Numerical Analysis and Misspecifications in Finance: From Model Risk to Localization Error Estimates for Nonlinear PDEs

Christophe Berthelot, Mireille Bossy and Denis Talay
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Christophe Berthelot: Bull Company, 1 rue de Provence, BP 208, 38432 Échirolles Cedex, France
Mireille Bossy: INRIA, 2004 Route des Lucioles, B.P. 93, 06902 Sophia-Antipolis, France
Denis Talay: INRIA, 2004 Route des Lucioles, B.P. 93, 06902 Sophia-Antipolis, France

Chapter 1 in Stochastic Processes and Applications to Mathematical Finance, 2004, pp 1-25 from World Scientific Publishing Co. Pte. Ltd.

Abstract: AbstractIn this paper we aim to illustrate the power of probabilistic techniques to analyze numerical errors arising in the study of financial issues and related to various misspecifications. We first describe the global convergence rate of approximation of quantiles of components of diffusion processes when one combines a time discretization of the model and a Monte Carlo simulation in view of computing VaR type quantities. We then present a worst case approach to the problem of controlling model risk; this approach leads to a stochastic game problem and a fully nonlinear PDE in an unbounded domain. To approximate its unique viscosity solution one needs to localize the PDE under consideration and to define artificial boundary conditions. We show that backward stochastic differential equations (BSDEs) are a useful tool to estimate the error due to misspecified Neumann boundary conditions on the artificial boundary.

Keywords: Stochastic Processes; Stochastic Differential Equations; Malliavin Calculus; Stochastic Control and Optimization; Functionals of Brownian Motions and Lévy Processes; Stochastic Models of Financial Market; Derivative Pricing; Hedging Problem (search for similar items in EconPapers)
Date: 2004
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