  We prove existence, uniqueness, and regularity of viscosity solutions to the
stationary and evolution obstacle problems defined by a class of nonlocal
operators that are not stable-like and may have supercritical drift. We give
sufficient conditions on the coefficients of the operator to obtain H\"older
and Lipschitz continuous solutions. The class of nonlocal operators that we
consider include non-Gaussian asset price models widely used in mathematical
finance, such as Variance Gamma Processes and Regular L\'evy Processes of
Exponential type. In this context, the viscosity solutions that we analyze
coincide with the prices of perpetual and finite expiry American options.
