A Barndorff-Nielsen and Shephard model with leverage in Hilbert space for commodity forward markets.

Abstract

We propose an extension of the model proposed by Barndorff-Nielsen and Shephard, based on stochastic processes of Ornstein-Uhlenbeck taking values in Hilbert spaces, including the leverage effect. We compute explicitly the characteristic function of the logreturn and the volatility processes. By introducing a measure change of Esscher type we provide a relation between the dynamics described with respect to the historical and to the risk-neutral measures. We discuss in detail the application of the proposed model in order to describe the commodity forward curve dynamics in a Heath-Jarrow-Morton framework, including the modelling of forwards with delivery period occurring in energy markets and pricing of options. For the latter, we show that a Fourier approach can be applied in this infinite-dimensional setting, relying on the attractive property of conditional Gaussianity of our stochastic volatility model. In our analysis, we study both arithmetic and geometric models of forward prices and provide appropriate martingale conditions in order to ensure a no-arbitrage dynamics.