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

Abstract

We prove the existence and pointwise lower and upper bounds for the fundamental solution of the degenerate second order partial differential equation related to Geman-Yor stochastic processes, that arise in models for option pricing theory in finance.Lower bounds are obtained by using repeatedly an invariant Harnack inequality and by solving an associated optimal control problem with quadratic cost. Upper bounds are obtained by the fact that the optimal cost satisfies a specific Hamilton-Jacobi-Bellman equation.

Harnack Inequalities and Bounds for Densities of Stochastic Processes

“…Clearly, the knowledge of the asymptotic behavior of the function Ψ is crucial for the application of our Theorem 7.1. In [9], it is shown that one can write the function Ψ in terms of the function g defined as follows…”

“…Hence, we obtain lower and upper bounds for the fundamental solution Γ of the variable coefficients operator L in terms of the fundamental solutions Γ ± of the constant coefficients operators L ± , whose expressions, up to some scaling parameters, agree with the function p in (7.30). We refer to the article [9] for the precise statement of the results of this section and for further details.…”

Existence of a fundamental solution of partial differential equations associated to Asian options

“…• The PDE approach, which has the aim to solve numerically the Cauchy problem associated with the no-arbitrage PDE. Related works following this line are those of [11], [7], [46]. In [13], the author applies a method on conditioning on the geometric mean price.…”

“…In [14], the author derives an accurate approximation formulae for Asian-rate Call options in the Black & Scholes model by a matched asymptotic expansion. In this work we rely on the results proved in [11], where the authors prove via probabilistic techniques the existence of the fundamental solution Γ for the operator L with smooth coefficients a and b. Moreover, we recall the existence and local regularity results proved by Lanconelli, Pascucci and Polidoro [31], under the assumption that the coefficients a and b belong to some space of Hölder continuous functions.…”

Local densities for a class of degenerate diffusions

“…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.…”