-71 option pricing, 106-7, 136 oversimplistic models, 22 principal component analvsis (PCA). 154-71 scenarios, 39 smile fitting, 106-7
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