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Index

Bank for International Settlements (BIS),

absolute returns volatility, 10“11

129

actual volatility, 101

Barone-Adesi and Whaley quadratic

ADC (asymmetric dynamic covariance)

approximation, 89“90, 92“5

model, 65“6

Basel Accords, 129“31

ARCH (autoregressive conditional

Basel Committee, 129“31

heteroscedasticity) models, 7, 10,

BEKK model, 66

17, 31, 37“44, 121, 122, 126

biased forecast, 29

Engle (1982), 37“8

bid“ask bounce, 16, 90, 91, 117

forecasting performance, 43“4

Black model, 127

LM-ARCH, 51, 53

Black-Scholes model, 49“50, 71“95

see also GARCH

binomial method, 80“6

autoregressive conditional

matching volatility with u and d,

heteroscedasticity models

83“4

see ARCH models

two-step binomial tree and

ARFIMA model, 34, 51, 53, 127

American-style options, 85“6

ARIMA model, 34

dividend and early exercise premium,

ARMA model, 7, 34

88“90

Asian ¬nancial crisis, 9

dividend yield method, 88“9

asymmetric dynamic covariance (ADC)

Heston price and, 99“102

model, 65“6

implied volatility smile, 74“7, 97

at the money (ATM), 76

investor risk preference, 91“2

implied volatility, 101, 106, 107,

known and ¬nite dividends, 88

116“17

measurement errors and bias, 90“2

options, 87, 91, 106, 117, 119

no-arbitrage pricing, 77“80

autocorrelation, 19

stock price dynamics, 77

of absolute returns, 8

partial differential equation, 77“8

asset returns, 7

solving the partial differential

lack of, 7

equation, 79“80

autocorrelation function, decay rate, 7

option price, 97

autoregressive model, 126

for pricing American-style options,

72, 85“6

backtest, 132“3

for pricing European call and put

three-zone approach to, 133“5

options, 71“7

backward induction, 86

216 Index

forecasting volatility, 16“17

Black-Scholes model (continued )

fractionally integrated model, 46, 50“4

risk, 91

positive drift in fractional integrated

testing option pricing model, 86“7

series, 52“3

Black-Scholes implied volatility (BSIV),

forecasting performance, 53“4

49, 50, 73“4, 91, 100

see also FIGARCH; FIEGARCH

break model, 46

fractionally integrated series, 54

Brownian motion, 18

GARCH, 18, 34, 38“9, 121, 122, 126

Capital Accord, 130

component GARCH (CGARCH), 55

CGARCH (component GARCH), 55

EVT-GARCH, 135

Chicago Board of Exchange (CBOE), 2

exponential (EGARCH), 34, 41, 43,

close-to-open squared return, 14

121, 122

component GARCH (CGARCH), 55

GARCH(1,1) model, 12, 17, 38,

conditional variance, 22

40, 46

conditional volatility, 10

GARCH(1,1) with occasional break

constant correlation model, 66

model, 55

GARCH-t, 135

deep-in-the-money call option, 76

GJR-GARCH, 41“2, 43, 44, 122

deep-out-of-the-money call option, 76

integrated (IGARCH), 39“40, 45

delta, 89, 90

quadratic (QGARCH), 42

Diebold and Mariano asymptotic test, 25,

regime switching (RSGARCH), 57“8,

26“7

122

Diebold and Mariano sign test, 25, 27

smooth transition (STGARCH), 58

Diebold and Mariano Wilcoxon sign-rank

TGARCH, 42

test, 27

generalized ARCH see GARCH

discrete price observations, 15

generalized method of moments (GMM),

60

EGARCH, 34, 41, 43, 121, 122

generalized Pareto distribution, 137

equal accuracy in forecasting models,

Gibbs sampling, 61

tests for, 25

GJR-GARCH, 41“2, 43, 44, 46, 122

error statistics, 25

Gram“Charlier class of distributions, 12

EVT-GARCH method, 135

EWMA (exponentially weighted moving

Heath-Jarrow-Morton model, 127

average), 31, 33, 40, 44

Heston formula, 99, 102

explained variability, proportion of,

Heston stochastic volatility option

29“30

pricing model, 98“9

exponential GARCH (EGARCH), 34, 41,

assessment, 102“4

43, 121, 122

zero correlation, 103

exponential smoothing method, 33

nonzero correlation, 103“4

exponentially weighted moving average

Black-Scholes implied and, 99“102

(EWMA), 31, 33, 40, 44

market price of volatility risk, 107“13

extreme value method see high-low

case of stochastic volatility, 107“8

method

constructing the risk-free strategy,

108“10

factor ARCH (FARCH) model, 66

correlated processes, 110“11

FARCH (factor ARCH) model, 66

Ito™s lemma for stochastic variables,

fat tails 7

107

FIEGARCH, 50“1, 52“4

market price of risk, 111“13

FIGARCH, 50, 51“2, 53, 122

volatility forecast using, 105“7

¬nancial market stylized facts, 3“9

heteroscedasticity-adjusted Mean Square

forecast biasedness, 117“19

Error (HMSE), 23“4

forecast error, 24

Index 217

long memory effect of volatility, 7

high-low method (H-L), 12“14, 17, 21

long memory SV models, 60

historical average method, 33

historical volatility models, 31“5

market risk, 130

forecasting performance, 35

Markov chain, 60

modelling issues, 31“2

maximum likelihood method, 11

regime switching, 34“5

MCMC (Monte Carlo Markov chain), 59

single-state, 32“4

MDH (mixture of distribution

transition exponential smoothing,

hypothesis), 148

34“5

mean 2

types, 32“5

Mean Absolute Error (MAE), 23, 44

HISVOL, 121, 122

Mean Absolute Percent Error (MAPE),

23, 44

implied standard deviation (ISD), 116,

Mean Error (ME), 23

121, 122

Mean Logarithm of Absolute Errors

implied volatility method, 30, 91, 101

(MLAE), 24

at the money, 101, 106, 107

Mean Square Error (MSE), 23

Black-Scholes (BSIV), 49, 50, 73“4,

MedSE (median standard error), 44

91, 100

mixture of distribution hypothesis

implied volatility smile, 74“7, 97

(MDH), 148

importance sampling, 60

Monte Carlo Markov chain (MCMC), 59,

independent and identically distributed

60“3

returns (iid), 16, 136

parameter w, 62“3

in-sample forecasts, 30

volatility vector H , 61“2

integrated cumulative sums of squares

moving average method, 33

(ICSS), 55

multivariate volatility, 65“9

integrated GARCH (IGARCH), 39“40,

applications, 68“9

45

bivariate example, 67

integrated volatility, 14, 39, 40

interest rate, level effect of, 58

nearest-to-the-money (NTM), 116“17

in-the-money (ITM) option, 87

near-the-money (NTM), 76, 106

intraday periodic volatility patterns,

negative risk premium, 102

15“16

nonsynchronous trading, 15

inverted chi-squared distribution, 62

normal distribution, 2

IOSCO (International Organization of

Securities Commissions), 130

operational risk, 131

Ito™s lemma, 77, 107, 108, 110

option forecasting power, 115“19

option implied standard deviation,

Jensen inequality, 11

115“16

option pricing errors, 30

kurtosis, 4, 17

stochastic volatility (SV) option pricing

model, 97“113

large numbers, treatment of, 17“19

Ornstein-Uhlenbeck (OU) process,

Legal & General, 4

108

level effect in interest rates, 58

orthogonality test, 28“30

leverage effect (volatility asymmetry), 8,

outliers, removal of, volatility persistence

37, 56, 148

and, 18“19

likelihood ratio statistic, 139

out-of-sample forecasts, 30

LINEX, 24

out-of-the-money (OTM) put option, 76,

liquidity-weighted assets, 130

87, 92, 119

LM-ARCH model, 51, 53

out-of-the money (OTM) implied

log-ARFIMA, 51, 53, 127

volatility, 101

lognormal distribution, 2

218 Index

MCMC approach, 60“3

quadratic GARCH (QGARCH), 42

parameter w, 62“3

quadratic variation, 14“15

volatility vector H , 61“2

quasi-maximum likelihood estimation

option pricing model, 97“113

(QMLE), 60, 136

stock market crash, October, 1987 18

strict white noise, 16

random walk (RW) model, 32“3, 44, 126

strike price effect, 101

range-based method, 12, 17

Student-t distribution, 17

realized bipower variation, 15

switching regime model, 54

realized volatility, 14“16

recursive scheme, 25

tail event, 135, 136

regime-switching GARCH (RSGARCH),

TAR model, 34, 43, 58

57“8, 122

Taylor effect, 8, 46

regime-switching model, 19, 46

TGARCH, 42

regression-based forecast ef¬ciency,

Theil-U statistic, 24

28“30

threshold autoregressive model (TAR),

risk 1

34, 43, 58

risk management, volatility models in,

time series, 4“7

129“41

trading volume weighted, 117

risk-neutral long-run level of volatility,

trimming rule, 18

98

risk-neutral price distribution, 101

UK Index for Small Capitalisation Stocks

risk-neutral probability measure, 83,

(Small Cap Index), 4

85

unbiased forecast, 29

risk-neutral reverting parameter, 98

unconditional volatility, 10

risk-neutrality, 49

risk-weighted assets, 130

value-at-risk (VaR), 129, 131“5

rolling scheme, 25

10-day VaR, 137“8

Root Mean Square Error (RMSE), 23, 44

evaluation of, 139“41

RSGARCH, 57“8, 122

extreme value theory and, 135“9

model, 136“7

serial correlation, 28

multivariate analysis, 138“9

Sharpe ratio, 1“2

variance as volatility measure, 1

sign test, 25

variance“covariance matrix, 65

simple regression method, 33“4

VAR-RV model, 51, 53

skewness, 4

VDAX, 126

smile effect, 75

VECH model, 66

smirk, 75

vega, 89, 90

smooth transition GARCH (STGARCH),

vega weighted, 117

58

VIX (volatility index), 126

smooth transition exponential smoothing,

new de¬nition, 143“4

34“5

old version (VXO), 144“5

squared percentage error, 24

reason for change, 146

squared returns models, 11

volatility asymmetry see leverage effect

Standard and Poor market index

volatility breaks model, 54“5

(S&P100), 4

volatility clustering, 7, 147

standard deviation as volatility measure,

volatility component model, 46, 54

1, 2

volatility dynamic, 132

STGARCH 58

volatility estimation, 9“17

stochastic volatility (SV), 31, 59“63, 121

realized volatility, quadratic variation

forecasting performance, 63

and jumps, 14“16

innovation, 59“60

Index 219

scaling and actual volatility, volatility index see, VIX

16“17 volatility long memory

using high“low measure, 12“14 break process, 54“5

using squared returns, 11“12 competing models for, 54“8

volatility forecasting records, 121“8 components model, 55“7

getting the right conditional variance de¬nition, 45“6

and forecast with the ˜wrong™ evidence and impact, 46“50

models, 123“4 forecasting performance, 58

predictability across different assets, regime-switching model

124“8 (RS-GARCH), 57“8

exchange rate, 126“7 volatility persistence, 17“19, 45, 136

individual stocks, 124“5 volatility risk premium, 119

other assets, 127“8 volatility skew, 76, 116

stock market index, 125“6 volatility smile, 101, 116

which model?, 121“3 volatility spillover relationships, 65

volatility forecasts, 21“30 volume-volatility, 147“8

comparing forecast errors of different

models, 24“8 weighted implied, 116“17

error statistics and form of µt , 23“4 weighted least squares, 117

form of X t , 21“2 Wilcoxon single-rank test, 25

wildcard option, 91

see also volatility forecasting

records

volatility historical average, 126 yen“sterling exchange rate, 4

Index compiled by Annette Musker