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A new general approach to optimal stopping problems in L\'evy models, regime switching L\'evy models and L\'evy models with stochastic volatility and stochastic interest rate is developed. For perpetual options, explicit solutions are found, for options with finite time horizon, time discretization is used, and explicit solutions are derived for resulting sequences of perpetual options.
The main building block is the option to abandon a monotone payoff stream. The optimal exercise boundary is found using the operator form of the Wiener-Hopf method, which is standard in analysis, and interpretation of the factors as {\em expected present value operators} (EPV-operators) under supremum and infimum processes.
Other types of options are reduced to the option to abandon a monotone stream. For regime-switching models, an additional ingredient is an efficient iteration procedure.
L\'evy models with stochastic volatility and/or stochastic interest rate are reduced to regime switching models using discretization of the state space for additional factors. The efficiency of the method for 2 factor L\'evy models with jumps and for 3-factor Heston model with stochastic interest rate is demonstrated. The method is much faster than Monte-Carlo methods and can be a viable alternative to Monte Carlo method as a general method for 2-3 factor models.
Joint work of Svetlana Boyarchenko,University of Texas at Austin and Sergei Levendorski\v{i},
University of Leicester