Computing Black Scholes with Uncertain Volatility-A Machine Learning Approach

TL;DR

This paper introduces a machine learning-enhanced numerical method to quantify the impact of uncertain volatility factors on Black Scholes derivative pricing, improving computational efficiency and accuracy.

Contribution

It develops a Bi-Fidelity approach combined with gPC and stochastic Galerkin methods to efficiently compute uncertainty in Black Scholes model prices due to volatility variability.

Findings

The method accurately quantifies volatility uncertainty effects.

Numerical examples demonstrate improved computational efficiency.

The approach effectively integrates machine learning with stochastic analysis.

Abstract

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.