  Treating high dimensionality is one of the main challenges in the development
of computational methods for solving problems arising in finance, where tasks
such as pricing, calibration, and risk assessment need to be performed
accurately and in real-time. Among the growing literature addressing this
problem, Gass et al. [14] propose a complexity reduction technique for
parametric option pricing based on Chebyshev interpolation. As the number of
parameters increases, however, this method is affected by the curse of
dimensionality. In this article, we extend this approach to treat
high-dimensional problems: Additionally exploiting low-rank structures allows
us to consider parameter spaces of high dimensions. The core of our method is
to express the tensorized interpolation in tensor train (TT) format and to
develop an efficient way, based on tensor completion, to approximate the
interpolation coefficients. We apply the new method to two model problems:
American option pricing in the Heston model and European basket option pricing
in the multi-dimensional Black-Scholes model. In these examples we treat
parameter spaces of dimensions up to 25. The numerical results confirm the
low-rank structure of these problems and the effectiveness of our method
compared to advanced techniques.
