  The classical linear Black--Scholes model for pricing derivative securities
is a popular model in financial industry. It relies on several restrictive
assumptions such as completeness, and frictionless of the market as well as the
assumption on the underlying asset price dynamics following a geometric
Brownian motion. The main purpose of this paper is to generalize the classical
Black--Scholes model for pricing derivative securities by taking into account
feedback effects due to an influence of a large trader on the underlying asset
price dynamics exhibiting random jumps. The assumption that an investor can
trade large amounts of assets without affecting the underlying asset price
itself is usually not satisfied, especially in illiquid markets. We generalize
the Frey--Stremme nonlinear option pricing model for the case the underlying
asset follows a Levy stochastic process with jumps. We derive and analyze a
fully nonlinear parabolic partial-integro differential equation for the price
of the option contract. We propose a semi-implicit numerical discretization
scheme and perform various numerical experiments showing influence of a large
trader and intensity of jumps on the option price.
