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Abstract
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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 the discretization
of the state space of additional factors. The efficiency of the method
for 2 factor models with jumps and for 3-factor Heston model with
stochastic interest rate is demonstrated.
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