Abstracts

Modeling and pricing of financial instruments as well as model calibration are some of the major tasks of mathematical finance. In particular, calibration to market prices requires efficient pricing methods. Commonly, calibration procedures are designed for European options and a number of efficient algorithms have been developed in academia and practice. However, most of the frequently traded stock options are of American type, which has to be incorporated in realistic calibration procedures. In contrast to European options, those of American type are path dependent. Therefore, efficient pricing methods for path dependent options are indispensable. Moreover, model uncertainty leads to the simultaneous use of various models ranging from stochastic volatility to models driven by pure jump processes. Essentially three approaches to compute option prices in these models are being used: Monte Carlo simulation, Fourier based valuation methods and solving the related partial differential equation (PDE). Monte Carlo simulation is typically too slow for calibration purposes. Fourier techniques have been proven to be efficient in Lévy models, especially for pricing European options. PDE methods, however, can be used for pricing not only European but also various path dependent options. Moreover, this can be achieved with the same computational and implementational effort. Applying PDE methods in models driven by jump processes, instead of partial differential equations, partial integro-differential equations (PIDEs) have to be solved. Due to the integral part, standard techniques generally lead to inefficient algorithms. Combining Fourier and PIDE-techniques with model reduction, we achieve flexibility of the pricing method towards different option and model types, and at the same time efficiency and computational feasibility. This also allows us to calibrate jump models to American options, thus avoiding de-Americanization procedures, which are typically used by practitioners. We present the theoretical framework for models driven by time-inhomogeneous Lévy processes. The potential of the method is illustrated by numerical results concerning the calibration of pure jump models to American option prices.