In this article it is shown that one is able to evaluate the price of perpetual calls, puts, Russian and integral options directly as the Laplace transform of a stopping time of an appropriate diffusion using standard fluctuation theory. This approach is offered in contrast to the approach of optimal stopping through free boundary problems. Following ideas of Carr [Rev. Fin. Studies 11 (1998) 597--626], we discuss the Canadization of these options as a method of approximation to their finite time counterparts. Fluctuation theory is again used in this case.
"Perpetual options and Canadization through fluctuation theory." Ann. Appl. Probab. 13 (3) 1077 - 1098, August 2003. https://doi.org/10.1214/aoap/1060202835