On a strongly continuous semigroup for a Black-Scholes integro-differential operator: European options under jump-diffusion dynamics

We consider a Black-Scholes integro-differential operator associated with a partial integro-differential equation for pricing European options with a jump-diffusion process for the underlying asset. Using the theory of one-parameter semigroups, we prove that the operator is the infinitesimal generator of a strongly continuous semigroup and express the semigroup explicitly as a convolution of a jump function, the Black-Scholes kernel and the payoff function. This is analogous to the Gauss-Weierstrass and Poisson semigroups. Then we investigate the pricing of European options under jump diffusion for two broad classes of payoff functions. A generalised put-call parity relating the functions from both classes is also obtained.