SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Numerical methods for pricing companies with PDE models and GPUs

C. Vázquez, D. Castillo, A. Ferreiro and J.A. García-Rodríguez

Abstract
We propose appropriate numerical methods for companies valuation models proposed in [1]. The models are formulated in terms of final-boundary value problems associated to Kolmogorov type equations, sometimes including an additional unilateral constraint. We also analyze the required boundary conditions so that the final-boundary value problem is well posed, thus allowing to remove unnecessary boundary conditions proposed in [1]. Numerical methods are mainly semilagrangian schemes in the direction without diffusion combined with implicit second order finite differences schemes in the direction where diffusion is present. This choice of numerical methods allows to develop an original parallelization strategy, which results to be specially efficient when using GPUs technologies [2]. This methodology can be applied to problems in mining industry or Asian options pricing [3].

Bibliography
[1] M.Z. Apabhai, N.I. Georgikoppoulos, D. Hasnip, R.K.D. Jamie, M. Kim and P. Wilmott, A model for the value of a business, some optimization problems in the operating procedures and the value of the debt, IMA J. Appl. Math., 59 (1997), pp. 273-285.
[2] D. Castillo, A. Ferreiro and J.A. García-Rodríguez, Numerical methods to solve PDE models for pricing business companies in different regimes and implementation in GPUs, Appl. Math. Comp., 219 (2013), pp. 11233-11257.
[3] Y. d'Halluin, P.A. Forsyth and G. Labahn, A semi-lagrangian approach for American Asian options under jump-diffusion, SIAM J. on Sci. Comp., 27 (2005), pp. 315-345.

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