In this paper we introduce efficient Monte Carlo estimators for the valuation of high-dimensional derivatives and their sensitivities ("Greeks"). These estimators are based on an analytical, usually approximative representation of the underlying density. We study approximative densities obtained by the WKB method. The results are applied in the context of a Libor market model.

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In a rather general setting of multivariate diffusion market models we derive global iterative probabilistic schemes for computing the free boundary and its Greeks for a generic class of American derivative models using front-fixing methods. The convergence of the scheme is closely linked to a proof of global regularity of the free boundary surface.1 imsart-aap ver.

We derive obstacle problems for pricing of American derivatives with multiple underlyings heuristically using only a few postulates such that classical (Brownian motion) models as well as models based on Levy processes can be considered in our frame. For the classical models we define a "signed measure" which allows to compute the exercise region near maturity and obtain a generic condition for continuity of the free boundary and prove some more general features of exercise regions for classical models. Especially, we investigate the exercise regions of the most important American derivatives with one and multiple underlyings where we include dependence of volatility and interest rates on time and the underlyings extending and recovering some classical results. Further applications include stochastic volatility models. It is shown that in classical stochastic volatility models where volatility is driven by an Ornstein-Uhlenbeck process an American compound call has a nonempty exercise region and compute the exercise region near expiration in a typical situation.