IDEAS home Printed from https://ideas.repec.org/a/inm/ordeca/v8y2011i3p220-232.html
   My bibliography  Save this article

A Discrete Time Approach for Modeling Two-Factor Mean-Reverting Stochastic Processes

Author

Listed:
  • Warren J. Hahn

    (Graziadio School of Business and Management, Pepperdine University, Malibu, California 90263)

  • James S. Dyer

    (McCombs School of Business, University of Texas at Austin, Austin, Texas 78712)

Abstract

Two-factor stochastic processes have been developed to more accurately describe the intertemporal dynamics of variables such as commodity prices. In this paper we develop an approach for modeling these types of stochastic processes in discrete time as two-dimensional binomial sequences. This approach facilitates the numerical solution of dynamic optimization problems such as investment decision making under uncertainty and option valuation related to commodities. We implement this approach in a two-dimensional lattice format, apply it to two hypothetical valuation problems discussed by Schwartz and Smith, and compare the results to those from simulation- and dynamic-programming-based methods.

Suggested Citation

  • Warren J. Hahn & James S. Dyer, 2011. "A Discrete Time Approach for Modeling Two-Factor Mean-Reverting Stochastic Processes," Decision Analysis, INFORMS, vol. 8(3), pages 220-232, September.
    • Handle: RePEc:inm:ordeca:v:8:y:2011:i:3:p:220-232
    DOI: 10.1287/deca.1110.0209
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/deca.1110.0209
    Download Restriction: no
    File URL: https://libkey.io/10.1287/deca.1110.0209?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    2. Eduardo Schwartz & James E. Smith, 2000. "Short-Term Variations and Long-Term Dynamics in Commodity Prices," Management Science, INFORMS, vol. 46(7), pages 893-911, July.
    3. Bessembinder, Hendrik, et al, 1995. "Mean Reversion in Equilibrium Asset Prices: Evidence from the Futures Term Structure," Journal of Finance, American Finance Association, vol. 50(1), pages 361-375, March.
    4. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    5. James E. Smith & Robert F. Nau, 1995. "Valuing Risky Projects: Option Pricing Theory and Decision Analysis," Management Science, INFORMS, vol. 41(5), pages 795-816, May.
    6. David G. Laughton & Henry D. Jacoby, 1993. "Reversion, Timing Options, and Long-Term Decision-Making," Financial Management, Financial Management Association, vol. 22(3), Fall.
    7. Luiz E. Brandão & James S. Dyer & Warren J. Hahn, 2005. "Response to Comments on Brandão et al. (2005)," Decision Analysis, INFORMS, vol. 2(2), pages 103-109, June.
    8. Ross A. Maller & David H. Solomon & Alex Szimayer, 2006. "A Multinomial Approximation For American Option Prices In Lévy Process Models," Mathematical Finance, Wiley Blackwell, vol. 16(4), pages 613-633, October.
    9. Schwartz, Eduardo, 1998. "Valuing long-term commodity assets," Journal of Energy Finance & Development, Elsevier, vol. 3(2), pages 85-99.
    10. San-Lin Chung & Pai-Ta Shih, 2007. "Generalized Cox-Ross-Rubinstein Binomial Models," Management Science, INFORMS, vol. 53(3), pages 508-520, March.
    11. Samuel Hikspoors & Sebastian Jaimungal, 2008. "Asymptotic Pricing of Commodity Derivatives using Stochastic Volatility Spot Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(5-6), pages 449-477.
    12. Schwartz, Eduardo S, 1997. "The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging," Journal of Finance, American Finance Association, vol. 52(3), pages 923-973, July.
    13. He, Hua, 1990. "Convergence from Discrete- to Continuous-Time Contingent Claims Prices," The Review of Financial Studies, Society for Financial Studies, vol. 3(4), pages 523-546.
    14. Gibson, Rajna & Schwartz, Eduardo S, 1990. "Stochastic Convenience Yield and the Pricing of Oil Contingent Claims," Journal of Finance, American Finance Association, vol. 45(3), pages 959-976, July.
    15. Reiter, Michael, 1999. "Solving higher-dimensional continuous-time stochastic control problems by value function regression," Journal of Economic Dynamics and Control, Elsevier, vol. 23(9-10), pages 1329-1353, September.
    16. Boyle, Phelim P., 1988. "A Lattice Framework for Option Pricing with Two State Variables," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 23(1), pages 1-12, March.
    17. Nelson, Daniel B & Ramaswamy, Krishna, 1990. "Simple Binomial Processes as Diffusion Approximations in Financial Models," The Review of Financial Studies, Society for Financial Studies, vol. 3(3), pages 393-430.
    18. Patrick Jaillet & Ehud I. Ronn & Stathis Tompaidis, 2004. "Valuation of Commodity-Based Swing Options," Management Science, INFORMS, vol. 50(7), pages 909-921, July.
    19. Hahn, Warren J. & Dyer, James S., 2008. "Discrete time modeling of mean-reverting stochastic processes for real option valuation," European Journal of Operational Research, Elsevier, vol. 184(2), pages 534-548, January.
    20. Luiz E. Brandão & James S. Dyer & Warren J. Hahn, 2005. "Using Binomial Decision Trees to Solve Real-Option Valuation Problems," Decision Analysis, INFORMS, vol. 2(2), pages 69-88, June.
    21. Eduardo S. Schwartz, 1998. "Valuing Long-Term Commodity Assets," Financial Management, Financial Management Association, vol. 27(1), Spring.
    22. Bastian-Pinto, Carlos & Brando, Luiz & Hahn, Warren J., 2009. "Flexibility as a source of value in the production of alternative fuels: The ethanol case," Energy Economics, Elsevier, vol. 31(3), pages 411-422, May.
    23. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    24. Evis Këllezi & Nick Webber, 2004. "Valuing Bermudan options when asset returns are Levy processes," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 87-100.
    25. Hull, John & White, Alan, 1993. "One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 28(2), pages 235-254, June.
    26. Arun Chockalingam & Kumar Muthuraman, 2011. "American Options Under Stochastic Volatility," Operations Research, INFORMS, vol. 59(4), pages 793-809, August.
    27. Chung-Li Tseng & Kyle Y. Lin, 2007. "A Framework Using Two-Factor Price Lattices for Generation Asset Valuation," Operations Research, INFORMS, vol. 55(2), pages 234-251, April.
    28. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    29. James E. Smith, 2005. "Alternative Approaches for Solving Real-Options Problems," Decision Analysis, INFORMS, vol. 2(2), pages 89-102, June.
    30. Ho, Teng-Suan & Stapleton, Richard C & Subrahmanyam, Marti G, 1995. "Multivariate Binomial Approximations for Asset Prices with Nonstationary Variance and Covariance Characteristics," The Review of Financial Studies, Society for Financial Studies, vol. 8(4), pages 1125-1152.
    31. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    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. Kuangyuan Zhang & Richard Olawoyin & Antonio Nieto & Andrew N. Kleit, 2018. "Risk of commodity price, production cost and time to build in resource economics," Environment, Development and Sustainability: A Multidisciplinary Approach to the Theory and Practice of Sustainable Development, Springer, vol. 20(6), pages 2521-2544, December.
    2. L. Robin Keller, 2011. "From the Editor ---Multiattribute and Intertemporal Preferences, Probability, and Stochastic Processes: Models and Assessment," Decision Analysis, INFORMS, vol. 8(3), pages 165-169, September.
    3. L. Robin Keller & Ali Abbas & J. Eric Bickel & Vicki M. Bier & David V. Budescu & John C. Butler & Philippe Delquié & Kenneth C. Lichtendahl & Jason R. W. Merrick & Ahti Salo & George Wu, 2011. "From the Editors ---Probability Scoring Rules, Ambiguity, Multiattribute Terrorist Utility, and Sensitivity Analysis," Decision Analysis, INFORMS, vol. 8(4), pages 251-255, December.
    4. Karl Inderfurth & Peter Kelle & Rainer Kleber, 2014. "The Effect of Material Price and Product Demand Correlations on Combined Sourcing and Inventory Management," FEMM Working Papers 140013, Otto-von-Guericke University Magdeburg, Faculty of Economics and Management.
    5. Lorenzo Reus, 2020. "Efficient selection of copper sales contracts for small‐ and medium‐sized mining," Managerial and Decision Economics, John Wiley & Sons, Ltd., vol. 41(4), pages 624-630, June.
    6. Nadarajah, Selvaprabu & Margot, François & Secomandi, Nicola, 2017. "Comparison of least squares Monte Carlo methods with applications to energy real options," European Journal of Operational Research, Elsevier, vol. 256(1), pages 196-204.
    7. Alon Dourban & Liron Yedidsion, 2017. "Optimal Purchasing Policy For Mean-Reverting Items in a Finite Horizon," Papers 1711.03188, arXiv.org.
    8. Elias, R.S. & Wahab, M.I.M. & Fang, L., 2016. "The spark spread and clean spark spread option based valuation of a power plant with multiple turbines," Energy Economics, Elsevier, vol. 59(C), pages 314-327.
    9. Hahn, Warren J. & DiLellio, James A. & Dyer, James S., 2018. "Risk premia in commodity price forecasts and their impact on valuation," Energy Economics, Elsevier, vol. 72(C), pages 393-403.

    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. Hahn, Warren J. & Dyer, James S., 2008. "Discrete time modeling of mean-reverting stochastic processes for real option valuation," European Journal of Operational Research, Elsevier, vol. 184(2), pages 534-548, January.
    2. Bastian-Pinto, Carlos & Brando, Luiz & Hahn, Warren J., 2009. "Flexibility as a source of value in the production of alternative fuels: The ethanol case," Energy Economics, Elsevier, vol. 31(3), pages 411-422, May.
    3. Secomandi, Nicola & Seppi, Duane J., 2014. "Real Options and Merchant Operations of Energy and Other Commodities," Foundations and Trends(R) in Technology, Information and Operations Management, now publishers, vol. 6(3-4), pages 161-331, July.
    4. Carlos Andrés Zapata Quimbayo, 2020. "OPCIONES REALES Una guía teórico-práctica para la valoración de inversiones bajo incertidumbre mediante modelos en tiempo discreto y simulación de Monte Carlo," Books, Universidad Externado de Colombia, Facultad de Finanzas, Gobierno y Relaciones Internacionales, number 138, April.
    5. Carlos de Lamare Bastian-Pinto & Alexandre Paula Silva Ramos & Luiz de Magalhães Ozorio & Luiz Eduardo Teixeira Brandão, 2015. "Uncertainty and Flexibility in the Brazilian Beef Livestock Sector: the Value of the Confinement Option," Brazilian Business Review, Fucape Business School, vol. 12(6), pages 100-120, November.
    6. Savolainen, Jyrki, 2016. "Real options in metal mining project valuation: Review of literature," Resources Policy, Elsevier, vol. 50(C), pages 49-65.
    7. Max F. Schöne & Stefan Spinler, 2017. "A four-factor stochastic volatility model of commodity prices," Review of Derivatives Research, Springer, vol. 20(2), pages 135-165, July.
    8. Tianyang Wang & James S. Dyer, 2010. "Valuing Multifactor Real Options Using an Implied Binomial Tree," Decision Analysis, INFORMS, vol. 7(2), pages 185-195, June.
    9. Sebastian Maier, 2021. "Re-evaluating natural resource investments under uncertainty: An alternative to limited traditional approaches," Annals of Operations Research, Springer, vol. 299(1), pages 907-937, April.
    10. Hendrik Kohrs & Hermann Mühlichen & Benjamin R. Auer & Frank Schuhmacher, 2019. "Pricing and risk of swing contracts in natural gas markets," Review of Derivatives Research, Springer, vol. 22(1), pages 77-167, April.
    11. Slade, Margaret E., 2001. "Valuing Managerial Flexibility: An Application of Real-Option Theory to Mining Investments," Journal of Environmental Economics and Management, Elsevier, vol. 41(2), pages 193-233, March.
    12. Seiji Harikae & James S. Dyer & Tianyang Wang, 2021. "Valuing Real Options in the Volatile Real World," Production and Operations Management, Production and Operations Management Society, vol. 30(1), pages 171-189, January.
    13. Insley, M.C. & Wirjanto, T.S., 2010. "Contrasting two approaches in real options valuation: Contingent claims versus dynamic programming," Journal of Forest Economics, Elsevier, vol. 16(2), pages 157-176, April.
    14. Li, Chenxu & Ye, Yongxin, 2019. "Pricing and Exercising American Options: an Asymptotic Expansion Approach," Journal of Economic Dynamics and Control, Elsevier, vol. 107(C), pages 1-1.
    15. Andrianos E. Tsekrekos & Mark B. Shackleton & Rafał Wojakowski, 2012. "Evaluating Natural Resource Investments under Different Model Dynamics: Managerial Insights," European Financial Management, European Financial Management Association, vol. 18(4), pages 543-575, September.
    16. Chung-Li Tseng & Daniel Wei-Chung Miao & San-Lin Chung & Pai-Ta Shih, 2021. "How Much Do Negative Probabilities Matter in Option Pricing?: A Case of a Lattice-Based Approach for Stochastic Volatility Models," JRFM, MDPI, vol. 14(6), pages 1-32, May.
    17. Lander, Diane M. & Pinches, George E., 1998. "Challenges to the Practical Implementation of Modeling and Valuing Real Options," The Quarterly Review of Economics and Finance, Elsevier, vol. 38(3, Part 2), pages 537-567.
    18. Andrea Gamba & Lenos Trigeorgis, 2007. "An Improved Binomial Lattice Method for Multi-Dimensional Options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(5), pages 453-475.
    19. Weihan Li & Jin E. Zhang & Xinfeng Ruan & Pakorn Aschakulporn, 2024. "An empirical study on the early exercise premium of American options: Evidence from OEX and XEO options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 44(7), pages 1117-1153, July.
    20. Björn Lutz, 2010. "Pricing of Derivatives on Mean-Reverting Assets," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-642-02909-7, July.

    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:inm:ordeca:v:8:y:2011:i:3:p:220-232. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.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.