An Optimal Extraction Problem with Price Impact

Abstract

A price-maker company extracts an exhaustible commodity from a reservoir, and sells it instantaneously in the spot market. In absence of any actions of the company, the commodity's spot price evolves either as a drifted Brownian motion or as an Ornstein-Uhlenbeck process. While extracting, the company affects the market price of the commodity, and its actions have an impact on the dynamics of the commodity's spot price. The company aims at maximizing the total expected profits from selling the commodity, net of the total expected proportional costs of extraction. We model this problem as a two-dimensional degenerate singular stochastic control problem with finite fuel. To determine its solution, we construct an explicit solution to the associated Hamilton-Jacobi-Bellman equation, and then verify its actual optimality through a verification theorem. On the one hand, when the (uncontrolled) price is a drifted Brownian motion, it is optimal to extract whenever the current price level is larger or equal than an endogenously determined constant threshold. On the other hand, when the (uncontrolled) price evolves as an Ornstein-Uhlenbeck process, we show that the optimal extraction rule is triggered by a curve depending on the current level of the reservoir. Such a curve is a strictly decreasing C ∞ -function for which we are able to provide an explicit expression. Finally, our study is complemented by a theoretical and numerical analysis of the dependency of the optimal extraction strategy and value function on the model's parameters.

Universal bounds and monotonicity properties of ratios of Hermite and parabolic cylinder functions

“…Since then, inequalities of this form have attracted a lot of attention, and have been proved to be valid for other polynomials such as Hermite (obtained from Hermite functions by taking ν ∈ N), Jacobi, Laguerre or ultraspherical polynomials (see [12,22], among others), and for special functions as (modified) Bessel, Gamma, parabolic cylinder or hypergeometric functions (see [2,4,5,6,7,23], among many others). Applications of Turán type inequalities can be found in many fields, ranging from biophysics (see [3] and the references therein) to information theory (see [15]) and stochastic control (see [8,11]).…”

Mean Field Games with Singular Controls

“…In the second example, we consider a MFG of optimal exploitation of exhaustible resource. A model with infinite horizon and without game nature was introduced in [18]. We introduce a model among infinite players with mean-field interaction and finite horizon.…”

Optimal installation of renewable electricity sources: the case of Italy

“…This is also analogous to other papers modeling price impact: for example, in problems of optimal execution, Becherer et al (2017) and Becherer et al (2018) take into account a multiplicative and transient price impact, whereas (Guo and Zervos 2015) consider an exponential parametrization in a geometric Brownian motion setting allowing for a permanent price impact. Also, a price impact model has been studied by Al Motairi and Zervos (2017), motivated by an irreversible capital accumulation problem with permanent price impact, and by Ferrari and Koch (2019), in which the authors consider an extraction problem with Ornstein-Uhlenbeck dynamics and transient price impact. In all of the aforementioned papers on price impact models dealing with singular stochastic controls (Al Motairi and Zervos 2017;Becherer et al 2017Becherer et al , 2018Ferrari and Koch 2019;Guo and Zervos 2015), the agents' actions can lead to an immediate jump in the underlying price process, whereas in our setting, it cannot.…”

“…Also, a price impact model has been studied by Al Motairi and Zervos (2017), motivated by an irreversible capital accumulation problem with permanent price impact, and by Ferrari and Koch (2019), in which the authors consider an extraction problem with Ornstein-Uhlenbeck dynamics and transient price impact. In all of the aforementioned papers on price impact models dealing with singular stochastic controls (Al Motairi and Zervos 2017;Becherer et al 2017Becherer et al , 2018Ferrari and Koch 2019;Guo and Zervos 2015), the agents' actions can lead to an immediate jump in the underlying price process, whereas in our setting, it cannot. Our model is instead analogous to , , which show how to incorporate a market impact due to cross-border trading in electricity markets, and to Rowińska et al (2018), which models the price impact of wind electricity production on power prices.…”