TY  - JOUR
A1  - Hellmuth, Kathrin
A1  - Klingenberg, Christian
T1  - Computing Black Scholes with uncertain volatility — a machine learning approach
JF  - Mathematics
N2  - 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.
KW  - numerical finance
KW  - Black Scholes equation
KW  - uncertainty quantification
KW  - uncertain volatility
KW  - polynomial chaos
KW  - Bi-Fidelity method
Y1  - 2022
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-262280
SN  - 2227-7390
VL  - 10
IS  - 3
ER  -