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

Abstract

We derive the joint density of a Skew Brownian motion, its last visit to the origin, its local and occupation times. The result allows to obtain explicit analytical formulas for pricing European options under both a two valued local volatility model and a displaced diffusion model with constrained volatility.

The Parametrix Method for Skew Diffusions

“…29 that |p T (x + h, y) − p T (x, y)| = |h| Using the inequality (a − b) 2 (7). For any (z, u) ∈ R 0 × R, p Z T (z, u) is differentiable as a function of z ∈ R 0 and we have…”

“…Gairat and Shcherbakov in [7] give explicitly the joint density function of a skew diffusion with constant diffusion coefficient and some of its functionals. They apply their results to a mathematical finance model of stock prices with switching coefficients.…”

Stratonovich stochastic differential equation with irregular coefficients: Girsanov’s example revisited

“…The joint distribution of B θ t and L t pB θ q, t ą 0, is well known and can be found, e.g. in Appuhamillage et al (2011); Étoré and Martinez (2012); Gairat and Shcherbakov (2017):…”

Limit Theorems for Local and Occupation Times of Random Walks and Brownian Motion on a Spider

“…We note in passing that the above definition of SBM is a special case of BMS as in (1.2) with N = 2, p 1 = p, p 2 = 1 − p = q, so that, in this case, the first one of the two legs is the positive half-line, and the second one is the negative half-line. In particular, we thus have Various results for local times are given in Walsh [33], Burdzy and Chen [5], Lyulko [28], Gairat and Shcherbakov [18].…”