Daily average temperature variations are modelled with a mean‐reverting Ornstein–Uhlenbeck process driven by a generalized hyperbolic Levy process and having seasonal mean and volatility. It is empirically demonstrated that the proposed dynamics fits Norwegian temperature data quite successfully, and in particular explains the seasonality, heavy tails and skewness observed in the data. The stability of mean‐reversion and the question of fractionality of the temperature data are discussed. The model is applied to derive explicit prices for some standardized futures contracts based on temperature indices and options on these traded on the Chicago Mercantile Exchange (CME).

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Stochastic Modelling of Temperature Variations with a View Towards 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.

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The volatility of temperature and pricing of weather derivatives

This article presents a general method for pricing weather derivatives. Specification tests find that a temperature series for Fresno, CA follows a mean-reverting Brownian motion process with discrete jumps and autoregressive conditional heteroscedastic errors. Based on this process, we define an equilibrium pricing model for cooling degree day weather options. Comparing option prices estimated with three methods: a traditional burn-rate approach, a Black-Scholes-Merton approximation, and an equilibrium Monte Carlo simulation reveals significant differences. Equilibrium prices are preferred on theoretical grounds, so are used to demonstrate the usefulness of weather derivatives as risk management tools for California specialty crop growers.

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Pricing Weather Derivatives

This paper analyzes the weather derivatives traded at the Chicago Mercantile Exchange (CME), with futures and options written on different temperature indices. We propose to model the temperature dynamics as a continuous-time autoregressive process with lag p and sea- sonal variation. The choice p - 3 turns out to be sufficient to explain the temperature dynamics observed in Stockholm, Sweden, where we fit the model to more than 40 years of daily observa- tions. The main finding is a clear seasonal variation in the regression residuals, where temperature shows high variability in winter, low in autumn and spring, and increasing variability towards the early summer. Our model allows for derivations of explicit prices for several futures and options. Note that the volatility term structure of futures written on the cumulative average temperature has a modified Samuelson effect, where the volatility prior to the measurement period increases, except for the last part, where it may decrease.

Putting a Price on Temperature

While significant progress in microcredit and microfinance has been made in low‐income countries, lending for small farming enterprises has been limited. This article reviews how innovative index‐based risk‐transfer products (IBRTPs) can be used to transfer the correlated natural disaster risks that often hamper the development of farm‐level microcredit. By linking lending to IBRTPs, access to microcredit can be enhanced while also providing opportunities to offer mutual sharing of the basis risk that remains after correlated risks are transferred into global markets. This opens the way for new thinking about developing agricultural insurance in low‐income countries.

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Enhancing microfinance using index‐based risk‐transfer products

The main objective of the work described is to find a pricing model for weather derivatives with payouts depending on temperature. Historical data are used to suggest a stochastic process that describes the evolution of the temperature. Since temperature is a non-tradable quantity, unique prices of contracts in an incomplete market are obtained using the market price of risk. Numerical examples of prices of some contracts are presented, using an approximation formula as well as Monte Carlo simulations.

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David Stillberger

Papers

On modelling and pricing weather derivatives