Cambridge University Press 0521843715 - Weather Derivative Valuation: The Meteorological, Statistical, Financial and Mathematical Foundations Stephen Jewson and Anders Brix Index More information Index ACF, 129 coefficient in ARFIMA, 140 actuarial VaR, 275 distribution, 83, 84 Anderson–Darling, 81, 299 Black equation, 252 anomaly Black–Scholes correlation, 202 adapted to weather, 257 definition, 128 assumptions, 250 forecast, 196 equation, 245 arbitrage pricing, 30, 241 extensions, 252 Black–Scholes equation, 245 greeks, 251 weather-adapted Black equation, 257 measure theory derivation, 245 weather-adapted Black–Scholes equation, PDE derivation, 243 257 Brownian motion ARFIMA, 140, 226 derivation for expected index, 234 ARMA, 135 for equity prices, 242 AROMA, 141 for expected index, 100, 255 average of average temperature, 15 linear imbalance model, 262 average temperature use for market prices, 279 definition, 10, 15 use in VaR calculations, 276 burn analysis backtesting, 54 assumptions, 63 balanced market model, 254, 257 comparison with index modelling, 109 banks, 1, 148 correlation with index modelling, 110 basis risk, 5 examples, 63 basket, 25 extended, 157 beta for options, 61 generalisation of delta to portfolios, for portfolios, 156 187 for swaps, 59 in the CAPM, 32 uncertainty, 68 bias in detrending, 55 call in forecasts expected pay-off example, 314 correction, 204 expected pay-off for the normal, 308 estimation, 200 greeks example, 332 binary greeks for normal, 327 expected pay-off example, 314 option pay-off definition, 21 expected pay-off for the normal, 312 pay-off distribution, 88, 303 greeks example, 332 pay-off variance example, 322 greeks for normal, 331 pay-off variance for normal, 317 option pay-off definition, 25 cap, 27 pay-off distribution, 88, 305 CAPM, 32 pay-off variance example, 322 CAT pay-off variance for normal, 322 index type, see cumulative average binomial temperature 369 © Cambridge University Press www.cambridge.org Cambridge University Press 0521843715 - Weather Derivative Valuation: The Meteorological, Statistical, Financial and Mathematical Foundations Stephen Jewson and Anders Brix Index More information 370 Index CCF, 164 jumps, 40 CDD, 13 reconstruction, 40 CDF trends in, 42 comparing, 78 degree days of pay-offs for standard contract types, cooling, 13 303 heating, 11 use in simulation, 353 delta chi-square, 81, 298 definition, 95, 97 Choleski, 159 for the normal, 324 climate change, 43 interpretation, 104–106 climate models, 43 with respect to temperature, 101 CMC, 194 delta hedging, 116, 243 CME, 7 derivatives collar equity, 29, 242 expected pay-off example, 314 financial, 29, 242 expected pay-off for the normal, 310 partial, 97 greeks example, 332 total, 101 greeks for normal, 328 design matrix, 292 pay-off definition, 23 dimension reduction, 164 pay-off distribution, 88, 304 discontinuity pay-off variance example, 322 in temperature data, 40 pay-off variance for normal, 319 discounting, 28 confidence intervals distributions on distributions, 79 adjusted kernel density, 87 copulas, 161 binomial, 84 correlation confidence intervals on, 79 and the CAPM, 32 fitting, 77 between weather and the stock market, 34 fitting using maximum likelihood, 296 correlations fitting using method of moments, 296 autocorrelations of temperature, 129 for event contracts, 83 between daily models and index models, 167 importance of choice, 108 between ENSO and weather, 215 kernel density, 86 between European cities, 153 multivariate, 153 between forecast and post-forecast, 226 negative binomial, 84 between indices, 153 non-parametric, 85 between temperatures and indices, 153 normal for seasonal contracts, 81 between the United States and Europe, 153 numerical goodness of fit tests, 81 between US cities, 153 of daily temperatures, 129 converting linear to rank, 160 of portfolio pay-offs, 157 cross-correlations for temperature, 163 parametric alternatives to normal, 83 forecasting, 210 parametric index, 77 from year to year, 63 Poisson, 83 in the pay-offs of a portfolio, 151 simulation from, 90 indices and pay-offs, 156 testing goodness of fit, 78 modelling linear, 158 variance estimation, 77 modelling rank, 160 downscaling, 195 rank, 160 dual-trigger contracts, 9, 266 Spearman, 160 costless swaps, see swaps ECMWF, 194 Cram´er–von Mises, 299 efficient forecasts, 211, 234 credit risk, 280 El Ni˜no, see ENSO critical days, see event indices end-to-end use of ensemble forecasts, 229 cumulative average temperature, 17 energy companies, 1 ensemble forecasts daily modelling, 74, 121 definition, 193 data ensemble means, 198 checking, 39 example, 199 cleaning, 37 models, 194 discontinuities, 40 use for making probabilistic forecasts, 209 enhancement, 40 use for predicting changes, 211 gap filling, 38 use for predicting correlations, 210 © Cambridge University Press www.cambridge.org Cambridge University Press 0521843715 - Weather Derivative Valuation: The Meteorological, Statistical, Financial and Mathematical Foundations Stephen Jewson and Anders Brix Index More information Index 371 ENSO, 213 implied volatility effects, 215 relation to risk loading, 263 forecasts and pricing, 240 index modelling, 73 impact on US temperatures, 216 combining with daily modelling, 167 mechanics, 213 comparison with burn, 109 predictions, 216 correlation with burn, 110 event indices, 17 for portfolios, 158 exchange trading, 7 incorporation of forecasts, 229 expiry VaR, 118, 274 index vega, see zeta exponential insurance distribution, 83, 297 and derivatives, 4 trend model, 48, 293 indemnity-based, 5 index-based, 4 fair premium (definition), 61 insurance companies, 1, 148 fair price (definition), 61 interest rates, see discounting fair strike (definition), 59 Ito’s lemma, 102, 243 fish farm, 4 forecasts, 192 jumps (in temperature data), 40 anomaly correlation, 202 bias, 200 kernel density, 76, 85 efficiency, 211 adjusted, 87 efficient forecast hypothesis, 211 basic, 86 ensemble forecasts, 209 closed-form solutions, 334 ensemble means, 198 Kolmogorov–Smirnov, 81, 299 for the expected temperature, 196 links with hedging, 8 La Ni˜na, see ENSO of changes, 211 leap years, 120, 127 of correlations, 210 limit, 20–25 probabilistic forecasts, 207 limited expected value function, 89 RMSE, 201 linear seasonal, 212, 240 contract, 19 skill, 198, 207 in covariates, 292 uncertainty, 207 index, 18 use of, 220 sensitivity analysis, 94 weather, 192, 221 trend, 45, 50, 55, 56 year ahead, 54 liquidation value, 269, 272, 279 liquidation VaR, 279 gamma liquidity, 5 definition, 95, 98 liquidity risk, 280 distribution, 83, 296 loess, 48, 51, 55, 292 for the normal, 325 log-normal (distribution), 83 interpretation, 105, 106 long (position), 19 gas price, 9, 32, 266 GoF, see goodness of fit mark to market, 5, 280 goodness of fit, 78 mark to model, 5, 121, 280 tests, 298 market making, 61, 62, 184 grafting, 232 market price of risk, 251, 259 greeks, 94 maximum likelihood, 48, 77, 134, 295 solutions for normal, 324 mean square error, see root mean square error HDD, 11 mean-variance, 174, 181, 182 hedge funds, 1, 148 mean-variance portfolio management, hedgers, 6 173 hedging, 116 measure theory, 245, 249, 259 delta hedging, 116, 241, 243 method of moments, 77, 295 dual-trigger contracts, 266 moneyness, 26 static hedging, 117 Monte Carlo, 59, 90, 229, 246, 293, 300 using swaps on a different location, moving average, 51, 292 266 MSE, see root mean square error hindcasting, see backtesting multi-year contracts, 112 hydropower, 3 multivariate modelling, 153 © Cambridge University Press www.cambridge.org Cambridge University Press 0521843715 - Weather Derivative Valuation: The Meteorological, Statistical, Financial and Mathematical Foundations Stephen Jewson and Anders Brix Index More information 372 Index NCEP, 194 arbitrage, 30, 241 negative binomial, 83, 84, 356 paradigms, 28, 31 non-parametric primary market, 6 daily models, 143 principal components analysis, 188 distribution transforms, 134 probabilistic forecasts, 207, 223, 224, 226, 228, distributions, 85 229, 239 trends, 51 pruning, 230 normal distribution, 297 put and seasonal contracts, 81 expected pay-off example, 314 closed-form expected pay-offs, 302 expected pay-off for the normal, 309 closed-form greeks, 324 greeks example, 332 closed-form pay-off variances, 315 greeks for normal, 327 multivariate simulations, 158 option pay-off definition, 22 simulation, 354 pay-off distribution, 88, 304 useful relations, 306, 315, 324 pay-off variance example, 322 numerical integration, 90 pay-off variance for normal, 318 options QQ plots, 78, 129 arbitrage pricing, 241 quadratic trend, 50 binary, 25, 305, 312, 322, 331 burn analysis, 61 random walk, see Brownian motion call, 21, 303, 308, 317, 327 regression put, 22, 304, 309, 318, 327 and forecast correction, 205 straddle, 24, 305, 310, 320, 329 and gap filling, 39 strangle, 24, 305, 311, 321, 330 and jump detection, 41 OTC, see over the counter and portfolio beta, 187 over the counter, 7 and probabilistic forecasting, 207 and value checking, 39 parametric reinsurance companies, 1, 148 daily models, 135 replication, 245, 249, 259 distributions, 77 rho trends, 50 and portfolios, 188 parity, 27 definition, 95 partial differential equation, 103, 245, 252, risk 257 budgeting, 186 pay-off functions, 19 credit, 280 binary, 25 liquidity, 280 call, 21 loading, 60, 62 collar, 23 management, 268 other, 25 measures, 151, 171 put, 22 neutrality, 247 straddle, 24 risk-adjusted return, 172, 181, 182, 186 strangle, 24 RMSE, see root mean square error swap, 19 root mean square error, 50, 55, 201 pay-off integrand, 113 PCA, see principal components