Sharp Estimates for Geman-Yor Processes and applications to Arithmetic Average Asian options

Cibelli, Gennaro; Polidoro, Sergio; Rossi, Francesco

doi:10.48550/arxiv.1610.07838

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“…Although more sophisticated models, with more flexible dynamics (local-stochastic volatility) for the price of the underlying asset, were proposed to price Asian options, the prototype process (1.3) is complex enough to exhibit some interesting mathematical properties. In fact, the problem of analytically characterizing its joint transition density is still partially open, and sharp upper/lower bounds were established only recently in [4]. It is easy to recognize in (1.4) the double degeneracy of the generator A that our framework allows for: on the one hand, the second order part of A is fully degenerate in that the partial derivative ∂ x2x2 is missing; on the other hand, the coefficient x 2 1 of the second order derivative ∂ x1x1 also degenerates near zero.…”

Section: Introductionmentioning

confidence: 99%

Local densities for a class of degenerate diffusions

Lanconelli¹,

Pagliarani²,

Pascucci³

2020

Ann. Inst. H. Poincaré Probab. Statist.

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We study a class of R d -valued continuous strong Markov processes that are generated, only locally, by an ultra-parabolic operator with coefficients that are regular w.r.t. the intrinsic geometry induced by the operator itself and not w.r.t. the Euclidean one. The first main result is a local Itô formula for functions that are not twice-differentiable in the classical sense, but only intrinsically w.r.t. to a set of vector fields, related to the generator, satisfying the Hörmander condition. The second main contribution, which builds upon the first one, is an existence and regularity result for the local transition density.

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Section: Introductionmentioning

confidence: 99%