Application of Neural Network Machine Learning to Solution of Black-Scholes Equation

Abstract

This paper presents a novel way to predict options price for one day in advance, utilizing the method of Quasi-Reversibility for solving the Black-Scholes equation.The Black-Scholes equation solved forwards in time with Tikhonov regularization as an ill-posed problem allows for extrapolation of option prices. This provides high-accuracy results, which can be further improved by applying Neural Network Machine Learning to the solution of the Black-Scholes equation as well as initial and boundary conditions and implied volatility.Using historical option and stock price data the results obtained from the method of Quasi-Reversibility and Machine Learning method are compared in terms of accuracy, precision and recall.It is shown that these methods can be applied to the real-world trading within a variety of trading strategies.

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

“…QRM is a regularization method for an ill-posed problem for the Black-Scholes equation. The goal of this paper is to address both analytically and numerically the following question: Why this algorithm has worked well for real market data in [10,13]? Our explanations are based on our new analytical results in the probability theory and are supported by our numerical results for the computationally simulated data generated by the geometrical Brownian motion.…”

“…Results of [10,Table 4] for real market data of [3] indicate that a combination of that mathematical model with that trading strategy has resulted in 72.83% profitable options out of 368 options for real market data. More recently, the model of [10] was used in [13] to forecast stock option prices in the case when results of QRM are enhanced by the machine learning approach, which was applied on the second stage of the price forecasting procedure. Market data of [3] for total 169,862 European call stock options were used in [13].…”

“…More recently, the model of [10] was used in [13] to forecast stock option prices in the case when results of QRM are enhanced by the machine learning approach, which was applied on the second stage of the price forecasting procedure. Market data of [3] for total 169,862 European call stock options were used in [13]. Following the machine learning approach, these data were divided in three sets [13, Table 1]: training (132,912 options), validation (13,401 options) and testing (23,549 options).…”

“…Total 23,549 options were tested by QRM, and good results on predictions of options with profits were obtained in [13, first lines in Tables 2,3]. Later the authors of [13] have tested the performance of QRM for all those 169,862 options, and results were almost the same as ones of [13, first lines in Tables 2,3]. However, since the latter results are not yet published, then we do not discuss them here.…”

“…We now present in Tables 1,2 the most recent results of [13], which were obtained using the method of [10] for the data consisting of 23,549 historical trades collected in 2018. The same market data of [3] were used in Tables 1,2.…”