kernel singularity up to the second order. Qualitative results using the notion of viscosity solutions were provided by M. Crandall and P.-L. Lions in [@CL92]. They were generalized to PIDEs by Awatif [@Sayahf2007] and Soner [@Soner1986] for the first order operators and by Alvarez and Tourin [@OlivierAgnes1996], Barles [*et al.*]{} [@GuyBarles1997], and Pham [@Huyen1998] for the second order operators. In [@Maria11],[@Maria12] Mariani and SenGupta proved existence of weak solutions of a generalized integro-differential equation using the Schaefer fixed point theorem. On other hand, in [@Amster12], Amster [*et al.*]{} proved the existence of solutions using the method of upper and lower solutions in a general domain in the case of several assets and for the regime-switching jump-diffusion model in [@Ionut2012]. In [@NBS15],[@NBS17] Arregui et al. applied the theory of abstract parabolic equations in Banach spaces (cf. [@Henry1981]) for the proof of existence and uniqueness of solutions of a system of nonlinear PDEs for pricing of XVA derivatives. In the recent paper Cruz and Ševčovič [@NBS19] investigated a nonlinear extension of the option pricing PIDE model (\[V-eq\]) from numerical point of view.

As a motivation we consider a model for pricing vanilla call and put options on underlying assets following Lévy stochastic 