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