normal distribution. However, the empirical distribution of stock returns exhibits fat tails. Furthermore, if we calibrate theoretical prices to the market prices, we realize that the implied volatility is neither constant as a function of strike nor as a function of time to maturity, contradicting thus assumptions of the Black-Scholes model. Several alternatives have been proposed in the literature for generalization of this model. The models with jumps can, at least in part, solve problems inherent to the Black-Scholes model. Jump–diffusion models also have an important role in derivative markets. In the classical Black-Scholes model the market is assumed to be complete, implying that every pay-off can be perfectly replicated. On the other hand, in jump–diffusion models there is no perfect hedge and this way options are not redundant.

taking into account jumps in the underlying asset process, the price $V(t,S)$ of an option on the underlying asset with a price $S$ and time $t\in[0,T]$ is a solution to the following nonlocal nonlinear partial integro-differential equation: $$\begin{aligned}
\frac{\partial V}{\partial t} &+&\frac{\sigma^2}{2} S^2 \frac{\partial^2 V}{\partial S^2} + r S \frac{\partial V}{\partial S}-rV \nonumber
\\
&+& \int_{\mathbb{R}}\left[ V(t,Se^z)-V(t,S)-(e^z-1) S \frac{\partial V}{\partial S}(t,S) \right] \nu({\mathrm{d}}z)=0,
\label{V-eq}\end{aligned}$$ $S>0, t\in (0,T)$. Here $\sigma>0$ is the 