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