European option pricing underlying two assets using PINN

[1] Azizi, E., Neisy, A., 2018, A New Approach in Geometric Brownian Motion Model, Fuzzy
Information and Engineering and Decision, Advances in Intelligent Systems and Computing
Springer International Publishing AG, 646, 334-350.
[2] Amilon, H., 2003, A Neural Network Versus BlackScholes: A Comparison of Pricing and
Hedging Performances, J. Forecast, 22, 317335.
[3] Bjork, T, 2009, Arbitrage Theory in Continuous Time, Oxford University Press , 546 pp.
[4] Chen, Y., 2017, Numerical Methods for Pricing Multi-Asset Options, Department of Computer Science, University of Toronto, 85.
[5] Duffy, J., 2022, Numerical Methods in Computational Finance: A Partial Differential Equation (PDE/FDM) Approach, Wiley, 544 pp.
[6] Esfahani, M. K., Neisy, A., De Marchi, S., 2021, An RBF approach for oil futures pricing
under the jump-diffusion model, Journal of Mathematical Modeling, 9, 81-92.
[7] Glasserman, P., 2004, Monte Carlo Methods in Financial Engineering, Springer Science and
Business Media, 596 pp.
[8] Grohs, P., Hornung, F., Jentzen, A., Wurstemberger, P.V., 2023, A proof that artificial
neural networks overcome the curse of dimensionality in the numerical approximation of
Black-Scholes partial differential equations, Memoirs of the American Mathematical Society,
47-109.
[9] Hagan, M., Demuth, H., Beale, M., De Jes´us, O., 1996, Neural Network Design, PWS Pub,
744 pp.
[10] Han, J., Jentzenb, A., E, W., 2018, Solving high-dimensional partial differential equations
using deep learning, Proc. Natl. Acad. Sci. USA, 34, 85058510.
[11] Hull, J., 2000, Options, Futures Other Derivatives, Volume 1, Prentice Hall, 698 pp.
[12] Kohler, M., Krzyzak, A., Todorovic, N., 2010, Pricing of highdimensional American options
by neural networks, Math. Financ, 20, 383410.
[13] Kollmannsberger, S., Angella, D., Jokeit, M., Herrmann, L., 2021, Deep Learning in Computational Mechanics, Springer, 108 pp.
[14] Liu, S., Oosterlee, C.W., Bothe, S.M., 2019, Pricing options and computing implied volatilities using neural networks, Risks, 7, 16.
[15] Margrabe, W, 1978, The value of an option to exchange one asset for another, J. Financ, 33,
177186.
[16] Neisy, A., Salmani Gharaei, K., 2017, Financial Engineering and Markets Modeling: A
MATLAB-based Approach, Allameh Tabatabaei University , 323 pp.
[17] Neisy, A., De Marchi, S., Jalili, R., 2018, A Radial Basis Function Method for Solving Bond
Pricing Model, Dolomites Research Week on Approximation (DRWA18), Alba di Canazei.
[18] Oksendal, B., 2003, Stochastic Differential Equations: An Introduction with Applications,
Springer, 406 pp.
[19] Pasquale, A. Cuomo, S. Mariapia, R., 2024, A physicsinformed deep learning approach for
solving strongly degenerate parabolic problems, Engineering with Computers Springer.
[20] Pettersson, U., Larsson, E., Marcusson, G., 2008, Improved radial basis function methods
for multi-dimensional option pricing, Computational and Applied Mathematics, 222, 8293.
[21] Raissi, M., Karniadakis, G.E., Perdikaris, P., 2017, Physics informed deep learning (Part I):
Data-driven solutions of nonlinear partial differential equations, arXiv, arXiv:1711.10561v1..
[22] Salvador, B., Oosterlee, C.W.,Van der Meer, R., 2021, Financial Option Valuation by Unsupervised Learning with Artificial Neural Networks, Mathematics MDPI, 9,1-20.
[23] Sirignano, J., Spiliopoulos, K., 2018, A Deep Learning Algorithm for Solving Partial Differential Equations, DGM: Journal of Computational Physics, Amsterdam, The Netherlands.
[24] Shahrokhabadi, M.A., Neisy, A., Perracchione, E., Polato, M., 2019, Learning with reduced
kernel-based methods: Environmental and financial applications, Dolomites Research Notes
on Approximation, 12, 12-27.
[25] Shidfar, A., Zakeri, A., Neisy, A., 2005, A two Dimension Inverse Heat Conduction Problem For Estimating Heat Source, International Journal of Mathematics and Mathematical
Science, 10, 1633-1641.
[26] Wilmott, P., Howison, S., Dewynne, J., 2012, The Mathematics of Financial Derivatives,
Cambridge University Press, 312 pp.
[27] Van der Meer, R., Oosterlee, C., Borovykh, A., 2020, Optimally weighted loss functions for
solving PDEs with Neural Netwoks, arXiv, arXiv:2002.06269.
[28] Yadav, N., Yadav, A., Kumar, M., 2015, Neural Network Methods for Solving Differential
Equations, Springer, 114 pp.
[29] Zurada, J.M., 1992, Introduction to Artificial Neural Systems, St. Paul : West, 812 pp.