Applications of δ-function perturbation to the pricing of derivative securities

Abstract

In the recent econophysics literature, the use of functional integrals is widespread for the calculation of option prices. In this paper, we extend this approach in several directions by means of δ−function perturbations. First, we show that results about infinitely repulsive δ−function are applicable to the pricing of barrier options. We also introduce functional integrals over skew paths that give rise to a new European option formula when combined with δ−function potential. We propose accurate closed-form approximations based on the theory of comonotonic risks in case the functional integrals are not analytically computable.

On the constructions of the skew Brownian motion

“…By a weak solution to (19), we mean a function u such that for all ψ ∈ C ∞ ([0, T ]; R) with ψ(T, x) = 0, then, integrating formally (19) with respect to ψ(t, x)ρ(x) dx with ρ(x) = 1/2a(x) and using integrations by parts,…”

“…Using an integration by parts on R * + and R * − and the freedom of choice of ψ 1 and ψ 2 , we are led to (18). Moreover, one knows that u(t, x) is continuous on (0, T ] × R. Thus, the weak solution of (19) is also a solution of (12) and the converse is also true. The article [49] and the book [48, § III.13, p. 224] contain accounts on the properties of the solution of the transmission problem.…”

“…Since, a few works have used the Skew Brownian motion as a tool for solving applied problems: in astrophysics [98], in ecology [15], in homogenization [50,92], in geophysics [56,57] and more recently in finance [19,20,21,22],...…”

Density of Skew Brownian Motion and Its Functionals With Application in Finance

“…A Skew Brownian motion (SBM) with parameter p is a Markov process that evolves as a standard Brownian motion reflected at the origin so that the next excursion is chosen to be positive with probability p. SBM was introduced in Ito and McKean (1963) and has been studied extensively in probability since then. The process naturally appears in diverse applications, e.g., Appuhamillage et al (2011aAppuhamillage et al ( , 2011b and Lejay (2006), and, in particular, in finance applications, e.g., Decamps, De Schepper, and Goovaerts (2004), Schoutens (2006a, 2006b), and Rossello (2012). In this paper, we derive the joint distribution of SBM and some of its functionals and apply this distribution to derivative pricing under both a local volatility model with discontinuity and a displaced diffusion model with constrained volatility.…”

A Donsker theorem to simulate one-dimensional processes with measurable coefficients

“…Indeed DFOs appear in the modelization of diffusion phenomena, and the irregularity of the coefficient can reflect the irregularity of the media the particle is evolving in. This is interesting in a wide variety of physical situations, for example in fluid mechanics in porous media (see [RTW05]), in the modelization of the brain (see [Fau99]), and can also be used in finance (see [DDG05]). …”