Publication: On properties of analytically solvable families of local volatility diffusion models

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Title On properties of analytically solvable families of local volatility diffusion models
Authors/Editors* G. Campolieti and R. Makarov
Where published* Mathematical Finance
How published* Journal
Year* 2006
Volume -1
Number -1
We present some recent developments in the construction and classification of new analytically solvable one-dimensional diffusion models for which the transition densities and other quantities that are fundamental to derivatives pricing are represented in closed form. Our approach is based on so-called diffusion canonical transformations that allow us to uncover new multiparameter processes that are mapped onto various simpler diffusions. Using an asymptotic analysis of the boundary behavior of the processes, we arrive at a rigorous characterization or classification of the newly constructed diffusions with respect to probability conservation and the martingale property. The approach is applicable to a fairly general class of nonlinear volatility diffusions. Specifically, we analyze and classify in detail four main families of diffusion models that arise from the underlying squared Bessel process (the Bessel family), CIR process (the confluent hypergeometric family), the Ornstein-Uhlenbeck diffusion (the OU family) and the Jacobi diffusion (hypergeometric family). We show that the Bessel family is a superset of the constant elasticity of variance (CEV) models. The Bessel family, in turn, is generalized by the confluent hypergeometric family. For these two families we find further subfamilies of conservative strict supermartingales and absorbed martingales. For the new classes of absorbed diffusions we also derive analytically exact first-passage time densities for paths hitting an absorbing endpoint boundary, with densities given in terms of generalized inverse gaussians and extensions that involve the modified Bessel functions and confluent hypergeometric (Kummer) functions. As for the two other models, we show that the OU family of processes are strictly conservative martingales, whereas the Jacobi family is formed by absorbed non-martingales. Considered as asset price diffusion models, we also show that these models demonstrate a large range of local volatility shapes including pronounced skew and smile patterns. A discussion of applications to option pricing concludes the paper.
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