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Reduced order models in PIDE constrained optimization

Sachs E. W., Schu M.

Control and Cybernetics

2010

Vol. 39, no 3

661-675

Mathematical models for option pricing often result in partial differential equations originally starting with the Black-Scholes model. In this context, recent enhancements are models driven by Levy processes, which lead to a partial differential equation with an additional integral term. If one solves the problems mentioned last numerically, this yields large linear systems of equations with dense matrices. However, by using the special structure and an iterative solver the problem can be handled efficiently. To further reduce the computational cost in the calibration phase we implement a reduced order model, like proper orthogonal decomposition (POD), which proves to be very efficient. In this paper we use a special multi-level trust region POD algorithm to calibrate the option pricing model and give numerical results supporting the gain in efficiency.

A proposal of portfolio choice for infinitely divisible distributions of asset returns

Kliber P.

Foundations of Computing and Decision Sciences

2008

Vol. 33, No. 1

43-52

In the paper \ve present a proposal of augmenting portfolio analysis for the infinitely divisible distributions of returns - so that the prices of assets can follow Levy processes. In the classical portfolio analysis (by Markovitz or Sharp) the portfolio is evaluated according to two criteria: mean return and variance of returns. Such an approach is cumbersome second moments of assets' returns do not exist or if the interdependence between the returns of different assets can not be described only by covariation. In this article we propose a model in which asset prices follow multidimensional Levy process and the interdependence between assets are described by covariance (Gaussian part) and multidimensional jump measure (Poisson pan). Then we propose to choose the optimal portfolio based on three criteria: mean return, total variance of diffusion and a measure of jump risk. We also consider augmenting this multi-criteria choice setup for the costs of possible portfolio adjustments.