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.

中文翻译：

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Heath-Jarrow-Morton框架中的均值回复加性能量正向曲线

在本文中，我们使传统的能源商品建模方法与无套利一致。实际上，传统上，能源价格在现实世界的概率测度\（\ mathbb {P} \）下被建模为均值回复过程，这与在风险中性测度下它们应该是mar是明显矛盾的。 （\ mathbb {Q} \）。这里的关键点是，只要在\（\ mathbb {P} \）和\（\ mathbb {Q} \）之间定义了适当的度量更改，这两种动力学就可以共存。。为此，我们设计了一个Heath-Jarrow-Morton框架，用于由任何交货期的远期合同组成的加性，均值回归，多商品市场。即使对于相对简单的动力学，我们也面临在\（\ mathbb {P} \）和\（\ mathbb {Q} \）之间找到密度的问题，使得远期合约等交易资产的价格是\下的真实under （\ mathbb {Q} \）和\（\ mathbb {P} \）下的均值还原。此外，我们还能够处理电力和天然气市场中远期合同的特殊交付机制，即远期合同的卖方承诺在一定时期内以其他商品或股票形式以实物或财务方式进行交付，市场，远期通常在到期日结算。通过假设远期价格可以表示为通用随机性的仿射函数，我们可以通过公式化\（\ mathbb {P} \）和\（ \ mathbb {Q} \）定义明确。在这方面，我们证明了随机指数的mar性质。第一种方法可以验证由两个部分组成的度量更改：Esscher型密度和具有随机且无界核的Girsanov变换。第二种使用不同的方法，适用于连续密度的情况。我们将介绍特别是通过引入广义的Lucia-Schwartz模型和跨商品的共同市场，来说明该框架如何提供一种描述各种模型的明确方法。