Stochastic Modelling of Temperature Variations with a View Towards Weather Derivatives

Abstract

Daily average temperature variations are modelled with a mean‐reverting Ornstein–Uhlenbeck process driven by a generalized hyperbolic Lévy 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).

Keywords:

• Temperature modelling
• stochastic processes
• Lévy processes
• mean‐reversion
• seasonality
• fractionality
• temperature futures and options

Acknowledgements

We are grateful to Kjersti Ulseth and Neil Shephard for valuable and inspiring discussions. Jūratė Šaltytė‐Benth acknowledges financial support by the Norwegian Research Council under grant NFR: 155120/432.

Notes

In their model, Brody et al. (2002) allow the speed of mean‐reversion κ to vary with time. However, they do not discuss the modelling of this any further and we find it most natural to leave it constant.

We believe there is a small misprint in Equations 2 and 2c in Campbell and Diebold (2002) in the presentation of the ARCH model

February 29 was removed from the sample in each leap year to have years of equal size

We used nlinfit in Matlab to perform this estimation.

The confidence band is given automatically by Splus.

Admittedly, this cannot be seen from the tables. The estimated numbers can be obtained from the authors by request

Recall that we had approximately 13 years of observations, which means that each group will constitute of 13 residuals for each particular day of the year.

This is efficiently done using the programming language Ox; see Doornik (1998).

See http://www.cme.com/prd/wec/ for more information about this trading.

The HDD index over the time interval [τ₁, τ₂] is defined in a continuous‐time setting as

. One considers 18°C to be the limit for switching on heating, thereby the name HDD.