Mean-reverting additive energy forward curves in a Heath–Jarrow–Morton framework

Abstract

In this paper, we make the traditional modeling approach of energy commodity forwards consistent with no-arbitrage. In fact, traditionally energy prices are modeled as mean-reverting processes under the real-world probability measure $$\mathbb {P}$$ , which is in apparent contradiction with the fact that they should be martingales under a risk-neutral measure $$\mathbb {Q}$$ . The key point here is that the two dynamics can coexist, provided a suitable change of measure is defined between $$\mathbb {P}$$ and $$\mathbb {Q}$$ . To this purpose, we design a Heath–Jarrow–Morton framework for an additive, mean-reverting, multicommodity market consisting of forward contracts of any delivery period. Even for relatively simple dynamics, we face the problem of finding a density between $$\mathbb {P}$$ and $$\mathbb {Q}$$ , such that the prices of traded assets like forward contracts are true martingales under $$\mathbb {Q}$$ and mean-reverting under $$\mathbb {P}$$ . Moreover, we are also able to treat the peculiar delivery mechanism of forward contracts in power and gas markets, where the seller of a forward contract commits to deliver, either physically or financially, over a certain period, while in other commodity, or stock, markets, a forward is usually settled on a maturity date. By assuming that forward prices can be represented as affine functions of a universal source of randomness, we can completely characterize the models which prevent arbitrage opportunities by formulating conditions under which the change of measure between $$\mathbb {P}$$ and $$\mathbb {Q}$$ is well defined. In this respect, we prove two results on the martingale property of stochastic exponentials. The first allows to validate measure changes made of two components: an Esscher-type density and a Girsanov transform with stochastic and unbounded kernel. The second uses a different approach and works for the case of continuous density. We show how this framework provides an explicit way to describe a variety of models by introducing, in particular, a generalized Lucia–Schwartz model and a cross-commodity cointegrated market.