Valuation of two-factor options under the Merton jump-diffusion model using orthogonal spline wavelets

summary:This paper addresses the two-asset Merton model for option pricing represented by non-stationary integro-differential equations with two state variables. The drawback of most classical methods for solving these types of equations is that the matrices arising from discretization are full and ill-conditioned. In this paper, we first transform the equation using logarithmic prices, drift removal, and localization. Then, we apply the Galerkin method with a recently proposed orthogonal cubic spline-wavelet basis combined with the Crank-Nicolson scheme. We show that the proposed method has many benefits. First, as is well-known, the wavelet-Galerkin method leads to sparse matrices, which can be solved efficiently using iterative methods. Furthermore, since the basis functions are cubic splines, the method is higher-order convergent. Due to the orthogonality of the basis functions, the matrices are well-conditioned even without preconditioning, computation is simplified, and the required number of iterations is less than for non-orthogonal cubic spline-wavelet bases. Numerical experiments are presented for European-style options on the maximum of two assets

Wavelet method for option pricing under the two-asset Merton jump-diffusion model

summary:This paper examines the pricing of two-asset European options under the Merton model represented by a nonstationary integro-differential equation with two state variables. For its numerical solution, the wavelet-Galerkin method combined with the Crank-Nicolson scheme is used. A drawback of most classical methods is the full structure of discretization matrices. In comparison, the wavelet method enables the approximation of discretization matrices with sparse matrices. Sparsity is essential for the efficient application of iterative methods in solving the resulting systems and the efficient computation of the matrices arising from the discretization of integral terms. To illustrate the efficiency of the method, we provide the results of numerical experiments concerning a European option on the maximum of two assets

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 $L^2$-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

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 Black–Scholes 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

Adaptive wavelet scheme for convection-diffusion equations

One of the most important part of adaptive wavelet methods is an efficient approximate multi- plication of stiffness matrices with vectors in wavelet coordinates. Although there are known algorithms to perform it in linear complexity, the application of them is relatively time con- suming and its implementation is very difficult. Therefore, it is necessary to develop a well- conditioned wavelet basis with respect to which both the mass and stiffness matrices are sparse in the sense that the number of nonzero elements in any column is bounded by a constant. Then, matrix-vector multiplication can be performed exactly with linear complexity. We present here a wavelet basis on the interval with respect to which both the mass and stiffness matrices corresponding to the one-dimensional Laplacian are sparse. Consequently, the stiffness matrix corresponding to the n-dimensional Laplacian in tensor product wavelet basis is also sparse. Moreover, the constructed basis has an excellent condition number. In this contribution, we shortly review this construction and show several numerical tests

Multiwavelets based on hermite cubic splines

the convection-diffusion equation. We use an implicit scheme for the time discretization an

Agency employment in the Czech Republic and problems connected with it

This dissertation drala with one one form of employing, that is on the labour market especially in the last days very sought, agency employing. It looks on this way of employing from the legal view and analyses other aspects of this employing way, law and obligation individual sides of this working relationship and last but not least possibilities of control and obligation sanction from the state side. Chapter one deals with definition of the term "agency employing" from the economic and law point of view. Next chapter deals with law aspects agency employing in conditions of the Czech labour market. The theme of the third chapter is other subject on the Czech labour market, it also can negotiate employment, employment office. Next chapters deals with other aspects of agency employing , under the Czech and also international conditions, law and obligations of agency employers and at least main problems agency employing and its possibilities of solution. The end is focused on the current difficult situation on the labour market

Cubic spline wavelets with four vanishing moments on the interval and their applications to option pricing under Kou model

The paper is concerned with the construction of a cubic spline wavelet basis on the unit interval and an adaptation of this basis to the first-order homogeneous Dirichlet boundary conditions. The wavelets have four vanishing moments and they have the shortest possible support among all cubic spline wavelets with four vanishing moments corresponding to B-spline scaling functions. We provide a rigorous proof of the stability of the basis in the space L-2 (0, 1) or its subspace incorporating boundary conditions. To illustrate the applicability of the constructed bases, we apply the wavelet-Galerkin method to option pricing under the double exponential jump-diffusion model and we compare the results with other cubic spline wavelet bases and with other methods