The Black-Scholes equation is frequently used and critical for assessment of the value of investments and projects, such as dealing with petroleum reservoir exploration, development, and production, involving significantly high capital-intensive investment and prospect values. This paper presents a functional analysis of the Black-Scholes equation and develops analytical solutions under various conditions. It is demonstrated that the analytical solutions besides the special Black and Scholes’ solutions can be derived by means of the method of Laplace transformation. The generic analytic solutions and the analytical solutions for two specific boundary conditions are presented. The functional behaviors of these analytic solutions are demonstrated by parametric studies and applications. The solutions presented here are expected to be instrumental in understanding and application of these solutions in future studies. The analyses and analytical solutions presented can be readily extended for valuation of forward contracts and commodity options.

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