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Applicable Analysis
An International Journal
Volume 102, 2023 - Issue 1
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Research Article

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

Pages 220-238 | Received 24 Feb 2021, Accepted 15 Jun 2021, Published online: 06 Jul 2021
 

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.

2010 Mathematics Subject Classifications:

Disclosure statement

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

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