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We present solutions to perpetual American option pricing problems in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. As examples, we consider barrier, lookback and credit options. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems and the latter are solved by using the normal-reflection and smooth- and continuous-fit conditions at the optimal stopping boundaries. It is known that the prices of perpetual options serve as upper estimations for the related American options with finite expiry.
In the case of perpetual American double barrier options (out-of-money puts or calls) the initial problem is reduced to an irregular optimal stopping problem. In the equivalent free-boundary problem the smooth-fit and continuous-fit conditions at the optimal exercise boundary (when the latter coincides with one of the barriers) may break down.
In the case of perpetual American lookback options (with fixed and floating strikes) the initial problem is reduced to an optimal stopping problem for the maximum process. In the equivalent two-dimensional free-boundary problem the normal-reflection condition may break down. It is shown that under some relationships on the parameters of the considered exponential jump-diffusion model the optimal exercise boundary can be uniquely determined as a component of a two-dimensional system of (first-order) nonlinear ordinary differential equations.
In the case of perpetual American credit options (put or call) the initial problem (formulated in a reduced form model where the default time is unknown) is reduced to an optimal stopping problem having a continuous but not absolutely continuous discounting factor. This yields that the normal reflection condition fails to hold in this case and the equivalent two-dimensional free-boundary problem can be solved by using the related generalized reflection condition.
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