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- ItemSimulation based policy iteration for American style derivatives : a multilevel approach(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Belomestny, Denis; Ladkau, Marcel; Schoenmakers, John G.M.This paper presents a novel approach to reduce the complexity of simulation based policy iteration methods for pricing American options. Typically, Monte Carlo construction of an improved policy gives rise to a nested simulation algorithm for the price of the American product. In this respect our new approach uses the multilevel idea in the context of the inner simulations required, where each level corresponds to a specific number of inner simulations. A thorough analysis of the crucial convergence rates in the respective multilevel policy improvement algorithm is presented. A detailed complexity analysis shows that a significant reduction in computational effort can be achieved in comparison to standard Monte Carlo based policy iteration.
- ItemRobust optimal stopping(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Krätschmer, Volker; Ladkau, Marcel; Laeven, Roger J.A.; Schoenmakers, John G.M.; Stadje, MitjaThis paper studies the optimal stopping problem in the presence of model uncertainty (ambiguity). We develop a method to practically solve this problem in a general setting, allowing for general time-consistent ambiguity averse preferences and general payoff processes driven by jump-diffusions. Our method consists of three steps. First, we construct a suitable Doob martingale associated with the solution to the optimal stopping problem using backward stochastic calculus. Second, we employ this martingale to construct an approximated upper bound to the solution using duality. Third, we introduce backward-forward simulation to obtain a genuine upper bound to the solution, which converges to the true solution asymptotically. We analyze the asymptotic behavior and convergence properties of our method. We illustrate the generality and applicability of our method and the potentially significant impact of ambiguity to optimal stopping in a few examples.