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Price options on investment project expansion under commodity price and volatility uncertainties using a novel finite difference method

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  • Li, Nan
  • Wang, Song
  • Zhang, Kai

Abstract

In this paper we develop a PDE-based mathematical model for valuing real options on the expansion of an investment project whose underlying commodity price and its volatility follow their respective geometric Brownian motions. This mathematical model is of the form of a 2-dimensional Black-Scholes equation whose payoff condition is determined also by a PDE system. A novel 9-point finite difference scheme is proposed for the discretization of the spatial derivatives and the fully implicit time-stepping scheme is used for the time discretization of the PDE systems. We show that the coefficient matrix of the fully discretized system is an M-matrix and prove that the solution generated by this finite difference scheme converges to the exact one when the mesh sizes approach zero. To demonstrate the usefulness and effectiveness of the mathematical model and numerical method, we present a case study on a real option pricing problem in the iron-ore mining industry. Numerical experiments show that our model and methods are able to produce numerical results which are financially meaningful.

Suggested Citation

  • Li, Nan & Wang, Song & Zhang, Kai, 2022. "Price options on investment project expansion under commodity price and volatility uncertainties using a novel finite difference method," Applied Mathematics and Computation, Elsevier, vol. 421(C).
    • Handle: RePEc:eee:apmaco:v:421:y:2022:i:c:s0096300322000236
    DOI: 10.1016/j.amc.2022.126937
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    References listed on IDEAS

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    1. Stewart C. Myers, 1984. "Finance Theory and Financial Strategy," Interfaces, INFORMS, vol. 14(1), pages 126-137, February.
    2. Moyen, Nathalie & Slade, Margaret & Uppal, Raman, 1996. "Valuing risk and flexibility : A comparison of methods," Resources Policy, Elsevier, vol. 22(1-2), pages 63-74.
    3. Mark. B. Garman., 1976. "A General Theory of Asset Valuation under Diffusion State Processes," Research Program in Finance Working Papers 50, University of California at Berkeley.
    4. Brennan, Michael J & Schwartz, Eduardo S, 1985. "Evaluating Natural Resource Investments," The Journal of Business, University of Chicago Press, vol. 58(2), pages 135-157, April.
    5. Costa Lima, Gabriel A. & Suslick, Saul B., 2006. "Estimating the volatility of mining projects considering price and operating cost uncertainties," Resources Policy, Elsevier, vol. 31(2), pages 86-94, June.
    6. 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.
    7. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    8. Haque, Md. Aminul & Topal, Erkan & Lilford, Eric, 2014. "A numerical study for a mining project using real options valuation under commodity price uncertainty," Resources Policy, Elsevier, vol. 39(C), pages 115-123.
    9. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    10. Brennan, Michael J. & Schwartz, Eduardo S., 1978. "Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 13(3), pages 461-474, September.
    Full references (including those not matched with items on IDEAS)

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