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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.