(\[PDE-u\]) in the abstract form as follows: $$\begin{aligned}
&&\frac{\partial u}{\partial \tau} + A u = 
\omega \frac{\partial u}{\partial x} + f[u] + g(\tau, u), \quad x\in \mathbb{R}, \tau\in(0,T),
\label{problem_transformed}
\\
&&u(0,x)=u_0(x), x\in \mathbb{R},
\nonumber\end{aligned}$$ where the linear operators $A$ and $f$ are defined by: $$\begin{aligned}
A u &=& - \frac{\sigma^2}{2}\frac{\partial^2 u}{\partial x^2},
\label{Au_def}
\\
f[u](x) &=& 
\int_{\mathbb{R}}\left[ u(x+z)-u(x)- (e^z-1) \frac{\partial u}{\partial x}(x)\, \right] \nu({\mathrm{d}}z),
\label{functional_f_def}\end{aligned}$$ and $g$ is a Hölder continuous mapping in the $\tau$ variable and it is Lipschitz continuous in the $u$ variable.

As a motivation for studying PIDE (\[problem\_transformed\]) we consider a model for pricing vanilla call and put options on underlying assets following Lévy stochastic processes. The classical linear Black-Scholes equation can be transformed into equation v(\[problem\_transformed\]) where $f\equiv 0, g\equiv 0$. A nontrivial integral part $f[u]$ arises from a generalization of the Black-Scholes model to the case when the underlying asset price follows a stochastic Lévy process with jumps (see Section 4). In Section 5 we will investigate equation (\[problem\_transformed\]) with a nontrivial integral part $f[u]$ and a nonlinearity $g$ corresponding to the penalization function. The resulting PIDE of the form (\[problem\_transformed\]) represents an approximation of a solution to the partial integro-differential variational inequality arising 