[Une nouvelle méthode pour résoudre les équations de Kolmogorov en finance mathématique]
Nous proposons une nouvelle méthode numérique, utilisant une technique de localisation originale, pour résoudre des équations de Fokker–Planck–Kolmogorov multi-dimensionnelles. Nous présentons des tests numériques extensifs qui démontrent l'intérêt pratique de cette approche pour les applications en finance. En particulier, cette approche nous permet de traiter les problèmes de calibration et de valorisation, ainsi que le calcul de mesures de risque variées.
For multi-dimensional Fokker–Planck–Kolmogorov equations, we propose a numerical method which is based on a novel localization technique. We present extensive numerical experiments that demonstrate its practical interest for finance applications. In particular, this approach allows us to treat calibration and valuation problems, as well as various risk measure computations.
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@article{CRMATH_2017__355_6_680_0,
author = {LeFloch, Philippe G. and Mercier, Jean-Marc},
title = {A new method for solving {Kolmogorov} equations in mathematical finance},
journal = {Comptes Rendus. Math\'ematique},
pages = {680--686},
publisher = {Elsevier},
volume = {355},
number = {6},
year = {2017},
doi = {10.1016/j.crma.2017.05.003},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2017.05.003/}
}
TY - JOUR AU - LeFloch, Philippe G. AU - Mercier, Jean-Marc TI - A new method for solving Kolmogorov equations in mathematical finance JO - Comptes Rendus. Mathématique PY - 2017 SP - 680 EP - 686 VL - 355 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2017.05.003/ DO - 10.1016/j.crma.2017.05.003 LA - en ID - CRMATH_2017__355_6_680_0 ER -
%0 Journal Article %A LeFloch, Philippe G. %A Mercier, Jean-Marc %T A new method for solving Kolmogorov equations in mathematical finance %J Comptes Rendus. Mathématique %D 2017 %P 680-686 %V 355 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2017.05.003/ %R 10.1016/j.crma.2017.05.003 %G en %F CRMATH_2017__355_6_680_0
LeFloch, Philippe G.; Mercier, Jean-Marc. A new method for solving Kolmogorov equations in mathematical finance. Comptes Rendus. Mathématique, Tome 355 (2017) no. 6, pp. 680-686. doi : 10.1016/j.crma.2017.05.003. http://www.numdam.org/articles/10.1016/j.crma.2017.05.003/
[1] Polar factorization and monotone re-arrangements of vector-valued functions, Commun. Pure Appl. Math., Volume 44 (1991), pp. 375-417
[2] Option pricing: valuation models and applications, Manag. Sci., Volume 50 (2004), pp. 1145-1177
[3] Meshfree methods (Rieth and, M.; Schommers, W., eds.), Handbook of Theoretical and Computational Nanotechnology, vol. 2, American Scientific Publishers, 2006
[4] Implied volatility surface: construction methodologies and characteristics (preprint unpublished work) | arXiv
[5] Revisiting the method of characteristics via a convex hull algorithm, J. Comput. Phys., Volume 298 (2015), pp. 95-112
[6] P.G. LeFloch, J.-M. Mercier, Tackling the curse of dimensionality, in preparation.
[7] Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator, ACM Trans. Model. Comput. Simul., Volume 8 (1998), pp. 3-30
[8] Optimally transported schemes and applications in mathematical finance, November 2008 http://www.crimere.com/blog/jean-marc/?p=336 (notes available at)
[9] Distribution of points in a cube and approximate evaluation of integrals, Ž. Vyčisl. Mat. Mat. Fiz., Volume 7 (1967), pp. 784-802 (in Russian)
[10] Optimal Transport: Old and New, Springer Verlag, 2006
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