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