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

Abstract

In 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.

Stochastic representations of derivatives of solutions of one-dimensional parabolic variational inequalities with Neumann boundary conditions

“…To obtain (13) we observe that we may substitute X to X into (11): indeed, in [14] the diffusion process is reflected at the boundary of a bounded domain whereas, here, the domain is the infinite interval (d, +∞); however, it is easy to see that Menaldi's proof of inequality (3.23) also applies in this latter case. 3 Therefore, in view of Proposition 2.6, for all t ≤ s ≤ T , P-a.s., ∂ x X t,x,n s converges weakly into some process that we denote by ∂ x X t,x s and X t,x s ∈ D. Suppose now that we have proven, for all x in (d, d ):…”

“…In all this section, in addition to the assumptions made in Theorem 3.4 we suppose that the function L is in C 1,2 ([0, T ]×R; R), bounded with bounded derivatives. Adapting a technique due to Cvitanić and Ma [11], Berthelot et al [3] have shown existence and uniqueness of an adapted solution (Y t,x , Z t,x , R t,x ), and that the function v :…”

“…From a numerical analysis point of view, one needs to estimate |V (t, x) − v(t, x)|. Berthelot, Bossy and Talay [3] have tackled this issue by using a stochastic approach based on Backward Stochastic Differential Equations (BSDE). Given the reflected forward Stochastic Differential Equation (SDE)…”

“…which means that O is reduced to a bounded interval (d, d ). Two main reasons explain the limitation to one-dimensional problems: first, we need to prove an explicit representation of the derivative ∂ x X t,x t , where X t,x is as in (3); this representation appears to be simple and of exponential type; exhibiting such an explicit formula seems difficult for general multi-dimensional flows 1 (Malliavin derivatives were also explicited by Lépingle, Nualart and Sanz [10] in the one-dimensional case only); second, in order to get stochastic representations for ∂ x v(t, x) when h = 0, that is, in the case of nonhomogeneous Neumann boundary conditions, we use an integration by parts technique which seems limited to the one-dimensional case (see Lemma 3.7).…”

“…The paper is organized as follows. In Section 2 we explicit the derivative of the flow of the reflected flow (X t,x ) defined in (3). In Section 3 we get two stochastic representations for derivatives of solutions of semilinear parabolic partial differential equations (which corresponds to the case where L(t, x) ≡ −∞ and R ≡ 0): the first representation involves the gradient of f , the second one does not involve it.…”