THE SPECTRAL DECOMPOSITION OF THE OPTION VALUE

Vadim Linetsky ()
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Vadim Linetsky: Department of Industrial Engineering and Management Sciences, McCormick School of Engineering and Applied Sciences, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA

International Journal of Theoretical and Applied Finance (IJTAF), 2004, vol. 07, issue 03, 337-384

Abstract: This paper develops a spectral expansion approach to the valuation of contingent claims when the underlying state variable follows a one-dimensional diffusion with the infinitesimal variancea2(x), driftb(x)and instantaneous discount (killing) rater(x). The Spectral Theorem for self-adjoint operators in Hilbert space yields the spectral decomposition of the contingent claim value function. Based on the Sturm–Liouville (SL) theory, we classify Feller's natural boundaries into two further subcategories: non-oscillatory and oscillatory/non-oscillatory with cutoffΛ≥0(this classification is based on the oscillation of solutions of the associated SL equation) and establish additional assumptions (satisfied in nearly all financial applications) that allow us to completely characterize the qualitative nature of the spectrum from the behavior ofa,bandrnear the boundaries, classify all diffusions satisfying these assumptions into the three spectral categories, and present simplified forms of the spectral expansion for each category. To obtain explicit expressions, we observe that the Liouville transformation reduces the SL equation to the one-dimensional Schrödinger equation with a potential function constructed froma,bandr. If analytical solutions are available for the Schrödinger equation, inverting the Liouville transformation yields analytical solutions for the original SL equation, and the spectral representation for the diffusion process can be constructed explicitly. This produces an explicit spectral decomposition of the contingent claim value function.

Keywords: Option pricing; barrier options; diffusion processes; spectral expansion; Sturm–Liouville equation (search for similar items in EconPapers)
Date: 2004
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Citations: View citations in EconPapers (36)

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DOI: 10.1142/S0219024904002451

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