Option pricing with Weyl-Titchmarsh theory

Yishen Li and Jin Zhang

Quantitative Finance, 2004, vol. 4, issue 4, 457-464

Abstract: In the Black-Merton-Scholes framework, the price of an underlying asset is assumed to follow a pure diffusion process. No-arbitrage theory shows that the price of an option contract written on the asset can be determined by solving a linear diffusion equation with variable coefficients. Applying the separating variable method, the problem of option pricing under state-dependent deterministic volatility can be transformed into a Schrodinger spectral problem, which has been well studied in quantum mechanics. With Weyl-Titchmarsh theory, we are able to determine the boundary condition and the nature of the eigenvalues and eigenfunctions. The solution can be written analytically in a Stieltjes integral. A few case studies demonstrate that a new analytical option pricing formula can be produced with our method.

Date: 2004
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DOI: 10.1080/14697680400008643

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