Banach space $X$ for both call and put option pay-off diagrams $\Phi$, i.e. $\Phi(S)=(S-K)^+$ and $\Phi(S)=(K-S)^+$. The idea how to formulate existence and uniqueness of a solution to the PIDE (\[PDE-u-BS\]) is based on the idea of shifting the solution $u$ by $u_{BS}$ where the function $u_{BS}(\tau,x) =e^{r\tau} V_{BS}(T-\tau, Ke^x)$ corresponds to transformation of the classical solution $V_{BS}$ to the linear Black-Scholes equation without PIDE part, i.e. $$\begin{aligned}
&&\frac{\partial V_{BS}}{\partial t} +\frac{\sigma^2}{2} S^2 \frac{\partial^2 V_{BS}}{\partial S^2} + r S \frac{\partial V_{BS}}{\partial S}-rV_{BS} =0, 
\\
&&V_{BS}(T,S)=\Phi(S).
\nonumber\end{aligned}$$ Recall that the solution $V_{BS}$ for a call or put option can be expressed explicitly: $$\begin{aligned}
V^{call}_{BS}(t,S) &=& S N(d_1) - K e^{-r(T-t)} N(d_2), 
\\
V^{put}_{BS}(t,S)  &=& K e^{-r(T-t)} N(-d_2) - S N(-d_1), \end{aligned}$$ where $$d_{1,2} = \frac{\ln(S/K) + (r\pm\sigma^2/2)(T-t)}{\sigma\sqrt{T-t}},
\ \ \hbox{and} \ \  N(d)=\int_{-\infty}^d \frac{e^{-\xi^2/2}}{\sqrt{2\pi}} {\mathrm{d}}\xi$$ is the cumulative distribution function of the normal distribution (cf. [@NBS5]). Furthermore, the transformed function $u_{BS}$ is a solution to the linear parabolic PDE: $$\begin{aligned}
&& \frac{\partial u_{BS}}{\partial \tau}
=\frac{\sigma^2}{2} \frac{\partial^2 u_{BS}}{\partial x^2} 
+ \left(r-\frac{1}{2}\sigma^2\right)\frac{\partial u_{BS}}{\partial x},
\label{PDE-uBS}
\\
&&u_{BS}(0,x)=\Phi(K e^{x}), \quad \tau\in(0,T), x\in \mathbb{R},
\nonumber\end{aligned}$$ where $\Phi(Ke^x)=K(e^x-1)^+$ for the call option and $\Phi(Ke^x)=K(1-e^x)^+$ for the put option.

In what follows, we shall provide important estimates 