Geometric step options and Lévy models: duality, PIDEs, and semi-analytical pricing

<p style='text-indent:20px;'>The present article studies geometric step options in exponential Lévy markets. Our contribution is manifold and extends several aspects of the option pricing literature. First, we provide symmetry duality relations derive various characterizations for both European-type American-type double barrier options. In particular, are able to obtain a jump-diffusion disentanglement early exercise premium contracts its maturity-randomized equivalent as well characterize diffusion jump contributions these premiums separately by means partial integro-differential equations ordinary equations. As an application our characterizations, semi-analytical results (regular) down-and-out call under hyper-exponential models. Lastly, use latter discuss structure once jumps added subsequently analysis impact on price hedging parameters (European-type American-type) contracts.</p>