Pricing Weather Derivatives Research Articles

Weather‐risk hedging by farmers

PurposeSmall farmers in developing countries have very little means of managing the weather‐risk to their agricultural produce. Weather derivatives could provide a solution, but the demand for such instruments and the willingness to invest in them needs to be researched. The purpose of this paper is to assess weather‐risk hedging by farmers, focusing on the willingness to pay in Rajasthan, India.Design/methodology/approachThe paper presents results of a contingent valuation study done on the findings of a survey carried out in six villages in the state of Rajasthan. The survey was done on over 500 farmers after explaining the concept of weather derivatives and how they would work to help them hedge their weather‐related yield risk. The survey included questions on factors, which could have a bearing on the farmers' willingness to pay, and a bidding game where responses were solicited to premiums in a hypothetical market. Probit and logit models are used to determine probabilities of “Yes” responses to various bids and the mean willingness‐to‐pay.FindingsThe paper brings out a model, which uses nine independent variables affecting the probability of a farmer saying “Yes” to a price quoted to him for a weather derivative. Using the results from the probit and logit models, the farmers' mean willingness‐to‐pay is determined to be around 8.8 per cent of the maximum possible payout of a weather derivative contract.Originality/valueWith weather derivatives being accepted as a means of risk management for agriculture in developing countries, the willingness‐to‐pay figures determined in this paper would provide an insight to the structuring and pricing of weather derivatives, especially in developing countries.

Indifference Pricing of Weather Derivatives

AbstractWeather derivatives are difficult to price due to the nontradability of weather and the absence of liquid secondary markets for these contracts. We use the concept of indifference pricing to develop a model for calculating the willingness to pay for weather insurance. Compared with other approaches, indifference pricing is less ambitious since it does not attempt to predict a transacted market price. The application of indifference pricing in the case of German crop producers shows that their willingness to pay for weather insurance depends on the production program and varies regionally. This suggests the development of tailored insurance products.

Temperature Models for Pricing Weather Derivatives

We present four models for predicting temperatures that can be used for pricing weather derivatives. Three of the models have been suggested in prior literature, and we suggest another model which uses splines to remove trend and seasonality effects from temperature time series in a flexible way. Using historical temperature data from 35 weather stations across the United States, we test the performance of the models by evaluating virtual heating degree days (HDD) and cooling degree days (CDD) contracts. We find that all models perform better when predicting HDD indices than predicting CDD indices. However, all models based on a daily simulation approach significantly underestimate the variance of the errors.

Weather Derivative Pricing and the Modelling of Trends: Objective Bayesian Versions of the Flat-Line, Linear Trend and Damped Linear Trend Models

Published methods for the pricing of weather derivatives are based on classical statistics, in that the predictions of the distributions of weather indices that they use are based on best estimates of model parameters. It is likely that such methods do not accurately capture the true uncertainty because they either ignore or approximate parameter and model uncertainty. In this article we derive the objective Bayesian versions of the flat-line and linear-trend models. The use of objective Bayesian statistics allows us to incorporate parameter uncertainty into the predictive distribution in a simple way. The result is a change in the shape of the predictive distribution, but no change in the predicted mean and variance, when compared with comparable classical statistical methods. We also derive a Bayesian version of the damped linear-trend model, which, in a limited sense, also incorporates model uncertainty.

Closed-Form Expressions for the Pricing of Weather Derivatives: The Expected Payoff for t-Distributed Indices

We derive closed-form expressions for the expected payoff of weather derivatives contracts for a t distributed weather index. There are three common situations in which t distributions might serve as a reasonable model for weather indices: first, some weather variables may be t distributed; second the t distribution can be used as a fatter-tailed alternative to the normal distribution as a stress test or model alternative;and third, objective Bayesian predictions of normally distributed data are t distributed.

Modelling the Temperature Time‐dependent Speed of Mean Reversion in the Context of Weather Derivatives Pricing

In this paper, in the context of an Ornstein–Uhlenbeck temperature process, we use neural networks to examine the time dependence of the speed of the mean reversion parameter α of the process. We estimate non‐parametrically with a neural network a model of the temperature process and then compute the derivative of the network output w.r.t. the network input, in order to obtain a series of daily values for α. To our knowledge, this is the first time that this has been done, and it gives us a much better insight into the temperature dynamics and temperature derivative pricing. Our results indicate strong time dependence in the daily values of α, and no seasonal patterns. This is important, since in all relevant studies performed thus far, α was assumed to be constant. Furthermore, the residuals of the neural network provide a better fit to the normal distribution when compared with the residuals of the classic linear models used in the context of temperature modelling (where α is constant). It follows that by setting the mean reversion parameter to be a function of time we improve the accuracy of the pricing of the temperature derivatives. Finally, we provide the pricing equations for temperature futures, when α is time dependent.

A time series analysis on the seasonality of air temperature anomalies

AbstractSurface air temperature anomalies relative to seasonal variations are of great concern from a long‐term forecasting perspective. This article is concerned with marginally normalized time series in which the original data of the temperatures are standardized using the mean values and variances of the estimated deterministic seasonal cycles. A particular parametric form of a non‐stationary auto‐regressive (AR) model to quantify the anomalies is considered, by applying it to the normalized data. This model fits substantially better than an ordinary AR model for normalized datasets of the surface air temperature obtained from almost all of the stations in Japan, and exhibits a significant seasonal structure in their auto‐correlation. The model is applied to high‐pass filtered data to investigate the relation between the seasonal structure and a high‐frequency variability in anomalies, which helps in determining the climatic influence on anomalies of surface air temperature in Japan. Furthermore, an illustrative example in the pricing of weather derivatives to achieve better performance of the non‐stationary model than that of the ordinary AR model is presented. Copyright © 2007 Royal Meteorological Society

The volatility of temperature and pricing of weather derivatives

We propose an Ornstein–Uhlenbeck process with seasonal volatility to model the time dynamics of daily average temperatures. The model is fitted to approximately 45 years of daily observations recorded in Stockholm, one of the European cities for which there is a trade in weather futures and options on the Chicago Mercantile Exchange. Explicit pricing dynamics for futures contracts written on the number of heating/cooling degree-days (so-called HDD/CDD futures) and the cumulative average daily temperature (so-called CAT futures) are calculated, along with a discussion on how to evaluate call and put options with these futures as underlying.

Weather Derivative Pricing and the Normal Distribution: Comparing Three Fitting Schemes using the Out-of-Sample Log-Likelihood Scoring System

Many common weather indices are very close to being normally distributed, and it may be reasonable to assume they are exactly normally distributed for the purpose of pricing weather derivatives. Given that assumption, how should the indices be modelled? We use the expected out-of-sample log-likelihood score to compare 3 schemes: standing normal fitting, adjusted variance normal fitting, and the t-distribution.

Weather Derivative Pricing and the Normal Distribution: Fitting the Variance to Maximise Expected Predictive Log-Likelihood

The normal distribution is commonly used to predict weather indices when pricing weather derivatives. The standard method for making such predictions involves calculating an unbiased estimator for the population variance. The variance of the prediction (the predictive variance) is then the unbiased estimator for the population variance with an adjustment to account for sampling error on the mean. This is not, however, the only way to model the predictive variance, and it is not necessarily the best way. We investigate an alternative method, based on adjusting the predictive variance so as to maximise the expected predictive log-likelihood. For the small sample sizes often used in weather derivative pricing the resulting predictive variances are significantly larger than those calculated using the standard method.

Valuation of Weather Derivatives

One of most distinguishing features of weather derivatives is that the market is incomplete This article presents a new method for pricing weather derivatives in an incomplete market. The proposed method has two advantages exhibited by only a few of models in other weather-related articles. First, the method for pricing assets in an incomplete market is employed to overcome the non-tradable feature of the underlying assets of the weather derivatives for which the no-arbitrage method breaks down. Second, an efficent analytical method is incorporated into the proposed model to make the Asian-type payout of weather derivatives much easier to evaluate than by other numerical methods. Pricing formulae and numerical results for weather derivatives other than weather options, including weather caps, weather floors, weather swaps and weather collars, are also presented.

Estimating Trends in Weather Series: Consequences for Pricing Derivatives

Predictions of future weather conditions play an important role in pricing weather derivatives. In many instances, the dates for which we require predictions are well beyond the point where physical forecasts have any skill. Under these circumstances, predictions are generated from statistical models of historic data. This paper derives conditions for which the predictive performance in regression is improved by ignoring or shrinking the contribution from some of the explanatory variables. We suggest methods for estimating the degree of shrinkage required in practice. We illustrate our methods using surface temperature data from fifteen stations in the United States.

Portfolio Effects and Valuation of Weather Derivatives

AbstractIn a mean‐variance framework, the indifference pricing approach is adopted to value weather derivatives, taking account of portfolio effects. Our analysis shows how the magnitude of portfolio effects is related to the correlation between weather indexes and other risky assets, the correlation between weather indexes, and the payoff structures of the existing weather derivatives in an investor's asset portfolio. We also conduct some preliminary empirical analysis. This study contributes to the weather derivative pricing literature by incorporating both the hedgeable and unhedgeable parts of weather risks in illustrating the portfolio effects on the indifference prices of weather derivatives.

Density forecasting for weather derivative pricing

Weather derivatives enable energy companies to protect themselves against weather risk. Weather ensemble predictions are generated from atmospheric models and consist of multiple future scenarios for a weather variable. They can be used to forecast the density of the payoff from a weather derivative. The mean of the density is the fair price of the derivative, and the distribution about the mean is important for risk management tools, such as value-at-risk models. In this empirical paper, we use 1- to 10-day-ahead temperature ensemble predictions to forecast the mean and quantiles of the density of the payoff from a 10-day heating degree day put option. The ensemble-based forecasts compare favourably with those based on a univariate time series GARCH model. Promising quantile forecasts are also produced using quantile autoregression to model the forecast error of an ensemble-based forecast for the expected payoff.

Hot or Cold? A Comparison of Different Approaches to the Pricing of Weather Derivatives

This article reviews six different temperature forecasting models proposed by prior literature for pricing weather derivatives. Simulation of these models is used to estimate daily temperature and, as a consequence, the metrics used for pricing temperature derivatives. The models that rely on an auto-regressive moving average (ARMA) process exhibit a better goodness-of-fit than those that are established under Monte Carlo simulations. However, the superiority of ARMA-type models is not reflected over the forecast horizon. Over that period, the models that rely on Monte Carlo simulations exhibit a tendency to over-forecast the monthly accumulated heating degree day (AccHDD) index and to under-forecast the monthly accumulated cooling degree day (AccCDD) index. Alternatively, models established under the ARMA approach both under-forecast and over-forecast the monthly accumulated indices. All models consistently over-forecast the average daily temperature. The most appropriate pricing model varies between cities and months. Finally, the models examined in this study generate a more accurate AccHDD futures price than the price traded on the market. However, the ability of these models to estimate the AccCDD futures price is significantly poorer than that of the market.

Weather Derivative Pricing and the Impact of El Nino on US Temperature: Empirical Tests of an Optimal Categorical Forecasting Scheme

Weather derivative pricing for US locations can potentially be improved through the judicious use of seasonal forecasts. However, the number of El Nino and La Nina events that appear in the historical record is small and for many locations the signals are weak. We run some simulation-based tests of an optimal categorical forecasting scheme that takes the small sample size and weak signal into account and tries to produce RMSE-optimal seasonal forecasts.

Weather Derivative Pricing and the Detrending of Meteorological Data: Three Alternative Representations of Damped Linear Detrending

It is often desirable to remove the trends from historical meteorological data prior to using that data for the pricing of weather derivatives. In previous articles we have introduced the method of damped linear detrending and argued that it is an effective way to remove trends when the trends are approximately linear. In this article we show that damped linear detrending can be interpreted in three different ways: as a mixture of the linear and flat-line models, as a linear model with the trend slope reduced, and as a linear model with reverse extrapolation.

Weather Derivative Pricing and the Impact of El Nino on US Temperature: The Statistics of Optimal Categorical Predictions

Seasonal forecasts of US temperature have the potential to improve weather derivative pricing. One way to produce such forecasts is to divide the range of possible Pacific ocean surface temperatures into three categories and look at the US temperatures that occur for each category. Unfortunately this approach is hampered by weak signals and lack of data. We perform a statistical analysis of this problem and derive expressions for optimal forecasts that take into account the weakness of the signal and the small sample sizes.

Weather Derivative Pricing and the Year-ahead Forecasting of Surface Air Temperature: An Empirical Evaluation of Damped Linear Detrending

The pricing of temperature derivatives is often based on statistical methods that predict the surface air temperature a year in advance. We perform an empirical comparison of three simple methods for such year-ahead temperature forecasting and draw clear conclusions about their relative merits. The three methods we consider are the standard flat-line and best-fit linear detrending methods and the recently introduced damped linear detrending method.

Weather Derivative Pricing and a Preliminary Investigation into a Decision Rule for Detrending

Weather derivative pricing is often based on year-ahead forecasts of the weather made using simple statistical models and past weather data. Two of the simplest models in common use are the flat line and linear trend models. We consider the question of how to choose between these two models and test a simple decision rule based on closed-form expressions for the RMSE.