3,146 research outputs found
Application of Operator Splitting Methods in Finance
Financial derivatives pricing aims to find the fair value of a financial
contract on an underlying asset. Here we consider option pricing in the partial
differential equations framework. The contemporary models lead to
one-dimensional or multidimensional parabolic problems of the
convection-diffusion type and generalizations thereof. An overview of various
operator splitting methods is presented for the efficient numerical solution of
these problems.
Splitting schemes of the Alternating Direction Implicit (ADI) type are
discussed for multidimensional problems, e.g. given by stochastic volatility
(SV) models. For jump models Implicit-Explicit (IMEX) methods are considered
which efficiently treat the nonlocal jump operator. For American options an
easy-to-implement operator splitting method is described for the resulting
linear complementarity problems.
Numerical experiments are presented to illustrate the actual stability and
convergence of the splitting schemes. Here European and American put options
are considered under four asset price models: the classical Black-Scholes
model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV
model with jumps
Pricing European and American Options under Heston Model using Discontinuous Galerkin Finite Elements
This paper deals with pricing of European and American options, when the
underlying asset price follows Heston model, via the interior penalty
discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM
space discretization with Rannacher smoothing as time integrator with nonsmooth
initial and boundary conditions are illustrated for European vanilla options,
digital call and American put options. The convection dominated Heston model
for vanishing volatility is efficiently solved utilizing the adaptive dGFEM.
For fast solution of the linear complementary problem of the American options,
a projected successive over relaxation (PSOR) method is developed with the norm
preconditioned dGFEM. We show the efficiency and accuracy of dGFEM for option
pricing by conducting comparison analysis with other methods and numerical
experiments
Differential quadrature method for space-fractional diffusion equations on 2D irregular domains
In mathematical physics, the space-fractional diffusion equations are of
particular interest in the studies of physical phenomena modelled by L\'{e}vy
processes, which are sometimes called super-diffusion equations. In this
article, we develop the differential quadrature (DQ) methods for solving the 2D
space-fractional diffusion equations on irregular domains. The methods in
presence reduce the original equation into a set of ordinary differential
equations (ODEs) by introducing valid DQ formulations to fractional directional
derivatives based on the functional values at scattered nodal points on problem
domain. The required weighted coefficients are calculated by using radial basis
functions (RBFs) as trial functions, and the resultant ODEs are discretized by
the Crank-Nicolson scheme. The main advantages of our methods lie in their
flexibility and applicability to arbitrary domains. A series of illustrated
examples are finally provided to support these points.Comment: 25 pages, 25 figures, 7 table
Numerical solving unsteady space-fractional problems with the square root of an elliptic operator
An unsteady problem is considered for a space-fractional equation in a
bounded domain. A first-order evolutionary equation involves the square root of
an elliptic operator of second order. Finite element approximation in space is
employed. To construct approximation in time, regularized two-level schemes are
used. The numerical implementation is based on solving the equation with the
square root of the elliptic operator using an auxiliary Cauchy problem for a
pseudo-parabolic equation. The scheme of the second-order accuracy in time is
based on a regularization of the three-level explicit Adams scheme. More
general problems for the equation with convective terms are considered, too.
The results of numerical experiments are presented for a model two-dimensional
problem.Comment: 21 pages, 7 figures. arXiv admin note: substantial text overlap with
arXiv:1412.570
Discrete maximal regularity of time-stepping schemes for fractional evolution equations
In this work, we establish the maximal -regularity for several time
stepping schemes for a fractional evolution model, which involves a fractional
derivative of order , , in time. These schemes
include convolution quadratures generated by backward Euler method and
second-order backward difference formula, the L1 scheme, explicit Euler method
and a fractional variant of the Crank-Nicolson method. The main tools for the
analysis include operator-valued Fourier multiplier theorem due to Weis [48]
and its discrete analogue due to Blunck [10]. These results generalize the
corresponding results for parabolic problems
Solving Nonlinear Parabolic Equations by a Strongly Implicit Finite-Difference Scheme
We discuss the numerical solution of nonlinear parabolic partial differential
equations, exhibiting finite speed of propagation, via a strongly implicit
finite-difference scheme with formal truncation error . Our application of interest is the spreading of
viscous gravity currents in the study of which these type of differential
equations arise. Viscous gravity currents are low Reynolds number (viscous
forces dominate inertial forces) flow phenomena in which a dense, viscous fluid
displaces a lighter (usually immiscible) fluid. The fluids may be confined by
the sidewalls of a channel or propagate in an unconfined two-dimensional (or
axisymmetric three-dimensional) geometry. Under the lubrication approximation,
the mathematical description of the spreading of these fluids reduces to
solving the so-called thin-film equation for the current's shape . To
solve such nonlinear parabolic equations we propose a finite-difference scheme
based on the Crank--Nicolson idea. We implement the scheme for problems
involving a single spatial coordinate (i.e., two-dimensional, axisymmetric or
spherically-symmetric three-dimensional currents) on an equispaced but
staggered grid. We benchmark the scheme against analytical solutions and
highlight its strong numerical stability by specifically considering the
spreading of non-Newtonian power-law fluids in a variable-width confined
channel-like geometry (a "Hele-Shaw cell") subject to a given mass
conservation/balance constraint. We show that this constraint can be
implemented by re-expressing it as nonlinear flux boundary conditions on the
domain's endpoints. Then, we show numerically that the scheme achieves its full
second-order accuracy in space and time. We also highlight through numerical
simulations how the proposed scheme accurately respects the mass
conservation/balance constraint.Comment: 36 pages, 9 figures, Springer book class; v2 includes improvements
and corrections; to appear as a contribution in "Applied Wave Mathematics II
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