Forecasting Stock Options Prices via the Solution of an Ill-Posed Problem for the Black-Scholes Equation

TL;DR

This paper presents a new method for forecasting stock options prices by solving an ill-posed Black-Scholes equation using the Quasi-Reversibility Method, supported by convergence analysis and simulation data.

Contribution

It introduces a novel application of QRM to the Black-Scholes equation for stock options forecasting, including convergence analysis and validation with simulated data.

Findings

QRM effectively forecasts stock options prices.

The method shows good accuracy on market data.

Convergence of the solution is established via Carleman estimates.

Abstract

In the previous paper (Inverse Problems, 32, 015010, 2016), a new heuristic mathematical model was proposed for accurate forecasting of prices of stock options for 1-2 trading days ahead of the present one. This new technique uses the Black-Scholes equation supplied by new intervals for the underlying stock and new initial and boundary conditions for option prices. The Black-Scholes equation was solved in the positive direction of the time variable, This ill-posed initial boundary value problem was solved by the so-called Quasi-Reversibility Method (QRM). This approach with an added trading strategy was tested on the market data for 368 stock options and good forecasting results were demonstrated. In the current paper, we use the geometric Brownian motion to provide an explanation of that effectivity using computationally simulated data for European call options. We also provide a convergence analysis for QRM. The key tool of that analysis is a Carleman estimate.