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13.02.2019 | Ausgabe 4/2019

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

Zeitschrift:
Mathematics and Financial Economics > Ausgabe 4/2019
Autoren:
Fred Espen Benth, Marco Piccirilli, Tiziano Vargiolu
Wichtige Hinweise
This work was partially done while M. Piccirilli visited the University of Oslo. F. E. Benth acknowledges financial support from the research project FINEWSTOCH funded by the Norwegian Research Council. T. Vargiolu acknowledges financial support from the research project CPDA158845 “Multidimensional polynomial processes and applications to new challenges in mathematical finance and in energy markets”, funded by the University of Padova. We are grateful for the valuable and constructive critics of two anonymuous referees.

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## 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.

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