Forward Prices in Markets Driven by Continuous-time Autoregressive Processes

We analyse the forward price dynamics for contracts written on a spot following a continuous-time autoregressive dynamics. Prime examples of such spots could be power or freight rates, or weather variables like temperature and wind speed. It is shown that the forward price evolves according to template term structure functions, which are scaled by the deseasonalized spot and its derivatives. These template term structure functions can be expressed as a series of exponentially decaying functions with rates given by the (real parts) of the eigenvalues of the autoregressive dynamics. Moreover, the continuous-time autoregressive spot dynamics is differentiable up to an order less than the autoregressive order, and this is precisely the derivatives needed in the representation. The template term structures may produce humps in the forward curve. We consider several empirical examples for illustration based on a model relevant for the temperature market. A particular result of our analysis is that the paths of the forward price are non-differentiable, although the underlying spot is smooth. Our results offer insight into the dynamics of forward and futures prices for contracts in the markets for weather, shipping and power.