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 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).Temperature modelling, stochastic processes, Levy processes, mean-reversion, seasonality, fractionality, temperature futures and options,

A critical view on temperature modelling for application in weather derivatives markets

“…Further, it was observed by Campbell and Diebold [14] that the autocorrelation function (ACF) of the squared residuals in many US cities has a seasonal structure. The same observation was made for several locations in Norway and Lithuania in the papers Benth andŠaltytė Benth [4] anď Saltytė Benth et al [28], respectively, for German temperature data in Härdle and Lopez Cabrera [20], and for Asian data in Benth et al [9]. Moreover, a characteristic seasonal pattern for the daily variance of temperature was observed (see Benth et al [5] for a detailed discussion in connection with Stockholm temperature data).…”

“…In Benth et al [4,7], the choice of K = 3 turned out to give a very good As already mentioned, in many locations one finds signs of GARCH effects in the residuals after removing the influence of σ BSB (t) (that is, in the data ε(t)/σ BSB (t)). Such effects are minor, but to explain them in the proposed model it is natural to assume that σ(t) = σ BSB (t)σ GARCH (t) with…”

“…Moreover, a characteristic seasonal pattern for the daily variance of temperature was observed (see Benth et al [5] for a detailed discussion in connection with Stockholm temperature data). In addition, in Benth et al [4] and [5] a small GARCH effect in the ACF of the squared residuals after explaining the seasonality in the daily variance was observed, but not modelled.…”

“…Dornier and Querel [17] suggest an AR(1) structure for temperatures observed at Chicago O'Hare airport, whereas Alaton et al [1] use a similar model for temperatures in Bromma, Sweden. Brody et al [11] suggest to model temperatures in central England by such a model, followed by Benth and Saltyte-Benth [4] for temperatures collected in several cities in…”

Modelling spatio-temporal variability of temperature

“…The maximum lag is determined by the AIC and denotes temperature volatility which Campbell and Diebold (2005) model as a seasonal GARCH process. In this paper, however, we follow Benth and Šaltytė-Benth (2005), who explain temperature variance by the distinctive seasonal pattern:…”