IDEAS home Printed from https://ideas.repec.org/p/bie/wpaper/603.html
   My bibliography  Save this paper

An optimal extraction problem with price impact

Author

Listed:
  • Ferrari, Giorgio

    (Center for Mathematical Economics, Bielefeld University)

  • Koch, Torben

    (Center for Mathematical Economics, Bielefeld University)

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 C1-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

  • Ferrari, Giorgio & Koch, Torben, 2018. "An optimal extraction problem with price impact," Center for Mathematical Economics Working Papers 603, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:603
    as

    Download full text from publisher

    File URL: https://pub.uni-bielefeld.de/download/2932622/2932623
    File Function: First Version, 2018
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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. 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).
    3. Dirk Becherer & Todor Bilarev & Peter Frentrup, 2016. "Optimal Liquidation under Stochastic Liquidity," Papers 1603.06498, arXiv.org, revised Nov 2017.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Giorgio Ferrari & Torben Koch, 2018. "An Optimal Extraction Problem with Price Impact," Papers 1812.01270, arXiv.org.
    2. Insley, Margaret, 2017. "Resource extraction with a carbon tax and regime switching prices: Exercising your options," Energy Economics, Elsevier, vol. 67(C), pages 1-16.
    3. Kheiravar, Khaled H, 2019. "Economic and Econometric Analyses of the World Petroleum Industry, Energy Subsidies, and Air Pollution," Institute of Transportation Studies, Working Paper Series qt3gj151w9, Institute of Transportation Studies, UC Davis.
    4. Ferrari, Giorgio & Yang, Shuzhen, 2016. "On an optimal extraction problem with regime switching," Center for Mathematical Economics Working Papers 562, Center for Mathematical Economics, Bielefeld University.
    5. Margaret Insley, 2013. "On the timing of non-renewable resource extraction with regime switching prices: an optimal stochastic control approach," Working Papers 1302, University of Waterloo, Department of Economics, revised Aug 2013.
    6. Galay, Gregory, 2018. "The impact of spatial price differences on oil sands investments," Energy Economics, Elsevier, vol. 69(C), pages 170-184.
    7. Davis, Rebecca J. & Sims, Charles, 2016. "To Frack or Not to Frack: Option Value Analysis on the U.S. Natural Gas Market," 2016 Annual Meeting, July 31-August 2, Boston, Massachusetts 235642, Agricultural and Applied Economics Association.
    8. Joachim Gahungu and Yves Smeers, 2012. "A Real Options Model for Electricity Capacity Expansion," RSCAS Working Papers 2012/08, European University Institute.
    9. Ferrari, Giorgio, 2018. "On a Class of Singular Stochastic Control Problems for Reflected Diffusions," Center for Mathematical Economics Working Papers 592, Center for Mathematical Economics, Bielefeld University.
    10. Pui Chan Lon & Mihail Zervos, 2011. "A Model for Optimally Advertising and Launching a Product," Mathematics of Operations Research, INFORMS, vol. 36(2), pages 363-376, May.
    11. Paulwin Graewe & Ulrich Horst, 2016. "Optimal Trade Execution with Instantaneous Price Impact and Stochastic Resilience," Papers 1611.03435, arXiv.org, revised Jul 2017.
    12. GAHUNGU, Joachim & SMEERS, Yves, 2011. "A real options model for electricity capacity expansion," LIDAM Discussion Papers CORE 2011044, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    13. Jacob LaRiviere & David Kling & James N Sanchirico & Charles Sims & Michael Springborn, 2018. "The Treatment of Uncertainty and Learning in the Economics of Natural Resource and Environmental Management," Review of Environmental Economics and Policy, Association of Environmental and Resource Economists, vol. 12(1), pages 92-112.
    14. 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.
    15. Nuno M. Brites, 2022. "Optimal Harvesting of Stochastically Fluctuating Populations Driven by a Generalized Logistic SDE Growth Model," Mathematics, MDPI, vol. 10(17), pages 1-15, August.
    16. Kobari, L. & Jaimungal, S. & Lawryshyn, Y., 2014. "A real options model to evaluate the effect of environmental policies on the oil sands rate of expansion," Energy Economics, Elsevier, vol. 45(C), pages 155-165.
    17. Salvatore Federico & Giorgio Ferrari & Patrick Schuhmann, 2019. "A Model for the Optimal Management of Inflation," Department of Economics University of Siena 812, Department of Economics, University of Siena.
    18. Federico, Salvatore & Ferrari, Giorgio & Schuhmann, Patrick, 2019. "A Model for the Optimal Management of Inflation," Center for Mathematical Economics Working Papers 624, Center for Mathematical Economics, Bielefeld University.
    19. 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.
    20. Zhuo Jin & George Yin & Chao Zhu, 2011. "Numerical Solutions of Optimal Risk Control and Dividend Optimization Policies under A Generalized Singular Control Formulation," Papers 1111.2584, arXiv.org.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bie:wpaper:603. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Bettina Weingarten (email available below). General contact details of provider: https://edirc.repec.org/data/imbiede.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.