Parametrix Approximation of Diffusion Transition Densities

Abstract

Abstract.A new analytical approximation tool, derived from the classical PDE theory, is introduced in order to build approximate transition densities of diffusions. The tool is useful for approximate pricing and hedging of financial derivatives and for maximum likelihood and method of moments estimates of diffusion parameters. The approximation is uniform with respect to time and space variables. Moreover, easily computable error bounds are available in any dimension.

A probabilistic interpretation of the parametrix method

“…It has been successfully extended to other situations for theoretical goals (see, e.g., [9][10][11][12] and [14]). In [6], the authors consider the parametrix as an analytical method for approximations for continuous diffusions. These analytical approximations may be used as deterministic approximations and are highly accurate in the cases where the sum converges rapidly.…”

Adjoint Expansions in Local Lévy Models

“…Analytical approximations and their applications to finance have been studied by several authors in the last decades because of their great importance in the calibration and risk management processes. The large body of the existing literature (see, for instance, [16], [18], [28], [15], [3], [8], [6]) is mainly devoted to purely diffusive (local and stochastic volatility) models or, as in [2] and [29], to local volatility (LV) models with Poisson jumps, which can be approximated by Gaussian kernels.…”

“…Our approach is also more general than the so-called "parametrix" methods recently proposed in [8] and [6] as an approximation method in finance. The parametrix method is based on repeated application of Duhamel's principle which leads to a recursive integral representation of the fundamental solution: the main problem with the parametrix approach is that, even in the simplest case of a LV model, it is hard to compute explicitly the parametrix approximations of order greater than one.…”

“…The parametrix method is based on repeated application of Duhamel's principle which leads to a recursive integral representation of the fundamental solution: the main problem with the parametrix approach is that, even in the simplest case of a LV model, it is hard to compute explicitly the parametrix approximations of order greater than one. As a matter of fact, [8] and [6] only contain first order formulae. The adjoint expansion method contains the parametrix approximation as a particular case, that is at order zero and in the purely diffusive case.…”

Analytical Expansions for Parabolic Equations

“…More recently, some of techniques from perturbation theory and heat kernel expansions have been applied to problems arising in mathematical finance: see, for instance, Hagan and Woodward (1999); Henry-Labordère (2009); Benhamou et al (2010); Cheng et al (2011);Fouque et al (2011). The authors of the present manuscript have also made recent contributions in mathematical finance with a focus on finding closed-form pricing approximations for models both without jumps Corielli et al (2010); Pagliarani et al (2013) and with jumps Lorig et al (2013a) ;Jacquier and Lorig (2013), as well as finding closed-form approximations for implied volatility Lorig et al (2013b,c); Lorig (2013).…”