31,050 research outputs found
Adjoint methods for computing sensitivities in local volatility surfaces
In this paper we present the adjoint method of computing sensitivities of option prices with respect to nodes in the local volatility surface. We first introduce the concept of algorithmic differentiation and how it relates to\ud
path-wise sensitivity computations within a Monte Carlo framework. We explain the two approaches available: forward mode and adjoint mode. We illustrate these concepts on the simple example of a model with a geometric Brownian motion driving the underlying price process, for which\ud
we compute the Delta and Vega in forward and adjoint mode. We then go on to explain in full detail how to apply these ideas to a model where the underlying has a volatility term defined by a local volatility surface. We provide source codes for both the simple and the more complex case and\ud
analyze numerical results to show the strengths of the adjoint approach
Multigrid Methods in Lattice Field Computations
The multigrid methodology is reviewed. By integrating numerical processes at
all scales of a problem, it seeks to perform various computational tasks at a
cost that rises as slowly as possible as a function of , the number of
degrees of freedom in the problem. Current and potential benefits for lattice
field computations are outlined. They include: solution of Dirac
equations; just operations in updating the solution (upon any local
change of data, including the gauge field); similar efficiency in gauge fixing
and updating; operations in updating the inverse matrix and in
calculating the change in the logarithm of its determinant; operations
per producing each independent configuration in statistical simulations
(eliminating CSD), and, more important, effectively just operations per
each independent measurement (eliminating the volume factor as well). These
potential capabilities have been demonstrated on simple model problems.
Extensions to real life are explored.Comment: 4
Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems
In this paper we propose a general framework for the uncertainty
quantification of quantities of interest for high-contrast single-phase flow
problems. It is based on the generalized multiscale finite element method
(GMsFEM) and multilevel Monte Carlo (MLMC) methods. The former provides a
hierarchy of approximations of different resolution, whereas the latter gives
an efficient way to estimate quantities of interest using samples on different
levels. The number of basis functions in the online GMsFEM stage can be varied
to determine the solution resolution and the computational cost, and to
efficiently generate samples at different levels. In particular, it is cheap to
generate samples on coarse grids but with low resolution, and it is expensive
to generate samples on fine grids with high accuracy. By suitably choosing the
number of samples at different levels, one can leverage the expensive
computation in larger fine-grid spaces toward smaller coarse-grid spaces, while
retaining the accuracy of the final Monte Carlo estimate. Further, we describe
a multilevel Markov chain Monte Carlo method, which sequentially screens the
proposal with different levels of approximations and reduces the number of
evaluations required on fine grids, while combining the samples at different
levels to arrive at an accurate estimate. The framework seamlessly integrates
the multiscale features of the GMsFEM with the multilevel feature of the MLMC
methods following the work in \cite{ketelson2013}, and our numerical
experiments illustrate its efficiency and accuracy in comparison with standard
Monte Carlo estimates.Comment: 29 pages, 6 figure
Parameter estimation by implicit sampling
Implicit sampling is a weighted sampling method that is used in data
assimilation, where one sequentially updates estimates of the state of a
stochastic model based on a stream of noisy or incomplete data. Here we
describe how to use implicit sampling in parameter estimation problems, where
the goal is to find parameters of a numerical model, e.g.~a partial
differential equation (PDE), such that the output of the numerical model is
compatible with (noisy) data. We use the Bayesian approach to parameter
estimation, in which a posterior probability density describes the probability
of the parameter conditioned on data and compute an empirical estimate of this
posterior with implicit sampling. Our approach generates independent samples,
so that some of the practical difficulties one encounters with Markov Chain
Monte Carlo methods, e.g.~burn-in time or correlations among dependent samples,
are avoided. We describe a new implementation of implicit sampling for
parameter estimation problems that makes use of multiple grids (coarse to fine)
and BFGS optimization coupled to adjoint equations for the required gradient
calculations. The implementation is "dimension independent", in the sense that
a well-defined finite dimensional subspace is sampled as the mesh used for
discretization of the PDE is refined. We illustrate the algorithm with an
example where we estimate a diffusion coefficient in an elliptic equation from
sparse and noisy pressure measurements. In the example, dimension\slash
mesh-independence is achieved via Karhunen-Lo\`{e}ve expansions
A sparse grid approach to balance sheet risk measurement
In this work, we present a numerical method based on a sparse grid
approximation to compute the loss distribution of the balance sheet of a
financial or an insurance company. We first describe, in a stylised way, the
assets and liabilities dynamics that are used for the numerical estimation of
the balance sheet distribution. For the pricing and hedging model, we chose a
classical Black & Scholes model with a stochastic interest rate following a
Hull & White model. The risk management model describing the evolution of the
parameters of the pricing and hedging model is a Gaussian model. The new
numerical method is compared with the traditional nested simulation approach.
We review the convergence of both methods to estimate the risk indicators under
consideration. Finally, we provide numerical results showing that the sparse
grid approach is extremely competitive for models with moderate dimension.Comment: 27 pages, 7 figures. CEMRACS 201
On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients
In this paper, we propose a novel non-standard Local Fourier Analysis (LFA)
variant for accurately predicting the multigrid convergence of problems with
random and jumping coefficients. This LFA method is based on a specific basis
of the Fourier space rather than the commonly used Fourier modes. To show the
utility of this analysis, we consider, as an example, a simple cell-centered
multigrid method for solving a steady-state single phase flow problem in a
random porous medium. We successfully demonstrate the prediction capability of
the proposed LFA using a number of challenging benchmark problems. The
information provided by this analysis helps us to estimate a-priori the time
needed for solving certain uncertainty quantification problems by means of a
multigrid multilevel Monte Carlo method
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