Abstract
The main aim of this study is to introduce a 2-layered artificial neural network (ANN) for solving the Black–Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed to change the model into a sequence of ordinary differential equations (ODE). Subsequently, each of these ODEs is solved with the aid of an ANN. Adam optimization is employed as the learning paradigm since it can add the foreknowledge of slowing down the process of optimization when getting close to the actual optimum solution. The model also takes advantage of fine-tuning for speeding up the process and domain mapping to confront the infinite domain issue. Finally, the accuracy, speed, and convergence of the method for solving several types of the Black–Scholes model are reported.
Highlights
Both partial differential equation (PDE) of the ordinary and fractional order play an important role in pricing of financial derivatives
Financial markets show fractal behavior [1,2,3,4] and fractional PDEs (FPDE) which can better reflect that the reality of them have gained a lot of interest recently
E most famous PDE in finance is the Black–Scholes (BS) model, which is broadly adopted for option pricing
Summary
Both PDEs of the ordinary and fractional order play an important role in pricing of financial derivatives. PDEs of the ordinary order are the basis of various models proposed for pricing of different types of options. Financial markets show fractal behavior [1,2,3,4] and fractional PDEs (FPDE) which can better reflect that the reality of them have gained a lot of interest recently. Studies have presented different approaches for finding the numerical solution of this model and its variation when the exact form does not exist [5,6,7,8,9,10,11,12,13]. Due to the high computational complexity of these solutions, numerical methods are often better alternatives for solving such mathematical models
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