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[EN] In this work, we apply the local Wendland radial basis function (RBF) for solving the time-dependent multi dimensional option pricing nonlinear PDEs. Firstly, cross derivative terms of the PDE are removed with a change of spatial variables based in LDLT factorization of the di usion matrix. Then, it is discussed that the valuation of a multi-asset option up to 4D can be computed using a modi fied shape parameter algorithm. In fact, several experiments containing of three and four assets are worked out showing that the results of the presented method are in good agreement with the literature and could be much more accurate once the shape parameter is chosen carefully. ; This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and the Ministerio de Economia y Competitividad Spanish grant MTM2013-41765-P. ; Company Rossi, R.; Egorova, V.; Jódar Sánchez, LA.; Soleymani, F. (2018). A local radial basis function method for high-dimensional American option pricing problems. Mathematical Modelling and Analysis. 23(1):117-138. https://doi.org/10.3846/mma.2018.008 ; S ; 117 ; 138 ; 23 ; 1

In this work, we apply the local Wendland radial basis function (RBF) for solving the time-dependent multi dimensional option pricing nonlinear PDEs. Firstly, cross derivative terms of the PDE are removed with a change of spatial variables based in LDLT factorization of the diffusion matrix. Then, it is discussed that the valuation of a multi-asset option up to 4D can be computed using a modified shape parameter algorithm. In fact, several experiments containing of three and four assets are worked out showing that the results of the presented method are in good agreement with the literature and could be much more accurate once the shape parameter is chosen carefully. ; This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and the Ministerio de Economía y Competitividad Spanish grant MTM2013-41765-P.

[EN] A system of coupled free boundary problems describing American put option pricing under regime switching is considered. In order to build numerical solution firstly a front-fixing transformation is applied. Transformed problem is posed on multidimensional fixed domain and is solved by explicit finite difference method. The numerical scheme is conditionally stable and is consistent with the first order in time and second order in space. The proposed approach allows the computation not only of the option price but also of the optimal stopping boundary. Numerical examples demonstrate efficiency and accuracy of the proposed method. The results are compared with other known approaches to show its competitiveness. ; This work has been partially supported by the European Union in the FP7- PEOPLE-2012-ITN program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and the Ministerio de Economia y Competitividad Spanish grant MTM2013-41765-P. ; Egorova, V.; Company Rossi, R.; Jódar Sánchez, LA. (2016). A New Efficient Numerical Method for Solving American Option under Regime Switching Model. Computers and Mathematics with Applications. 71:224-237. https://doi.org/10.1016/j.camwa.2015.11.019 ; S ; 224 ; 237 ; 71

We propose a simple model for the behaviour of long-time investors on stock markets, consisting of three particles, which represent the current price of the stock, and the opinion of the buyers, or sellers resp., about the right trading price. As time evolves both groups of traders update their opinions with respect to the current price. The update speed is controled by a parameter $\gamma$, the price process is described by a geometric Brownian motion. The stability of the market is governed by the difference of the buyers' opinion and the sellers' opinion. We prove that the distance

[EN] This paper presents an explicit finite-difference method for nonlinear partial differential equation appearing as a transformed Black-Scholes equation for American put option under logarithmic front fixing transformation. Numerical analysis of the method is provided. The method preserves positivity and monotonicity of the numerical solution. Consistency and stability properties of the scheme are studied. Explicit calculations avoid iterative algorithms for solving nonlinear systems. Theoretical results are confirmed by numerical experiments. Comparison with other approaches shows that the proposed method is accurate and competitive. ; This paper has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN Program under Grant Agreement no. 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance). ; Company Rossi, R.; Egorova, V.; Jódar Sánchez, LA. (2014). Solving American Option Pricing Models by the Front Fixing Method: Numerical Analysis and Computing. Abstract and Applied Analysis. 2014:1-9. https://doi.org/10.1155/2014/146745 ; S ; 1 ; 9 ; 2014 ; Feng, L., Linetsky, V., Luis Morales, J., & Nocedal, J. (2011). On the solution of complementarity problems arising in American options pricing. Optimization Methods and Software, 26(4-5), 813-825. doi:10.1080/10556788.2010.514341 ; Van Moerbeke, P. (1976). On optimal stopping and free boundary problems. Archive for Rational Mechanics and Analysis, 60(2), 101-148. doi:10.1007/bf00250676 ; GESKE, R., & JOHNSON, H. E. (1984). The American Put Option Valued Analytically. The Journal of Finance, 39(5), 1511-1524. doi:10.1111/j.1540-6261.1984.tb04921.x ; BARONE-ADESI, G., & WHALEY, R. E. (1987). Efficient Analytic Approximation of American Option Values. The Journal of Finance, 42(2), 301-320. doi:10.1111/j.1540-6261.1987.tb02569.x ; Ju, N. (1998). Pricing by American Option by Approximating its Early Exercise Boundary as a Multipiece Exponential Function. Review of Financial Studies, 11(3), ...