Stephen Jewson, Anders Brix and Christine Ziehmann discuss the relative merits of current weather derivative pricing techniques and speculate on how weather contracts might be priced in the future.

The aim of this article is to give a brief introduction to weather derivative pricing, risk management and the use of forecasts. Depth is sacrificed for breadth: more details will be found in the authors' forthcoming book on the same subject. We start with a discussion of the methods available for pricing single contracts during the pre-forecast period. We review three actuarial methods currently in use, and comment on their relative merits.

The discussion then extends from single contracts to portfolios, and to the use of forecasts. Finally, we speculate on the role that arbitrage pricing may have in the future, and describe the basic features of a pari-mutuel trading framework. We will focus on temperature contracts throughout, since these make up the majority of contracts traded in the weather market. However, similar pricing methods apply to contracts based on other variables.

**Single contract pricing**

How should weather contracts be priced? There are two pricing paradigms that are potentially relevant to weather: actuarial methods, which involve estimating the distribution of possible financial outcomes of a contract (often, but not solely, from an analysis of historical data) and arbitrage methods, which rely on estimates of the cost of writing and hedging a contract or portfolio.1 Which is relevant for weather?

For weather swap contracts, the only possibility is actuarial pricing: there is nothing available with which to hedge the risk in a swap since the underlying index is not traded. For weather option contracts it would, in principle, be possible to hedge the risk using swaps on the same index, but in practice the swap market is not at present sufficiently liquid for that to be cost-effective, and so actuarial pricing should be used for these contracts too. The bulk of this article will thus describe actuarial pricing for swaps and options. Only at the very end will we speculate how options might be priced if the swap market were liquid, and give a brief description of how a pari-mutuel system works.

Actuarial pricing of an individual weather contract consists of estimating the distribution of possible outcomes of the contract using historical data, trends and forecasts. The price is then derived from this distribution, often as the expectation plus a loading proportional to the risk. We will break the estimation process down into three steps:

**Data cleaning**

Data cleaning is a set of algorithms that is applied to raw historical data obtained, for example, from one of the national meteorological services. It is a necessary first step in weather pricing because raw data is not generally of very high quality. Collecting long historical records of accurate and consistent data has not been a particularly high priority for the world's meteorological services, which have instead tended to focus on the development of complex forecasting models. As a result, historical data records contain many problems, such as gaps, erroneous values, and, most alarmingly, large jumps due to changes at the measuring stations.

Data cleaning attempts to correct these problems, and typically consists of the following stages:

1. replacement of missing values using information from surrounding locations.

2.identification and correction of erroneous values; and

3. identification and correction of the jumps in data due to station changes.

Data which has been corrected in this way is generally only available from the private sector. The cleaning procedures applied, especially in step 3, are necessarily complex and consist of many steps. State-of- the-art methods identify jumps using a combination of information available in the historical meta-data and statistical analysis.

**Detrending**

Even once data has been cleaned as described, further processing is still necessary before it can be considered consistent with present day observations. The major remaining issue is the presence of trends due to urbanisation and global warming. Detrending (or `re-centering') is the empirical process of estimating the size and shape of these trends, and removing them in such a way that the adjusted data is consistent with present day measurements. Trends can also be extrapolated into the future as part of a forecast of future warming. Over short periods of data (e.g. 20 years), linear trends are often used. With longer periods it may be necessary to use more complex trends (piecewise-linear, parametric non-linear or non-parametric) to account for the time-varying nature of the increase in temperature.

The decision of how many years of data to use is a difficult one, and affects the final price significantly. Backtesting studies by the authors indicate that, on average over many stations, using somewhere between 12 and 20 years of detrended data works well. However, individual stations may not behave like the average, and the data from each station needs to be inspected visually.

Detrending can be done either on daily temperatures or annual contract index values. For short contracts (with lengths of one day to one month), the former is preferable because it makes better use of the historical data.

One of the pitfalls of detrending is that removing a trend also removes part of the variability of the time-series: it is essential to compensate for this before moving on to the modelling stage.2

**Burn analysis**

Burn analysis is the simplest method for estimating the distribution of outcomes of a weather contract. We break it down into two stages. Note that daily detrending would occur before step 1, while annual index detrending would occur between steps 1 and 2. Detrending is not always used if only ten years of historical data are being considered.

1.Calculation of historical annual values of the settlement index for the weather contract over a period of history (such as the last 10, or last 30 years).

2.Conversion of these values into contract payoffs.

The resulting payoff values are then considered representative of the distribution of likely payoffs for the upcoming contract. The mean payoff can be calculated and used as the `fair price'. The distribution of payoffs can be used to give an indication of the probabilities of the various outcomes and to measure the risk.

Burn analysis is simple enough to implement with a spreadsheet and has the advantage that it does not involve any significant assumptions except for those made in the detrending stage and the assumption of independence of years.

**Index modelling**

Index modelling is an extension of burn analysis, where the historical annual index values are modelled statistically in order to produce a smooth distribution and to extrapolate the tails. The distribution is chosen empirically: winter HDD contracts in mid latitudes and summer CDD contracts in low latitudes can be modelled well with a normal distribution. Summer CDD contracts in higher latitudes tend not to be normally distributed (because of the common occurrence of days with no degree days) and hence other distributions such as the gamma may be used. Average temperature contracts can often be modelled with the normal distribution, and event contracts with the poisson or negative binomial distributions. An alternative to using these parametric distributions is to use a non-parametric distribution such as a kernel density. This has the advantage that it can be applied reasonably blindly to most degree day and average temperature contracts.

The disadvantages of index modelling are that there is no theory to indicate which distribution to use, and that the estimated parameters of the distribution will have large errors because of the limited amounts of historical data. We have performed simulation studies to evaluate the relative accuracy of burn and index modelling, and have come to the following conclusions:

**Daily modelling**

Some of the inadequacies of burn analysis and index modelling can in principle be overcome by daily modelling of the temperature time series. Suppose we had a statistical model that described temperature variability well. It would have the following advantages over burn and index models:

In practice, daily modelling of temperature for weather pricing is difficult because of the complex behaviour of observed temperatures and because the final prices depend sensitively on the details of the model.

The first stage of modelling is the mathematical description of the seasonal cycle in mean and variance. This is relatively straightforward, although there are two schools of thought even at this stage: the seasonal cycle can either be modelled deterministically (i.e. the seasonal cycle is the same every year, and only the weather changes), or stochastically (both the seasonal cycle and the weather change randomly).

The residual from the seasonal cycle (usually referred to as `anomalies' in meteorology) can then be modelled using time-series models. This is difficult: the anomalies may well not be normally distributed, and the distribution may change with time of year. The anomalies are strongly autocorrelated in time, and in ways that cannot be captured using the well-known ARMA models. The autocorrelations may also change with time of year. Models have been developed which cope with some, but not all, of these difficulties. The complete inadequacy of ARMA models (at least in the deterministic seasonal cycle case) is well known in the industry, and proprietary models have been developed which overcome the basic ARMA problem of underestimation of index variance. But the issues of seasonality and non-normality are less easy to solve.

**Summary of single contract pricing**

We have described three models for estimating the distribution of outcomes from a single weather contract. None of the models is perfect: burn is simple and light on assumptions, but may not make good use of available data. Daily modelling makes better use of the data, but relies on a number of assumptions that may not always hold. Index modelling is somewhere in between. A practical approach is to use a number of methods for pricing each contract to get a good estimate of the sensitivity of the results to choice of model.

**Portfolios**

Up to now we have considered how to evaluate the distribution of outcomes of a single contract. In practice, most issuers of weather contracts will write a large portfolio in order to take advantage of diversification and hedging effects. For them, the challenge is to estimate the distribution of financial outcomes of the whole portfolio. Single contracts are then priced in the context of the marginal changes they cause to the portfolio, rather than on their stand-alone performance. Burn analysis, index modelling and daily modelling can all be extended to portfolios.

**Burn analysis**

Burn analysis for portfolios is little different from burn analysis for individual contracts. The historical index values for each contract in the portfolio are converted to historical payoff values, and these are aggregated to calculate the historical performance of the portfolio.

**Index modelling**

Portfolio index modelling is also reasonably straightforward. Unlike the single contract case, however, the evaluation of the payoffs can only be done using simulations because of the non-linearity of the payoff structure of option contracts and the presence of non-normal marginal index distributions. In the special case where all the indices are assumed to be normally distributed, well-known simulation methods such as singular value or Choleski decomposition can be used. Thousands of surrogate index values for the contracts are simulated, with linear correlations that are the same as the observed. In the general case where the indices may have any distribution, one can generalise this procedure by using rank correlations or copulas.

**Daily modelling**

Daily modelling for portfolios is extremely challenging, because, in addition to the difficulties of modelling single locations as mentioned above, one also has to represent cross-lag correlations between locations. A compromise is to use a daily model for each location separately, derive index distributions, and use observed index correlations to combine these index distributions.

**Understanding portfolios**

There is much more to understanding the performance of a portfolio than just modelling the aggregate distribution of outcomes using one of the methods described above. We can also ask questions like:

All of these questions can be answered quite readily in a good portfolio modelling system.

**Forecasts**

So far we have been considering the evaluation of weather contracts and portfolios on the basis of historical data alone. This is relevant only when there are no useful forecasts, which is generally the case up to six months before the start of a contract in the US, and up to two weeks before the start of a contract in Europe. From then onwards additional information is available from weather forecasts in Europe and weather and seasonal forecasts in the US. Because of the errors in forecasts, they cannot completely replace the use of historical data.

Rather, historical data and forecasts need to be combined in such a way as to extract the information in both. Three simple methods for doing this are described below.

**All or nothing**

The `all or nothing' method for inclusion of forecasts is the simplest. The user of the forecast (which we will assume in this case is a weather forecast) decides that the forecast is reasonably accurate out to, for example, four days. They use the forecast for this four day period, and from then on use only historical data.

The advantage of this method is extreme simplicity. The disadvantage is that it does not use all the information in the forecast and does not account for forecast errors.

**Index distribution forecasts**

Forecasts of the distribution for the most common weather indices such as degree days and average temperatures can be obtained from private meteorological services. One can then combine these with the index distributions derived from historical data, typically by adding weighted pdfs. The advantage of this method is that it allows the entire period of the forecast to be used. The weights, which balance the relative contribution of the historical data and the forecast, can be optimised on the basis of back-testing, or guessed on the basis of intuition.

**Pruning**

The most advanced method for incorporation of forecasts into pricing models goes by the name of `pruning'. It involves taking simulations from a daily model and weighting them according to consistency with a probabilistic forecast (which can be derived either from an ensemble forecast or from a single forecast with error statistics). It has the advantages that it uses all the available information and can be applied to any index type at both the weather and the seasonal timescales.

**Non-actuarial methods**

Up to now we have discussed actuarial pricing of weather derivatives, which is the relevant method given current market conditions. In the subsequent two sections we describe two other possible ways in which weather contracts might be priced in the future.

**Arbitrage pricing**

Many financial derivatives are priced on the basis that the writer of the derivative will endeavour to hedge the risk they have assumed by dynamically trading in the underlying asset.

The arbitrage price of the derivative is then the discounted expected payoff of the derivative, plus the discounted expected loss on the asset.3 Can this theory be extended to weather? As we saw in the introduction, this cannot be applied to weather swaps because the underlying asset, the weather, cannot be traded. However it can certainly be applied to weather options.

Even in the current market it would be possible to delta-hedge the risk inherent in a weather option by trading a linear swap based on the same underlying index. The catch is that such trading would be extremely expensive, and the reduction in overall risk would not justify the extra cost. However, we now imagine a future scenario in which trading on linear swaps is sufficiently liquid that the cost of such trading is minimal. Arbitrage pricing would then apply.

To calculate the arbitrage price we need to understand the statistical dynamics of the swap price. The simplest model is to assume that the swap market is well balanced by hedging demand on both sides of the contract, and hence that it would trade at the discounted expected value. The line of argument can then proceed in one of two ways:

To calculate this arbitrage price, one could integrate the swap price process forward in time, using volatility calculated from historical forecast errors, and derive the expected payoff of the option from the simulated swap settlement values.

It is tempting to think that this arbitrage price can also be calculated using the actuarial methods described. However, this should not be done. We have seen that the actuarial methods suffer from a lot of uncertainty in the estimate of the mean payoff, and so an actuarially calculated value could differ significantly from that based on integration of the swap price process.

If a trader were to make a market in an option quoted at a price different from the swap price-based result, they would create arbitrage opportunities.

Other arbitrage models are also possible. We could relax the assumption that the swap market is balanced, and replace that with the assumption that there is more demand on one side of the contract. This would presumably lead to a non-zero market price for risk, and the swap price would drift more quickly than the risk-free rate. One might then be able to model the swap price with a different stochastic process, which in turn would lead to a no arbitrage price different from that derived above.

**Pari-mutuel systems**

Finally, we briefly mention the possibility of pari-mutuel systems for weather option trading. Pari-mutuel systems are used by bookmakers to set betting odds in such a way that the losers of a bet will pay the winners while the bookmaker pays nothing and takes a cut. A similar approach could be used for the trading of weather derivatives.

A weather derivative `bookmaker' trades options with a large number of counterparties: the premiums (or the limiting payoffs) are adjusted so that the total of all the payouts is exactly balanced by the premiums taken in (minus a cut). For the bookmakers this is arbitrage: they make a profit without taking on any risk except the credit risk due to the possibility that one or more counterparties might go bankrupt. The actual distribution of outcomes of a contract only affects the price indirectly via the level of demand for that contract relative to other contracts.

**Summary**

We have reviewed various methods for pricing weather derivatives. Only the actuarial methods are relevant at the present time, although arbitrage and pari-mutuel pricing may become more important in the future.

A number of variations of the actuarial approach were described and it was emphasised that none of these methods is perfect. A sensible and practical pricing approach is to use a selection of methods in order to gain a feeling for the uncertainty inherent in weather pricing.

1 In fact these two paradigms are not entirely different after all: one can think of arbitrage pricing as a special case of the actuarial concept of pricing in the context of a portfolio. In the arbitrage case the portfolio is a specially constructed one that is changed dynamically to reduce the total risk to zero, and so the actuarial price is just the discounted natural expectation of outcomes of the whole portfolio with no risk loading, which is also the arbitrage price. However, we will consider actuarial and arbitrage pricing as separate paradigms for our present purposes.

2 Note that a mathematically more consistent approach to the detrending and modelling stages would be to perform them simultaneously.

3 This simple intuitive result is equivalent to the formulation that is more often used in financial textbooks. To see the connection, introduce a change of measure in which natural probabilities are replaced by artificial ones chosen such that the expected loss on the asset is set to zero: the price is then the discounted expected payoff on the derivative using the artificial probabilities. This is the standard formula.

4 An efficient forecast is one for which the day-to-day changes of the forecast cannot be predicted in advance. Real forecasts are close to being efficient.

By Stephen Jewson, Anders Brix and Christine Ziehmann

Stephen Jewson is director of business development at Risk Management Solutions (RMS) in London. He is responsible for marketing and sales of RMS weather derivative products in Europe, and is a regular speaker and writer on weather risk quantification.

Anders Brix is lead modeller in the weather derivatives modelling group at RMS's London office. He recently completed the development of the RMS daily simulation model and has written a number of papers on statistics and probability.

Also based in London, Christine Ziehmann is senior modeller at RMS where she is responsible for the development of weather derivative pricing algorithms and software. She has published many academic papers in physics and meteorology.