  In this paper, a class of nonlinear option pricing models involving
transaction costs is considered. The diffusion coefficient of the nonlinear
parabolic equation for the price $V$ is assumed to be a linear function of the
option's underlying asset price and the Gamma Greek $V_{xx}$. The main aim of
this work is to study the governing PDE of the Delta Greek. The existence of
viscosity solutions is proved using the vanishing viscosity method.
Regularizing the equation by adding a small perturbation to the initial
problem, a sequence of approximate solutions $u^{\varepsilon}$ is constructed
and then the method of weak limits is applied to prove the convergence of the
sequence to the viscosity solution of the Delta equation. The approximate
problems constructed are shown to have good regularity, which allows the use of
efficient and robust numerical methods.
