A forward model in electricity price with fractional dynamics

In the last decade, a new generation of stochastic models has emerged in finance, built upon the fractional Brownian motion originally introduced by Mandelbrot in 1948. This revolution started with the groundbreaking article by Gatheral, Jaisson and Rosenbaum "Volatility is rough", where the authors improve the forecasts of an asset realized volatility by employing a rough fractional (stochastic volatility) model. In our work, we aim to extend the use of fractional dynamics to energy markets, with the goal of capturing their distinctive structural and stochastic features. In these kinds of market, the spot price is not tradeable, because in general there are very limited storage possibilities, which implies a strong seasonality in the underlying price. In addition, basic tradeable assets in electricity markets can typically be described as futures and forward contracts with electricity delivery over a period of time. We develop a consistent model to describe a forward market avoiding arbitrage, to ensure that we work within the Heath-Jarrow-Morton paradigm. The model we are going to propose for the evolution of forward prices in the energy market is additive and mean-reverting, and it is obtained by modifying the Lucia-Schwartz model with the introduction of fractional dynamics. In line with the existing literature, forward prices are regulated by two factors, X_1 and X_2, hidden in the dynamics. As in the Lucia-Schwartz model, the first one is stationary and mean-reverting to represent the short-term movements of the price curve. It is also responsible for the Samuelson effect; in fact, as time to maturity decreases, volatility increases. The novelty in our model lies in the second factor that includes a fractional martingale as introduced by Norros, Valkeila and Virtamo. The idea behind this choice is that forward's fractional dynamics is due to the non-Markovian behavior of energy spot prices. Also we mean to keep the price's seasonality features hoping to achieve it by a simpler description since it often requires a high number of parameters to be properly described (as in Vargiolu et al. 2018). The choice of a fractional martingale in the spot price allows keeping the martingale property of the instantaneous forwards also in this extended model. By introducing a fractional process, capturing the non-Markovian nature of the price, we aim at reducing the number of parameters when calibrating the model on actual market data. Clearly we will have to calibrate also the Hurst parameter H on the time series, exploiting the quadratic variation/covariation of the martingale components. Summarizing, the goal is to obtain a model that does not allow arbitrage and which recovers some known results in energy markets by reducing the number of parameters to be estimated. Calibration will be conducted within a one-year temporal interval, utilizing data from the time series of Phelix Base forward prices in the Italian market. Within this temporal window, multiple forwards covering the same time period are observed, including monthly, quarterly, and calendar year contracts. In this context, our model aims to mitigate arbitrage possibilities, ensuring its robustness in capturing the dynamics of the energy market. We limit our analysis to the one-dimensional case. A natural extension of this work would involve generalizing the model to multiple dimensions and exploring its application in describing co-movements among commodities.