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Название: Weather Derivative Valuation: The Meteorological, Statistical, Financial and Mathematical Foundations
Авторы: Jewson S., Brix A., Ziehmann C.
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.