arXiv subjects

Tiziano Vargiolu

Publications and source records attributed to Tiziano Vargiolu.

16 recordsLinked to original sources

arXiv2026

Nash equilibrium in a singular stochastic game between two renewable power producers with price impact

We study the singular stochastic game, formulated in Awerkin and Vargiolu (Decis. Econ. Finance 44(2), 2021), between two agents aiming at maximizing their profits by installing photovoltaic panels and selling the produced electricity, net of installation costs, in the case that their cumulative installations have an impact on power prices. We first solve explicitly the static, one-step, version of the game, and find that Nash equilibria divide the state-space into four regions: one where both players are idle, two where only one player installs new panels, and one where both players install. In some particular regimes, we find that the latter may not be uniquely distinguished from the previous two. We then consider the dynamic, continuous-time, problem. Led by the intuition garnered in the static case, we assume a free-boundary structure similar to that arising in the one-step game and provide a rigorous verification theorem for the corresponding system of free-boundary HJB equations, also taking into account the lack of smoothness of the value functions near the free boundaries. Finally, for each regular solution of the HJB system, we show that there exists a unique equilibrium strategy, which is obtained as the solution to the Skorokhod-type problem associated with the free boundary.

math.OC↗

arXiv2025

Parameter estimation of integrated fractional Brownian motion

Fractional Brownian motion (fBm) is a canonical model for long-memory phenomena. In the presence of large amounts of potentially memory-bearing data, the data are often averaged, which can change the structure of the underlying relationships and affect standard estimation procedures. To address this, we introduce the normalized integrated fractional Brownian motion (nifBm), defined as the average of fBm over a fixed interval. We derive its covariance structure, investigate the stationarity and self-similarity, and extend the framework to linear combinations of independent nifBms and models with deterministic drift. For such linear combinations, we establish stationarity of increments, investigate the asymptotic behavior of the autocovariance function, and prove an ergodic theorem essential for statistical inference. We consider two statistical models: one driven by a single nifBm and another by a linear combination of two independent nifBms, including cases with deterministic drift. For both models, we propose estimators that are strongly consistent and asymptotically normal for both the drift and the full parameter set. Numerical simulations illustrate the theoretical findings, providing a foundation for modeling averaged fractional dynamics, with potential applications in finance, energy markets, and environmental studies.

math.PR↗

arXiv2025

Optimal Investment and Fair Sharing Rules of the Incentives for Renewable Energy Communities

The focus on Renewable Energy Communities (REC) is fastly growing after the European Union (EU) has introduced a dedicated regulation in 2018. The idea of creating local groups of citizens, small- and medium-sized companies, and public institutions, which self-produce and self-consume energy from renewable sources is at the same time a way to save money for the participants, increase efficiency of the energy system, and reduce CO$_2$ emissions. Member states inside the EU are fixing more detailed regulations, which describe, how public incentives are measured. A natural objective for the incentive policies is of course to promote the self-consumption of a REC. A sophisticated incentive policy is that based on the so called 'virtual framework'. Under this framework all the energy produced by a REC is sold to the market, and all the energy consumed must be paid to retailers: self-consumption occurs only 'virtually', thanks a money compensation (paid by a central authority) for every MWh produced and consumed by the REC in the same hour. In this context, two problems have to be solved: the optimal investment in new technologies and a fair division of the incentive among the community members. We address these problems by considering a particular type of REC, composed by a representative household and a biogas producer, where the potential demand of the community is given by the household's demand, while both members produce renewable energy. We set the problem as a leader-follower problem: the leader decide how to share the incentive for the self-consumed energy, while the followers decide their own optimal installation strategy. We solve the leader's problem by searching for a Nash bargaining solution for the incentive's fair division, while the follower problem is solved by finding the Nash equilibria of a static competitive game between the members.

econ.GN↗

arXiv2025

Optimal energy storage management for self-consumption groups

We study the optimal management of a photovoltaic system's battery owned by a self-consumption group that aims to minimize energy consumption costs. We assume that the photovoltaic system is composed of a photovoltaic panel and a battery, where the photovoltaic panel produces energy according to a certain stochastic process. The management of the battery is the responsibility of a group administrator, who makes the joint decision to either store part of the photovoltaic energy production and sell the remaining energy at the electricity spot price, or discharge part of the energy stored in the battery and sell it in the electricity market. Inspired by European Union and Italian legislation, which promote incentives for energy transition and renewable energy production, we assume that the group receives a monetary incentive for the virtual self-consumed energy, defined as the minimum between the power bought from the grid to satisfy the group's power demand and the energy sold to the market. In this case, the energy sold by the group is a mix of part of the photovoltaic production that is not stored and part of the energy discharged from the battery. We model the problem as a stochastic optimal control problem, where the optimal strategy is the joint charge-discharge decision that minimizes the group's energy consumption costs. We find the solution numerically by applying a finite difference scheme to solve the Hamilton-Jacobi-Bellman equation associated with the value function of the optimal control problem.

econ.GN↗

arXiv2024

Gaussian Volterra processes as models of electricity markets

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. We also 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. Finally, we formulate a portfolio optimization problem for an agent who invests in an electricity market, and we solve it explicitly in the case of CRRA utility functions. We also find closed formulas for the price of options written on forward contracts, together with the hedging strategy.

math.PR↗

arXiv2022

Optimal installation of renewable electricity sources: the case of Italy

Starting from the model in Koch-Vargiolu (2019), we test the real impact of current renewable installed power in the electricity price in Italy, and assess how much the renewable installation strategy which was put in place in Italy deviated from the optimal one obtained from the model in the period 2012--2018. To do so, we consider the Ornstein-Uhlenbeck (O-U) process, including an exogenous increasing process influencing the mean reverting term, which is interpreted as the current renewable installed power. Using real data of electricity price, photovoltaic and wind energy production from the six main Italian price zones, we estimate the parameters of the model and obtain quantitative results, such as the production of photovoltaic energy impacts the North zone, while wind is significant for Sardinia and the Central North zone does not present electricity price impact. Then we implement the solution of the singular optimal control problem of installing renewable power production devices, in order to maximize the profit of selling the produced energy in the market net of installation costs. We extend the results of \cite{KV} to the case when no impact on power price is presented, and to the case when $N$ players can produce electricity by installing renewable power plants. We are thus able to describe the optimal strategy and compare it with the real installation strategy that was put in place in Italy.

math.OC↗

arXiv2020

Efficient representation of supply and demand curves on day-ahead electricity markets

Our paper aims to model supply and demand curves of electricity day-ahead auction in a parsimonious way. Our main task is to build an appropriate algorithm to present the information about electricity prices and demands with far less parameters than the original one. We represent each curve using mesh-free interpolation techniques based on radial basis function approximation. We describe results of this method for the day-ahead IPEX spot price of Italy.

q-fin.PR↗

arXiv2019

Optimal Installation of Solar Panels with Price Impact: a Solvable Singular Stochastic Control Problem

We consider a price-maker company which generates electricity and sells it in the spot market. The company can increase its level of installed power by irreversible installations of solar panels. In absence of the company's economic activities, the spot electricity price evolves as an Ornstein-Uhlenbeck process, and therefore it has a mean-reverting behavior. The current level of the company's installed power has a permanent impact on the electricity price and affects its mean-reversion level. The company aims at maximizing the total expected profits from selling electricity in the market, net of the total expected proportional costs of installation. This problem is modeled as a two-dimensional degenerate singular stochastic control problem in which the installation strategy is identified as the company's control variable. We follow a guess-and-verify approach to solve the problem. We find that the optimal installation strategy is triggered by a curve which separates the waiting region, where it is not optimal to install additional panels, and the installation region, where it is. Such a curve depends on the current level of the company's installed power, and is the unique strictly increasing function which solves a first-order ordinary differential equation (ODE). Finally, our study is complemented by a numerical analysis of the dependency of the optimal installation strategy on the model's parameters.

math.OC↗

arXiv2019

Capturing the power options smile by an additive two-factor model for overlapping futures prices

In this paper we introduce an additive two-factor model for electricity futures prices based on Normal Inverse Gaussian Lévy processes, that fulfills a no-overlapping-arbitrage (NOA) condition. We compute European option prices by Fourier transform methods, introduce a specific calibration procedure that takes into account no-arbitrage constraints and fit the model to power option settlement prices of the European Energy Exchange (EEX). We show that our model is able to reproduce the different levels and shapes of the implied volatility (IV) profiles displayed by options with a variety of delivery periods.

q-fin.MF↗

arXiv2019

Optimal management of pumped hydroelectric production with state constrained optimal control

We present a novel technique to solve the problem of managing optimally a pumped hydroelectric storage system. This technique relies on representing the system as a stochastic optimal control problem with state constraints, these latter corresponding to the finite volume of the reservoirs. Following the recent level-set approach presented in O. Bokanowski, A. Picarelli, H. Zidani, "State-constrained stochastic optimal control problems via reachability approach", SIAM J. Control and Optim. 54 (5) (2016), we transform the original constrained problem in an auxiliary unconstrained one in augmented state and control spaces, obtained by introducing an exact penalization of the original state constraints. The latter problem is fully treatable by classical dynamic programming arguments.

math.OC↗

arXiv2019

Pricing Reliability Options under different electricity prices' regimes

Reliability Options are capacity remuneration mechanisms aimed at enhancing security of supply in electricity systems. They can be framed as call options on electricity sold by power producers to System Operators. This paper provides a comprehensive mathematical treatment of Reliability Options. Their value is first derived by means of closed-form pricing formulae, which are obtained under several assumptions about the dynamics of electricity prices and strike prices. Then, the value of the Reliability Option is simulated under a real-market calibration, using data of the Italian power market. We finally perform sensitivity analyses to highlight the impact of the level and volatility of both power and strike price, of the mean reversion speeds and of the correlation coefficient on the Reliability Options' value.

q-fin.PR↗

arXiv2018

Nonzero-sum stochastic differential games with impulse controls: a verification theorem with applications

We consider a general nonzero-sum impulse game with two players. The main mathematical contribution of the paper is a verification theorem which provides, under some regularity conditions, a suitable system of quasi-variational inequalities for the value functions and the optimal strategies of the two players. As an application, we study an impulse game with a one-dimensional state variable, following a real-valued scaled Brownian motion, and two players with linear and symmetric running payoffs. We fully characterize a Nash equilibrium and provide explicit expressions for the optimal strategies and the value functions. We also prove some asymptotic results with respect to the intervention costs. Finally, we consider two further non-symmetric examples where a Nash equilibrium is found numerically.

math.PR↗

arXiv2018

Optimal Portfolio in Intraday Electricity Markets Modelled by Lévy-Ornstein-Uhlenbeck Processes

We study an optimal portfolio problem designed for an agent operating in intraday electricity markets. The investor is allowed to trade in a single risky asset modelling the continuously traded power and aims to maximize the expected terminal utility of his wealth. We assume a mean-reverting additive process to drive the power prices. In the case of logarithmic utility, we reduce the fully non-linear Hamilton-Jacobi-Bellman equation to a linear parabolic integro-differential equation, for which we explicitly exhibit a classical solution in two cases of modelling interest. The optimal strategy is given implicitly as the solution of an integral equation, which is possible to solve numerically as well as to describe analytically. An analysis of two different approximations for the optimal policy is provided. Finally, we perform a numerical test by adapting the parameters of a popular electricity spot price model.

q-fin.PM↗

arXiv2018

Additive energy forward curves in a Heath-Jarrow-Morton framework

One of the peculiarities of power and gas markets is the delivery mechanism of forward contracts. The seller of a futures contract commits to deliver, say, power, over a certain period, while the classical forward is a financial agreement settled on a maturity date. Our purpose is to design a Heath-Jarrow-Morton framework for an additive, mean-reverting, multicommodity market consisting of forward contracts of any delivery period. The main assumption is that forward prices can be represented as affine functions of a universal source of randomness. This allows us to completely characterize the models which prevent arbitrage opportunities: this boils down to finding a density between a risk-neutral measure $\mathbb{Q}$, such that the prices of traded assets like forward contracts are true $\mathbb{Q}$-martingales, and the real world probability measure $\mathbb{P}$, under which forward prices are mean-reverting. The Girsanov kernel for such a transformation turns out to be stochastic and unbounded in the diffusion part, while in the jump part the Girsanov kernel must be deterministic and bounded: thus, 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 apply this framework to two models: a generalized Lucia-Schwartz model and a cross-commodity cointegrated market.

q-fin.MF↗

arXiv2017

On the Singular Control of Exchange Rates

Consider the problem of a central bank that wants to manage the exchange rate between its domestic currency and a foreign one. The central bank can purchase and sell the foreign currency, and each intervention on the exchange market leads to a proportional cost whose instantaneous marginal value depends on the current level of the exchange rate. The central bank aims at minimizing the total expected costs of interventions on the exchange market, plus a total expected holding cost. We formulate this problem as an infinite time-horizon stochastic control problem with controls that have paths which are locally of bounded variation. The exchange rate evolves as a general linearly controlled one-dimensional diffusion, and the two nondecreasing processes giving the minimal decomposition of a bounded-variation control model the cumulative amount of foreign currency that has been purchased and sold by the central bank. We provide a complete solution to this problem by finding the explicit expression of the value function and a complete characterization of the optimal control. At each instant of time, the optimally controlled exchange rate is kept within a band whose size is endogenously determined as part of the solution to the problem. We also study the expected exit time from the band, and the sensitivity of the width of the band with respect to the model's parameters in the case when the exchange rate evolves (in absence of any intervention) as an Ornstein-Uhlenbeck process, and the marginal costs of controls are constant. The techniques employed in the paper are those of the theory of singular stochastic control and of one-dimensional diffusions.

math.OC↗

arXiv2016

Utility indifference pricing and hedging for structured contracts in energy markets

In this paper we study the pricing and hedging of structured products in energy markets, such as swing and virtual gas storage, using the exponential utility indifference pricing approach in a general incomplete multivariate market model driven by finitely many stochastic factors. The buyer of such contracts is allowed to trade in the forward market in order to hedge the risk of his position. We fully characterize the buyer's utility indifference price of a given product in terms of continuous viscosity solutions of suitable nonlinear PDEs. This gives a way to identify reasonable candidates for the optimal exercise strategy for the structured product as well as for the corresponding hedging strategy. Moreover, in a model with two correlated assets, one traded and one nontraded, we obtain a representation of the price as the value function of an auxiliary simpler optimization problem under a risk neutral probability, that can be viewed as a perturbation of the minimal entropy martingale measure. Finally, numerical results are provided.

q-fin.MF↗