  In financial mathematics, it is a typical approach to approximate financial
markets operating in discrete time by continuous-time models such as the Black
Scholes model. Fitting this model gives rise to difficulties due to the
discrete nature of market data. We thus model the pricing process of financial
derivatives by the Black Scholes equation, where the volatility is a function
of a finite number of random variables. This reflects an influence of uncertain
factors when determining volatility. The aim is to quantify the effect of this
uncertainty when computing the price of derivatives. Our underlying method is
the generalized Polynomial Chaos (gPC) method in order to numerically compute
the uncertainty of the solution by the stochastic Galerkin approach and a
finite difference method. We present an efficient numerical variation of this
method, which is based on a machine learning technique, the so-called
Bi-Fidelity approach. This is illustrated with numerical examples.
