Gaussian Volterra processes as models of electricity markets

Mishura Yuliya, Taras Shevchenko National University of Kyiv
Ottaviano Stefania, University of Padova
Vargiolu Tiziano, University of Padova

Soon after the liberalization of electricity markets in Europe, empirical evidence emerged that electricity prices show stylized properties (most notably, antipersistence and self-similarity) well explained by fractional Brownian motion (fBm). However, the wide application of fBm in energy markets, and more in general in financial markets, was hindered by the fact that fBm is not a semimartingale, thus (roughly speaking) it produces the possibility of arbitrage. Partial remedies proposed often lead to models that are difficult to treat analytically. The aim of this work is to show that it is indeed possible to use fBm, and even more general Gaussian Volterra processes, in electricity markets without introducing arbitrage. Precisely, we introduce a non-Markovian model for electricity markets where the spot price of electricity is driven by several Gaussian Volterra processes, which can be e.g., fractional Brownian motions (fBms), Riemann-Liouville processes or Gaussian-Volterra driven Ornstein-Uhlenbeck processes. Since in energy markets the spot price is not a tradeable asset, due to the limited storage possibilities, forward contracts are considered as traded products. We ensure necessary and sufficient conditions for the absence of arbitrage that, in this kind of market, reflects the fact that the prices of the forward contracts are (Gaussian) martingales under a risk-neutral measure. Moreover, we characterize the market completeness in terms of the number of forward contracts simultaneously considered and of the kernels of the Gaussian-Volterra processes. Then, we formulate a portfolio optimization problem for an agent who invests in an electricity market. We also find closed formulas for the price of options written on forward contracts, together with the hedging strategy. Finally, we provide a novel representation of Ornstein-Uhlenbeck (OU) processes driven by Gaussian Volterra processes. Also exploiting this result, we show analytically that, for some kinds of Gaussian-Volterra processes driving the spot prices, under conditions ensuring the absence of arbitrage, the market is complete.

Area: CS48 - Probabilistic models for energy transition (Tiziano Vargiolu and Athena Picarelli)

Keywords: fractional Brownian motion, Gaussian Volterra process, electricity markets, forward prices

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