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An Optimal Extraction Problem with Price Impact

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  • Giorgio Ferrari
  • Torben Koch

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 $C^{\infty}$-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.

Suggested Citation

  • Giorgio Ferrari & Torben Koch, 2018. "An Optimal Extraction Problem with Price Impact," Papers 1812.01270, arXiv.org.
  • Handle: RePEc:arx:papers:1812.01270
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    References listed on IDEAS

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    1. Alvarez, Luis H. R., 2000. "Singular stochastic control in the presence of a state-dependent yield structure," Stochastic Processes and their Applications, Elsevier, vol. 86(2), pages 323-343, April.
    2. Dirk Becherer & Todor Bilarev & Peter Frentrup, 2016. "Optimal Liquidation under Stochastic Liquidity," Papers 1603.06498, arXiv.org, revised Nov 2017.
    3. Abdullah Almansour and Margaret Insley, 2016. "The Impact of Stochastic Extraction Cost on the Value of an Exhaustible Resource: An Application to the Alberta Oil Sands," The Energy Journal, International Association for Energy Economics, vol. 0(Number 2).
    4. Dirk Becherer & Todor Bilarev & Peter Frentrup, 2017. "Stability for gains from large investors' strategies in M1/J1 topologies," Papers 1701.02167, arXiv.org, revised Mar 2018.
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    Cited by:

    1. Yong-Chao Zhang & Na Zhang & Qinglong Zhou, 2023. "The Closed-Form Solution of an Extraction Model and Optimal Stopping Problems with Regime Switching," Mathematics, MDPI, vol. 11(20), pages 1-16, October.
    2. Koch, Torben, 2019. "Universal Bounds and Monotonicity Properties of Ratios of Hermite and Parabolic Cylinder Functions," Center for Mathematical Economics Working Papers 615, Center for Mathematical Economics, Bielefeld University.
    3. Ferrari, Giorgio & Schuhmann, Patrick & Zhu, Shihao, 2022. "Optimal dividends under Markov-modulated bankruptcy level," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 146-172.
    4. Ferrari, Giorgio & Rodosthenous, Neofytos, 2018. "Optimal Management of Debt-To-GDP Ratio with Regime-Switching Interest Rate," Center for Mathematical Economics Working Papers 589, Center for Mathematical Economics, Bielefeld University.

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