Convergence of option rewards for Markov type price processes modulated by stochastic indices. I

Abstract

A general price process represented by a two-component Markov process is considered. Its first component is interpreted as a price process and the second component as an index process modulating the price component. American type options with pay-off functions admitting power type upper bounds are studied. Both the transition characteristics of the price processes and the pay-off functions are assumed to depend on a perturbation parameter $\delta \geq 0$ and to converge to the corresponding limit characteristics as $\delta \rightarrow 0$. In the first part of the paper, asymptotically uniform skeleton approximations connecting reward functionals for continuous and discrete time models are given. In the second part of the paper, these skeleton approximations are used for obtaining results about the convergence of reward functionals of American type options with perturbed price processes for both cases of discrete and continuous time. Examples related to modulated exponential price processes with independent increments are given.