57 research outputs found
Numerical method for pricing governing American options under fractional Black-Scholes model
In this paper we develop a numerical approach to a fractional-order differential linear complementarity problem (LCP) arising in pricing European and American options under a geometric LĂ©vy process. The (LCP) is first approximated by a penalized nonlinear fractional Black-Scholes (fBS) equation. To numerically solve this nonlinear (fBS), we use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of Crank-Nicolson method and a cubic spline collocation method, respectively. This method exhibits a second order of convergence in space, in time and also has an acceptable speed in comparison with some existing methods. We will compare our results with some recently proposed approaches. Keywords: Geometric LĂ©vy process, fractional Black-Scholes, Crank-Nicolson scheme, Spline collocation, Free Boundary Value Problem
An efficient numerical method based on exponential B-splines for time-fractional Black-Scholes equation governing European options
In this paper a time-fractional Black-Scholes model (TFBSM) is considered to
study the price change of the underlying fractal transmission system. We
develop and analyze a numerical method to solve the TFBSM governing European
options. The numerical method combines the exponential B-spline collocation to
discretize in space and a finite difference method to discretize in time. The
method is shown to be unconditionally stable using von-Neumann analysis. Also,
the method is proved to be convergent of order two in space and is
time, where is order of the fractional derivative. We implement the
method on various numerical examples in order to illustrate the accuracy of the
method, and validation of the theoretical findings. In addition, as an
application, the method is used to price several different European options
such as the European call option, European put option, and European double
barrier knock-out call option.Comment: 34 pages, 12 figure
Collocation method based on modified âcubicâ B-spline âfor option pricing âmodels
Collocationââ âmethod âbased âon âmodifiedâ cubic B-spline functions âhas âbeen âdevelopedâ âfor âthe âvaluation âââof Europeanâ, âAmerican and Barrier options of single âasset. âThe ânew âapproach âcontains ââdiscretizing âofâ tââemporal âderivativeâ âusing âfinite âdifference âapproximations âand âapproximatingâ the option price with the âmodifiedâ B-spline functionsâ. âStability of this method has been discussed and shown that it is unconditionally stableâ. âThe âefficiency âof âtheâ âproposed âmethod âis âtested âby âdifferent âexamplesâââ.
Numerical method for pricing American options under regime-switching jump-diffusion models
Our concern in this paper is to solve the pricing problem for American options in a Markov-modulated jump-diffusion model, based on a cubic spline collocation method. In this respect, we solve a set of coupled partial integro-differential equations PIDEs with the free boundary feature by using the horizontal method of lines to discretize the temporal variable and the spatial variable by means of Crank-Nicolson scheme and a cubic spline collocation method, respectively. This method exhibits a second order of convergence in space, in time and also has an acceptable speed in comparison with some existing methods. We will compare our results with some recently proposed approaches. Keywords: American Option, Regime-Switching, Crank-Nicolson scheme, Spline collocation, Free Boundary Value Problem
New collocation path-following approach for the optimal shape parameter using Kernel method
The goal of this work is to develop a numerical method combining Radial Basic Functions (RBF) kernel and a high order algorithm based on Taylor series and homotopy continuation method. The local RBF approximation applied in strong form allows us to overcome the difficulties of numerical integration and to treat problems of large deformations. Furthermore, the high order algorithm enables to transform the nonlinear problem to a set of linear problems. Determining the optimal value of the shape parameter in RBF kernel is still an outstanding research topic. This optimal value depends on density and distribution of points and the considered problem for e.g. boundary value problems, integral equations, delay-differential equations etc. These have been extensively attempts in literature which end up choosing this optimal value by tests and error or some other ad-hoc means. Our contribution in this paper is to suggest a new strategy using radial basis functions kernel with an automatic reasonable choice of the shape parameter in the nonlinear case which depends on the accuracy and stability of the results. The computational experiments tested on some examples in structural analysis are performed and the comparison with respect to the state of art algorithms from the literature is given
Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance
Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.South Afric
Review of modern numerical methods for a simple vanilla option pricing problem
Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the BlackâScholes model usually serve as a bench-mark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are depend-ent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better
Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance
Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from
the value of some underlying security. Most options can be grouped into either of
the two categories: European options which can be exercised only on the expiration
date, and American options which can be exercised on or before the expiration date.
American options are much harder to deal with than European ones. The reason being
the optimal exercise policy of these options which led to free boundary problems. Ever
since the seminal work of Black and Scholes [J. Pol. Bean. 81(3) (1973), 637-659],
the differential equation approach in pricing options has attracted many researchers.
Recently, numerical singular perturbation techniques have been used extensively for
solving many differential equation models of sciences and engineering. In this thesis,
we explore some of those methods which are based on spline approximations to solve
the option pricing problems. We show a systematic construction and analysis of these
methods to solve some European option problems and then extend the approach to
solve problems of pricing American options as well as some exotic options. Proposed
methods are analyzed for stability and convergence. Thorough numerical results are
presented and compared with those seen in the literature
Option Pricing under Multifactor Black-Scholes Model Using Orthogonal Spline Wavelets
The paper focuses on pricing European-style options on several underlying
assets under the Black-Scholes model represented by a nonstationary partial
differential equation. The proposed method combines the Galerkin method with
-orthogonal sparse grid spline wavelets and the Crank-Nicolson scheme with
Rannacher time-stepping. To this end, we construct an orthogonal cubic spline
wavelet basis on the interval satisfying homogeneous Dirichlet boundary
conditions and design a wavelet basis on the unit cube using the sparse tensor
product. The method brings the following advantages. First, the number of basis
functions is significantly smaller than for the full grid, which makes it
possible to overcome the so-called curse of dimensionality. Second, some
matrices involved in the computation are identity matrices, which significantly
simplifies and streamlines the algorithm, especially in higher dimensions.
Further, we prove that discretization matrices have uniformly bounded condition
numbers, even without preconditioning, and that the condition numbers do not
depend on the dimension of the problem. Due to the use of cubic spline
wavelets, the method is higher-order convergent. Numerical experiments are
presented for options on the geometric average.Comment: 43 pages, 10 figure
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