Variational sensitivity analysis of parametric Markovian market models

Abstract

Parameter sensitivities of prices for derivative contracts play an important role in model calibration as well as in quantification of model risk. In this paper a unified approach to the efficient numerical computation of all sensitivities for Markovian market models is presented. Variational approximations of the integro-differential equations corresponding to the infinitesimal generators of the market model differentiated with respect to the model parameters are employed. Superconvergent approximations to second and higher derivatives of prices with respect to the price process' state variables are extracted from approximate, computed prices with low, C 0 regularity by postprocessing. The extracted numerical sensitivities are proved to converge with optimal rates as the mesh width tends to zero. Numerical experiments for uni-and multivariate models with sparse tensor product discretization confirm the theoretical results.

Numerical Analysis of Additive, Lévy and Feller Processes with Applications to Option Pricing

“…In this setting, V = V = H 1 (D), if ' > 0, and V = H m/2+A (D) ⊂ H m/2 (D) = V , ∀A > 0, if ' = 0. We refer to [40] for more details. Figure 10 shows the sensitivity in this model w.r. to the parameter 5 = m .…”

Numerical methods for Lévy processes

“…To this end, write u(η 0 ) for a fixed realization η 0 of η in order to emphasize the dependence of u on η 0 in (4.4). Then, as shown in [66], the derivativeũ(δη) of u with respect to η, i.e., u(δη) := lim s→0 + where D η A is the derivative of A with respect to η. Therefore, the derivative of u with respect to η can be obtained as a solution of the same PIDE as the price u itself, where now the right hand side depends on u.…”

“…Thus, sensitivities with respect to model parameters can be calculated with the same computational effort as the price itself. Furthermore, it is shown in [66] that all computed sensitivities converge with the same rate as the original price u. For sensitivities with respect to a variation of the state space, a finite difference-like differentiation procedure is presented in [66] which allows to obtain the sensitivities from the finite element forward price with the same convergence rate but without additional work.…”

“…Furthermore, it is shown in [66] that all computed sensitivities converge with the same rate as the original price u. For sensitivities with respect to a variation of the state space, a finite difference-like differentiation procedure is presented in [66] which allows to obtain the sensitivities from the finite element forward price with the same convergence rate but without additional work. Using a similar approach, also Fourier methods are capable of calculating sensitivities with respect to state space variation efficiently; see e.g.…”

Quadratic Spline Wavelets for Sparse Discretization of Jump–Diffusion Models

“…Then, the Galerkin method combined with the Crank-Nicolson scheme is used for its numerical solution. The stability of the scheme and convergence of the method have already been investigated (see [4][5][6] and references therein). This paper focuses on the structure of the discretization matrices.…”