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

Abstract

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.

Keywords:

• One-parameter semigroups
• Black-Scholes theory
• jump-diffusion process
• partial integro-differential equation
• Mellin transform

2010 Mathematics Subject Classifications:

• 47D06
• 91G20
• 91G80
• 35A22
• 65R20

Disclosure statement

No potential conflict of interest was reported by the author(s).