Fast and reliable pricing of American options with local volatility

We present globally convergent multigrid methods for the nonsymmetric obstacle problems as arising from the discretization of Black—Scholes models of American options with local volatilities and discrete data. No tuning or regularization parameters occur. Our approach relies on symmetrization by transformation and data recovery by superconvergence

Fast valuation and calibration of credit default swaps under Lévy dynamics

In this paper we address the issue of finding an efficient and flexible numerical approach for calculating survival/default probabilities and pricing Credit Default Swaps under advanced jump dynamics. We have chosen to use the firm’s value approach, modeling the firm’s value by an exponential Levy model. For this approach the default event is defined as a first passage of a barrier and it is therefore possible to exploit a numerical technique developed to price barrier options under Levy models to calculate the default probabilities. The method presented is based on the Fourier-cosine series expansion of the underlying model’s density function

Structured Adaptive Finite-Volume Multigrid for Compressible Flows

this paper we give an outline of the theory necessary to derive first and second-order accurate discretisations on a structured, adaptive finite-volume mesh. The mesh is constructed so that the equations can be defined on a rather arbitrary domain, and the usual nonlinear multigrid techniques can be used for the solution of the discrete system. During the solution process the mesh can be adapted to the solution and to the accuracy of the discrete equations. This requires a sufficiently accurate estimate of the local truncation error. After formally introducing the geometric structure and notations, we discuss the discretisation and we study the various contributions to the local discretisation error. Emphasis is put on the discretisation involving the interfaces between the coarse and the fine parts of the grid. Our analysis leads to a small set of requirements, to be satisfied in order to attain a discretisation which is first or second-order accurate (in a sense that will be specified) with respect to the mesh size of the partitioning. Then interpolations are presented which satisfy these conditions

Geometric multigrid for an implicit-time immersed boundary method

The immersed boundary (IB) method is an approach to fluid-structure interaction that uses Lagrangian variables to describe the structure and Eulerian variables to describe the fluid. Explicit time stepping schemes for the IB method require solvers only for Eulerian equations, for which fast Cartesian grid solution methods are available. Such methods are relatively straightforward to develop and are widely used in practice but often require very small time steps to maintain stability. Implicit-time IB methods permit the stable use of large time steps, but efficient implementations of such methods require significantly more complex solvers that effectively treat both Lagrangian and Eulerian variables simultaneously. Several different approaches to solving the coupled Lagrangian-Eulerian equations have been proposed, but a complete understanding of this problem is still emerging. This paper presents a geometric multigrid method for an implicit-time discretization of the IB equations. This multigrid scheme uses a generalization of box relaxation that is shown to handle problems in which the physical stiffness of the structure is very large. Numerical examples are provided to illustrate the effectiveness and efficiency of the algorithms described herein. These tests show that using multigrid as a preconditioner for a Krylov method yields improvements in both robustness and efficiency as compared to using multigrid as a solver. They also demonstrate that with a time step 100--1000 times larger than that permitted by an explicit IB method, the multigrid-preconditioned implicit IB method is approximately 50--200 times more efficient than the explicit method