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- ItemLibor model with expiry-wise stochastic volatility and displacement(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Ladkau, Marcel; Schoenmakers, John G.M.; Zhang, JianingWe develop a multi-factor stochastic volatility Libor model with displacement, where each individual forward Libor is driven by its own square-root stochastic volatility process. The main advantage of this approach is that, maturity-wise, each square-root process can be calibrated to the corresponding cap(let)vola-strike panel at the market. However, since even after freezing the Libors in the drift of this model, the Libor dynamics are not affine, new affine approximations have to be developed in order to obtain Fourier based (approximate) pricing procedures for caps and swaptions. As a result, we end up with a Libor modeling package that allows for efficient calibration to a complete system of cap/swaption market quotes that performs well even in crises times, where structural breaks in vola-strike-maturity panels are typically observed
- 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.
- ItemSDE based regression for random PDEs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Anker, Felix; Bayer, Christian; Eigel, Martin; Ladkau, Marcel; Neumann, Johannes; Schoenmakers, John G.M.A simulation based method for the numerical solution of PDE with random coefficients is presented. By the Feynman-Kac formula, the solution can be represented as conditional expectation of a functional of a corresponding stochastic differential equation driven by independent noise. A time discretization of the SDE for a set of points in the domain and a subsequent Monte Carlo regression lead to an approximation of the global solution of the random PDE. We provide an initial error and complexity analysis of the proposed method along with numerical examples illustrating its behaviour.
- 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.