Many businesses, particularly ones that specialize in energy and agriculture, can be subjected to severe financial impact by changes in the weather. Techniques from financial engineering can be used to manage the financial risk involved in these changes, and are structured as swap, call, and put contracts based on `weather indices'. These financial instruments are referred to as `weather derivatives' and are the subject of this book. The weather indices are typically the `heating degree day' (HDD), and the `cooling degree day' (CDD), but could also be such quantities as precipitation, snowfall, and humidity. The HDD (for a particular day) is usually defined as the maximum difference (bounded below by zero) between a chosen baseline temperature and the average temperature. The CDD is the difference between the average temperature and a baseline (again bounded below by zero).
A perfect weather derivative would be designed so as to eliminate all risk due to the weather. For example, if the temperature is to be the index of choice, then one would like to be able to `hedge' so successfully so as to make, as far as the affected industry is concerned, the weather effectively irrelevant. This of course is not possible, due to the unavailability of perfect forecasts. However, one can enter into weather derivative contracts that will enable the affected industry to manage their weather risk in a manner that that makes use of what can actually be predicted in weather forecasts, with the remaining uncertainty being hedged. Possible lost revenue due to adverse weather can be hedged for example by a weather derivative that will give a revenue stream that is based on the forecast error.
Like all other financial instruments, there will be a cost associated with weather derivative contracts. Within the scope of propriety, the authors have given an excellent introduction to the methodologies used to price weather derivatives, and how to perform risk management of portfolios based on weather derivatives. Since the underlying weather indices are not traded, pricing based on arbitrage is more involved for the case of weather derivatives. The authors though show how arbitrage pricing can be done, and also give in-depth discussion on other pricing strategies, these being classified as `actuarial' and `market-based' pricing. Actuarial pricing, as the name implies, involves calculating the probabilities of all future outcomes of a contract or portfolio of contracts, while market-based pricing is based on the actual prices that are observed in the market. Arbitrage pricing can be done in locations where the option is actively traded. Otherwise, the authors show how a swap contract defined on the index can be used to obtain dynamic hedging. However, they remark that this pricing strategy is not widely done at the time of writing. Actuarial pricing thus dominates the discussions in the book.
The mathematical modeling involved in weather derivatives can be difficult, due in part to the fact that the underlying weather indices are nonstationary, i.e. they are characterized by variations and trends with scales greater than the length of the historical record. In addition, the weather indices exhibit a high degree of autocorrelation. Also, the actual measurement of volatility can be problematic, due to sparse data sets or even the unavailability of data. Further, arriving at a general method for estimating volatility is difficult since the exposure to weather risk is highly variable between different companies.
One method of valuing single contracts discussed early on in the book is called `burn analysis', and can be viewed as a step above a quick back-of-the-envelope calculation. It attempts to value a contract based on how it would have performed in the past. The authors estimate the fair strike for a swap, i.e. the strike that gives an expected value of zero, using burn analysis. This involves using the (detrended) historical index values and the calculation of the mean of this data to estimate the expected index. The authors show how to incorporate `risk loading' to model more closely what is actually going on in the trading of swaps. They also show how to apply the burn analysis to options, calculating the `fair premium' by using the historical pay-offs, with the mean of this data being the expected pay-off.
The most interesting part of the book is the one on arbitrage pricing models. The price charged for a weather contract will be influenced by the possibility of hedging, which is different from actuarial pricing, which is based on diversification. The arbitrage pricing mechanism that the authors discuss is restricted to weather swaps, and they review arbitrage theory both from the standpoint of partial stochastic differential equations and from measure theory. The swaps are all assumed to be linear and based on linear degree days. The authors derive the stochastic differential equations for the swap price to obtain a version of the Black-Scholes equation for weather swaps trading with a premium. They also derive, using a hedging strategy based on forward contracts, the partial differential equation satisfied by the price of the weather option. The solution of this equation gives the arbitrage price, which interestingly turns out to be the same as the actuarial fair price without risk loading. This is due to the absence of drift in the discounted swap price and also the fact that there is no expected loss on the swaps. The authors' algorithm for calculating the arbitrage price for options consists of taking the market swap price to be the expected index, using this to calculate the expected pay-off, and then discounting this quantity to give the arbitrage price. This algorithm is done assuming knowledge of the standard deviation of the settlement index. Their algorithm is interesting, but its validation is not discussed in the book. Readers will have to consult the references for further discussion on this important issue. To gain confidence in the efficacy of the algorithm will of course require it be used in real-life trading or risk management.
Author(s): Stephen Jewson, Anders Brix, Christine Ziehmann
Publisher: Cambridge University Press
Year: 2005
Language: English
Pages: 392
0521843715......Page 1
Title......Page 4
Copyright......Page 5
Contents......Page 6
Figures......Page 11
Tables......Page 16
Acknowledgements......Page 18
1.1 Introduction......Page 20
1.1.1 The impact of weather on business and the rationale for hedging......Page 21
1.1.2 Examples of weather hedging......Page 22
1.1.4 Insurance and derivatives......Page 23
1.1.5 Liquidity and basis risk......Page 24
1.1.6 Hedgers and speculators, primary and secondary markets......Page 25
Secondary trading and the Pareto optimum......Page 26
1.1.8 Hedging and forecasts......Page 27
1.1.9 Hedging weather and price......Page 28
1.2 Weather variables and indices......Page 29
Heating degree days......Page 30
Cooling degree days......Page 32
Relations between HDDs and CDDs......Page 34
1.2.2 Average of average temperature indices......Page 35
1.2.3 Cumulative average temperature indices......Page 36
1.2.5 A general classification of indices......Page 37
1.3.1 Swaps......Page 38
1.3.2 Call options......Page 40
1.3.3 Put options......Page 41
1.3.4 Collars......Page 42
1.3.6 Strangles......Page 43
1.3.11 Baskets......Page 44
1.3.14 Long and short......Page 45
Mean and expectation......Page 46
1.4 Principles of valuation......Page 47
Equity option valuation......Page 48
Actuarial,market-based and arbitrage pricing......Page 49
1.4.1 Other paradigms for weather pricing......Page 50
1.4.2 CAPM and the price of weather derivatives......Page 51
1.6 Overview of contents......Page 53
1.7 Notes on citations......Page 54
1.8 Further reading......Page 55
2.1 Data cleaning......Page 56
2.1.1 Gap fulling......Page 57
2.1.2 Value checking......Page 58
2.1.3 Jump detection......Page 59
2.2 The sources of trends in meteorological data......Page 61
2.2.1 The spatial structure of trends......Page 64
2.2.3 Urbanisation studies......Page 65
2.3 Removing trends in practice......Page 66
2.3.1 Detrending index time series......Page 67
Non-parametric trends......Page 70
2.3.2 The sensitivity of trends......Page 71
2.4 What kind of trend and how many years of historical data to use?......Page 72
2.4.1 Backtesting......Page 73
2.4.2 Detrending daily time series......Page 74
2.5 Conclusions......Page 76
2.6 Further reading......Page 77
Estimating the fair strike for linear swaps......Page 78
Adding a risk loading......Page 79
3.1.2 Burn analysis for options......Page 80
3.1.3 The distribution of pay-offs......Page 81
3.1.5 Examples......Page 82
A capped swap example......Page 83
3.1.6 Trading simulations, and the benefits of trading large portfolios......Page 85
Uncertainty on the expected index......Page 87
Uncertainty on the option premium......Page 88
Linear theory for option pricing uncertainty......Page 89
3.2 Further reading......Page 91
4.1 Statistical modelling methods......Page 92
4.2 Modelling the index distribution......Page 93
Discrete or continuous distributions?......Page 94
Parametric or non-parametric distributions?......Page 95
4.3.2 Variance estimation......Page 96
4.3.3 Testing goodness of fit......Page 97
Confidence intervals......Page 98
Arguments for and against the normal distribution......Page 100
The Poisson distribution......Page 102
Parametric alternatives to the Poisson distribution for event contracts......Page 103
4.4 Non-parametric distributions......Page 104
4.4.1 The basic kernel density......Page 105
4.5 Estimating the pay-off distribution and the expected pay-offs......Page 106
4.5.1 Closed-form expressions for the pay-off distribution......Page 107
4.5.3 Use of the limited expected value function......Page 108
Simulation......Page 109
4.6 Further reading......Page 111
5.1 Linear sensitivity analysis: the greeks......Page 113
Delta......Page 116
Gamma......Page 117
Theta......Page 118
Modelling the volatility......Page 119
Total derivatives of the expected pay-off......Page 120
5.1.1 Estimating the greeks......Page 122
5.2.1 Delta and the probability of being in the money......Page 123
5.2.2 Gamma and the curvature of the pay-off function......Page 124
5.3 A summary of the interpretation of the greeks......Page 125
5.4 Examples of the greeks......Page 126
5.5 The relative importance of choosing data, trends and distributions......Page 127
5.6 Comparing the accuracy of burn analysis and index modelling for option pricing......Page 128
5.7 The correlation between the results from burn and index modelling......Page 129
5.8 Pricing costless swaps......Page 130
5.9 Multi-year contracts......Page 131
5.11 The pay-off integrand......Page 132
5.12 Pricing options using the swap price......Page 134
5.13.1 Delta hedges......Page 135
5.13.2 Static hedging......Page 136
5.14 Sampling error and structuring......Page 137
5.16 Further reading......Page 139
6.1 The advantages of daily modelling......Page 140
More complete use of the available historical data......Page 141
Better representation of the index distribution......Page 142
6.1.3 Use of one model for all contracts on one location......Page 143
6.3 Modelling daily temperatures......Page 144
6.3.1 Modelling the seasonal cycle......Page 145
The DFT method......Page 146
The results of deseasonalisation......Page 147
6.4 The statistical properties of the anomalies......Page 148
6.4.1 The inherited properties of the index......Page 150
6.5.1 Transforming temperature anomalies to a normal distribution......Page 153
6.5.3 ARMA models......Page 154
6.5.4 ARFIMA models......Page 158
6.5.5 AROMA and SAROMA models......Page 160
Extension to SAROMA......Page 161
6.6 Non-parametric daily modelling......Page 162
6.6.1 Sliding window resampling......Page 163
6.8 The potential accuracy of daily models versus index models......Page 164
6.10 Acknowledgements......Page 166
7 Modelling portfolios......Page 167
7.1 Portfolios, diversification and hedging......Page 168
7.2.1 Relating index and temperature correlations......Page 172
7.3 Burn analysis for portfolios......Page 175
Extended burn analysis......Page 176
7.4.1 Normally distributed indices......Page 177
7.4.2 Rank correlations......Page 179
7.4.4 Conversion to pay-offs......Page 180
7.4.5 The consistency of simulations and constraints......Page 181
7.6 Parametric models for multivariate temperature variability......Page 182
7.7 Dimension reduction......Page 183
7.8 A general portfolio aggregation method......Page 186
7.9 Further reading......Page 187
Defining return......Page 188
Defining risk......Page 189
8.1.1 Risk-adjusted return......Page 191
8.1.2 Problems with mean-variance and mean-standard deviation approaches......Page 193
8.1.3 Utility theory......Page 194
Dominance......Page 196
Possible outcomes of stochastic dominance testing......Page 197
Comparing RAR, utility theory and SDT......Page 198
Second-order stochastic dominance......Page 199
Applying SDT......Page 200
Applying risk-adjusted return......Page 201
Sources of uncertainty......Page 202
8.5 Efficient implementation methods for adding single contracts to a portfolio......Page 203
Pay-off regression......Page 204
8.6.1 Breaking down risk and return (risk budgeting)......Page 205
8.6.2 Portfolio beta......Page 206
8.6.4 The dominant patterns of risk......Page 207
8.7 Reducing portfolio risk......Page 208
The portfolio start-up problem......Page 209
8.8 Further reading......Page 210
9.1 Weather forecasts......Page 211
9.1.2 Forecasting methods......Page 212
9.1.3 The leading models......Page 213
9.1.6 Forecast terminology: lead time, target day and forecast day......Page 214
9.2 Forecasts of the expected temperature......Page 215
9.2.1 The interpretation of single forecasts......Page 216
9.3 Forecast skill......Page 217
9.3.1 Skill measures based on full temperature values......Page 218
9.3.2 Bias in the mean......Page 219
9.3.3 Mean square error and mean absolute error......Page 220
9.3.5 Anomaly correlation......Page 221
9.4 Improving forecasts of the expected temperature......Page 222
9.4.1 Bias correction......Page 223
9.4.3 Combinations of single forecasts......Page 224
9.4.4 A summary of the strategy for evaluating single forecasts......Page 225
9.5.2 Measuring the skill of probabilistic forecasts......Page 226
Other skill measures......Page 227
9.6 The use of ensemble forecasts for making probabilistic forecasts......Page 228
9.6.1 Predicting correlations......Page 229
9.6.3 Predicting changes in weather forecasts......Page 230
9.7 Seasonal forecasts......Page 231
9.7.1 The physical background......Page 232
9.7.2 The effiects of El Niño......Page 234
9.8.2 Predicting the impact of El Niño......Page 235
9.9 Other sources of seasonal predictability......Page 237
9.10 Further reading......Page 238
10 The use of meteorological forecasts in pricing......Page 239
10.1 The use of weather forecasts......Page 240
10.2 Linear swaps on separable linear indices......Page 241
10.3 Linear swaps on separable indices......Page 242
10.4 The general case: any contract, any index......Page 243
10.4.1 Estimating the index standard deviation......Page 244
10.4.2 Estimating the size of the covariance term......Page 245
10.4.3 Short contracts......Page 247
10.4.4 Long contracts: methods based on index modelling......Page 248
Pruning......Page 249
10.4.6 Methods based on Brownian motion......Page 251
10.4.7 The stochastic process for the expected index......Page 252
10.4.8 The trapezium model......Page 255
10.4.9 Which method to use?......Page 258
10.7 Acknowledgements......Page 259
11 Arbitrage pricing models......Page 260
11.1 Standard arbitrage theory......Page 261
11.1.1 Delta hedging and the PDE approach......Page 262
11.1.2 Replication and the measure theory approach......Page 264
Risk neutrality......Page 266
Intuitive arguments and the relation to actuarial pricing......Page 267
Delta hedging versus replication......Page 268
Volatility and standard deviation......Page 269
11.2.1 The Black (76) model......Page 270
Discrete time hedging......Page 271
Shares can trade in any amount......Page 272
11.4.1 The balanced market model......Page 273
11.4.2 A toy model for swap prices......Page 275
11.4.3 Option pricing in the balanced market model......Page 276
11.4.4 Pricing weather options......Page 278
11.4.5 The linear imbalance model......Page 279
11.4.7 The stochastic imbalance model......Page 280
11.4.8 Stochastic volatility issues......Page 281
11.4.9 Volatility and risk loading......Page 282
Extensions to the standard theory for the weather case......Page 283
11.6 Further reading......Page 285
12.1 Risk management in liquid markets......Page 287
12.2 Marking positions......Page 288
12.2.1 Expected expiry value......Page 289
The use of market data......Page 290
The liquidation value of non-traded contracts......Page 291
12.3 Expiry risk......Page 292
The temperature-based approach......Page 294
The index-based approach......Page 295
The greeks-based approach......Page 297
12.5 Liquidation value at risk......Page 298
12.7 Liquidity risk......Page 299
12.9 Further reading......Page 300
13.1 Precipitation......Page 301
13.1.1 Precipitation index modelling......Page 302
13.1.2 Daily precipitation modelling......Page 304
13.2 Wind......Page 306
13.2.2 High-frequency wind modelling......Page 308
13.3 Further reading......Page 310
A.1 A general theory for trend modelling and the uncertainty of trend estimates......Page 311
A.1.1 Monte Carlo methods......Page 312
B.2 Parameter estimation......Page 314
B.2.2 Maximum likelihood......Page 315
C.1.1 The chi-square test......Page 317
C.1.3 The Anderson–Darling and Cramér–von Mises tests......Page 318
C.1.5 Monte Carlo tests......Page 319
D.1.1 The general form......Page 321
D.2.2 Call options......Page 322
D.2.4 Collars......Page 323
D.2.7 Binary options......Page 324
D.3 Useful relations for deriving expressions for the expected pay-off......Page 325
D.4 Closed-form expressions for the expected pay-off......Page 326
D.4.2 Call options......Page 327
D.4.3 Put options......Page 328
D.4.5 Straddles......Page 329
D.4.6 Strangles......Page 330
D.4.8 The general form......Page 331
D.5 Numerical examples......Page 332
E.1.1 Derivation strategy......Page 334
E.2.1 Swaps......Page 335
E.2.2 Call options......Page 336
E.2.3 Put options......Page 337
E.2.4 Collars......Page 338
E.2.5 Straddles......Page 339
E.2.6 Strangles......Page 340
E.3 Numerical examples......Page 341
F.1 Useful relations for deriving expressions for the greeks......Page 343
F.2 Closed-form expressions for the greeks......Page 344
F.2.1 Swaps......Page 345
F.2.3 Put options......Page 346
F.2.4 Collars......Page 347
F.2.5 Straddles......Page 348
F.2.6 Strangles......Page 349
F.2.8 The general form......Page 350
F.3 Numerical examples......Page 351
G.1 Closed-form solutions for the expected pay-off on a kernel density......Page 353
G.2 Closed-form solutions for the delta on a kernel density......Page 354
G.4 Closed-form solutions for the pay-off variance on a kernel density......Page 356
G.5 An example......Page 357
H.1.1 Derivation strategy......Page 359
H.1.2 Useful expressions......Page 360
H.3 Closed-form expressions for the beta......Page 363
H.3.1 Swap-swap covariance......Page 364
H.3.2 Swap-call covariance......Page 366
H.3.3 Swap-put covariance......Page 367
H.4 Discussion......Page 368
H.5 Numerical examples......Page 369
The general approach to simulation from a given CDF......Page 372
The Polar algorithm for normal distributions......Page 373
Algorithms for the gamma distribution......Page 374
Simulation from a negative binomial distribution......Page 375
The simulation of time series......Page 376
J.1.1 Index regression......Page 377
J.1.2 Pay-off regression......Page 378
References......Page 379
Index......Page 388
