Abstract: This chapter discusses the development and application of exact solution methodologies for pricing derivatives under state-dependent asset price processes. A fairly general mathematical framework is presented for obtaining pricing kernels satisfying various boundary conditions. The kernels are used to obtain new families of analytically exact closed-form pricing formulas for standard, as well as barrier-style European options under various types of multiparameter state-dependent volatility models. The approach for tackling state-dependent models is related to the general nature, whereby the most fundamental quantities, including the pricing kernels or transition probability density functions are solved. This, in turn, is achieved by introducing a new and special type of “mapping” of the original state-dependent diffusion problem onto a related diffusion problem corresponding to an appropriately chosen, simpler underlying process. The method of Green's functions is presented for solving the Kolmogorov partial differential equations for the kernel. The chapter considers an underlying x-space diffusion process and shows how analytical formulas for the time-dependent transition probability density for absorbing boundary conditions, as well as nonabsorbing conditions are generated via the Green's functions.
Publication Year: 2006
Publication Date: 2006-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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Cited By Count: 1
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