Finite Differences

Black-Scholes Model, comparison between Analytical Solution and Numerical Analysis ArXiv ID: 2510.27277 “View on arXiv” Authors: Francesco Romaggi Abstract The main purpose of this article is to give a general overview and understanding of the first widely used option-pricing model, the Black-Scholes model. The history and context are presented, with the usefulness and implications in the economics world. A brief review of fundamental calculus concepts is introduced to derive and solve the model. The equation is then resolved using both an analytical (variable separation) and a numerical method (finite differences). Conclusions are drawn in order to understand how Black-Scholes is employed nowadays. At the end a handy appendix (A) is written with some economics notions to ease the reader’s comprehension of the paper; furthermore a second appendix (B) is given with some code scripts, to allow the reader to put in practice some concepts. ...

Exact Terminal Condition Neural Network for American Option Pricing Based on the Black-Scholes-Merton Equations ArXiv ID: 2510.27132 “View on arXiv” Authors: Wenxuan Zhang, Yixiao Guo, Benzhuo Lu Abstract This paper proposes the Exact Terminal Condition Neural Network (ETCNN), a deep learning framework for accurately pricing American options by solving the Black-Scholes-Merton (BSM) equations. The ETCNN incorporates carefully designed functions that ensure the numerical solution not only exactly satisfies the terminal condition of the BSM equations but also matches the non-smooth and singular behavior of the option price near expiration. This method effectively addresses the challenges posed by the inequality constraints in the BSM equations and can be easily extended to high-dimensional scenarios. Additionally, input normalization is employed to maintain the homogeneity. Multiple experiments are conducted to demonstrate that the proposed method achieves high accuracy and exhibits robustness across various situations, outperforming both traditional numerical methods and other machine learning approaches. ...

Some PDE results in Heston model with applications ArXiv ID: 2504.19859 “View on arXiv” Authors: Edoardo Lombardo Abstract We present here some results for the PDE related to the logHeston model. We present different regularity results and prove a verification theorem that shows that the solution produced via the Feynman-Kac theorem is the unique viscosity solution for a wide choice of initial data (even discontinuous) and source data. In addition, our techniques do not use Feller’s condition at any time. In the end, we prove a convergence theorem to approximate this solution by means of a hybrid (finite differences/tree scheme) approach. ...