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,

The aim of this study was to evaluate the 1-year-period prevalence of tension-type headache in a large population based sample. The study population included 33,764 twins aged 12-41 years old from the population based new Danish Twin Registry. They received a posted headache questionnaire and the response rate was 83.5%. The self-reported 1-year-period prevalence of tension-type headache was 86.0%; 78.9% among men and 92.5% among women. The 1-year-period prevalence of infrequent episodic, frequent episodic and chronic tension-type headache was 63.5, 21.6 and 0.9%, respectively. Frequent episodic and chronic tension-type headache was significantly more frequent in women than men. The prevalence of frequent episodic tension-type headache increased slightly in men until age 39 then it declined, while it increased about 20% point in women from age 12 years to age 20-39 years old and then it declined. Congruently, the prevalence of chronic tension-type headache increased until age 39 and declined thereafter in both sexes. Chronic tension-type headache is rare in persons 12-14 years old. These effects were confirmed by age trends of the different subtypes of tension-type headache using a regression model. The prevalence of migraine varied from 7.0 to 16.8% in men and from 8.2 to 31.0% in women. It increased from age 12 to 34 years and then declined. The risk and frequency of tension-type headache was significantly higher in those with migraine than those who had never had migraine. Future large longitudinal follow-up studies are required.

We model spot prices in energy markets with exponential non-Gaussian Ornstein–Uhlenbeck processes. We generalize the classical geometric Brownian motion and Schwartz' mean-reversion model by introducing Lévy processes as the driving noise rather than Brownian motion. Instead of modelling the spot price dynamics as the solution of a stochastic differential equation with jumps, it is advantageous from a statistical point of view to model the price process directly. Imposing the normal inverse Gaussian distribution as the statistical model for the Lévy increments, we obtain a superior fit compared to the Gaussian model when applied to spot price data from the oil and gas markets. We also discuss the problem of pricing forwards and options and outline how to find the market price of risk in an incomplete market.