a power penalty method for a 2d fractional partial differential linear complementarity problem governing two asset american option pricing in this paper we propose a power penalty method for a linear complementarity problem lcp involving a fractional partial differential operator in two spatial dimensions arising in pricing american options on two underlying assets whose prices follow two independent geometric lvy processes we first approximate the lcp by a nonlinear 2d fractional partial differential equation fpde with a penalty term we then prove that the solution to the fpde converges to that of the lcp in a sobolev norm at an exponential rate depending on the parameters used in the penalty term the 2d fpde is discretized by a 2nd order finite difference method in space and cranknicolson method in time numerical experiments on a model basket option pricing problem were performed to demonstrate the convergent rates and the effectiveness of the penalty method

comments on exact solutions of electromagnetic fields in both near and far zones radiated by thin circular loop antennas a general representation with reply this paper presents an alternative vector analysis of the electromagnetic em fields radiated from thin circular loop antennas of arbitrary radius a this method which employs the dyadic green s function in the derivation of the em radiated fields makes the analysis more general compact and straightforward than those two methods published recently by werner 1996 and overfelt 1996 both near and far zones are considered so that the em radiated fields are expressed in terms of the vector wave eigenfunctions not only the exact solution of the em fields in the near and far zones outside the region where r a is derived by the use of the spherical hankel function of the first kind but also the closed series form of the em fields radiated in the near zone inside the region 0 spl les r a is obtained in series of the spherical bessel functions of the first kind as an example a fourier cosine series is used to expand an arbitrary current distribution along the loop and the exact representations of the em radiated fields due to the loop everywhere are obtained in closed form the closed form reduces to those for the sinusoidal current loop and further for the uniform current loop validity of the approximate formulas is discussed and clarified error analysis based on numerical computations of the radiated fields is also given to show the accuracy of the limiting cases

european option pricing with transaction costs the authors consider the problem of pricing european options in a market model similar to the black scholes one except that proportional transaction charges are levied on all sales and purchases o

exact field expressions for circular loop antennas using spherical functions expansion a new technique is introduced for evaluating the vector potential of circular loop antennas clas with arbitrary current distribution and dimensions which can be used to obtain the exact electromagnetic fields in near and far regions of the antenna the technique is based on the spherical bessel functions sbfs and associated legendre polynomials alps expansion this expansion is used to decouple the azimuth variables of the integrand of the vector potential integrals from the other variables in a separate functions that depend exponentially on azimuth variables this technique reduces greatly the mathematical operations for evaluation of the vector potential to only one simple integration the technique is used to obtain exact electromagnetic fields expressions for a thin wire cla of arbitrary current distribution and dimension which is valid everywhere except on the loop the well known far fields for the cla of arbitrary current distribution as well as near and far fields for uniform current loop and small loop are obtained from the general field expressions presented in this paper as an special cases

new exact solution procedure for the near fields of the general thin circular loop antenna an analytical method is presented which gives the various fields including the scalar potential of a general circular loop antenna in cylindrical coordinates in terms of six separate series three of the series are elementary and the other three involve legendre functions of the second kind and half integral order the results obtained are compared with direct numerical integration and with two far field formulae the standard textbook expression and what appears to be a new formula which gives more accurate results the series solution is more robust than numerical integration for the higher fourier terms of the general solution close to the loop the solution has also been compared with the solution of werner and it agrees exactly the solution presented here gives better convergence close to the loop whereas werner s solution is better far from the loop sample calculations are given for the axisymmetric case

penalty approach to a nonlinear obstacle problem governing american put option valuation under transaction costs we propose a penalty method for a finite dimensional nonlinear complementarity problem ncp arising from the discretization of the infinite dimensional free boundary obstacle problem governing the valuation of american options under transaction costs in this method the ncp is approximated by a system of nonlinear equations containing a power penalty term we show that the mapping involved in the system is continuous and strongly monotone thus the unique solvability of both the ncp and the penalty equation and the exponential convergence of the solution to the penalty equation to that of the ncp are guaranteed by an existing theory numerical results will be presented to demonstrate the convergence rates and usefulness of this penalty method

an exact formulation for the vector potential of a cylindrical antenna with uniformly distributed current and arbitrary radius an exact formulation is given for the vector potential and corresponding electric fields associated with a uniform current cylindrical dipole antenna of arbitrary radius the exact expressions converge rapidly in the induction and near field regions of the antenna they are completely general and independent of the usual restrictions involving the wavelength field point distance dipole radius and length the properties of these expressions allow them to be readily used for the efficient as well as accurate computational modeling of cylindrical wire antennas an exact expression for the cylindrical wire kernel which is analytically as well as numerically convenient to work with is presented it is demonstrated that the integral of the kernel can be expressed in terms of a series which involves a generalized exponential integral and higher order associated integrals several methods for evaluating the generalized exponential integral are discussed including some useful asymptotic expansions a numerically stable forward recurrence relation for the higher order associated integrals is also presented

on the numerical solution of nonlinear black scholes equations nonlinear black scholes equations have been increasingly attracting interest over the last two decades since they provide more accurate values by taking into account more realistic assumptions such as transaction costs risks from an unprotected portfolio large investor s preferences or illiquid markets which may have an impact on the stock price the volatility the drift and the option price itself in this paper we will focus on several models from the most relevant class of nonlinear black scholes equations for european and american options with a volatility depending on different factors such as the stock price the time the option price and its derivatives due to transaction costs we will analytically approach the option price by transforming the problem for a european call option into a convection diffusion equation with a nonlinear term and the free boundary problem for an american call option into a fully nonlinear nonlocal parabolic equation defined on a fixed domain following sevcovic s idea finally we will present the results of different numerical discretization schemes for european options for various volatility models including the leland model the barles and soner model and the risk adjusted pricing methodology model

a penalty method for a fractional order parabolic variational inequality governing american put option valuation a power penalty method is proposed for a parabolic variational inequality or linear complementarity problem lcp involving a fractional order partial derivative arising in the valuation of american options whose underlying stock prices follow a geometric levy process we first approximate the lcp with a nonlinear fractional partial differential equation fpde with a penalty term we then prove that the solution to the nonlinear fpde converges to that of the lcp in a sobolev norm at an exponential rate depending on the parameters used in the penalty term numerical results are presented to demonstrate the convergence rates and usefulness of the penalty method for pricing american put options of this type

irreducible affine varieties over a free group i irreducibility of quadratic equations and nullstellensatz the object of this paper which consists of two parts is to describe irreducible varieties over free groups and to characterize finitely generated fully residually free groups we prove that any variety over a free group f can be defined by a finite number of systems of equations s 1 in triangular form where quadratic words play the role of leading terms algebraically irreducible varieties are exactly the varieties whose coordinate groups are fully residually f the crucial point of the classification of fully residually free groups is to prove that the coordinate groups of irreducible varieties are embeddable into lyndon s free z x group f since every finitely generated fully residually free group is a free factor of the coordinate group of an irreducible variety and the group f is fully residually f we obtain a characterization of finitely generated fully residually free groups as subgroups of f the group f and its subgroups have been studied extensively during the last several years in particular every finitely generated subgroup of f hence every finitely generated fully residually free group can be obtained from free abelian groups of finite rank by finitely many free products with amalgamation and hnn extensions of the type where amalgamated and associated subgroups are free abelian of finite rank in particular this implies that every finitely generated fully residually free group is finitely presented there are three parts to this paper algebraic geometry over free groups the theory of free exponential groups and makanin razborov s machinery to deal with equations over free groups 11 14 13 the algebraic geometry approach has been shown to be very usefull in dealing with equations over groups it provides necessary topological means and a method to transcribe geometric notions into pure group theoretic language following baumslag myasnikov remeslennikov 1 we use the standard algebraic geometry notions such as variety zariski topology irreducibility of varieties radicals and coordinate groups some of the ideas of the algebraic geometry approach go back to r lyndon 9 e rips j stallings 16

convergence analysis of a monotonic penalty method for american option pricing abstract this paper is devoted to study the convergence analysis of a monotonic penalty method for pricing american options a monotonic penalty method is first proposed to solve the complementarity problem arising from the valuation of american options which produces a nonlinear degenerated parabolic pde with black scholes operator based on the variational theory the solvability and convergence properties of this penalty approach are established in a proper infinite dimensional space moreover the convergence rate of the combination of two power penalty functions is obtained

a filter algorithm for nonlinear systems of equalities and inequalities abstract in this paper we propose a filter algorithm for nonlinear systems of equalities and inequalities in the algorithm the systems of nonlinear equalities and inequalities are transformed into a nonlinear programming problem the new algorithm uses the non monotone technique in conjunction with filter idea and a global line search strategy that ensures convergence from any starting point under mild conditions global convergence and local superlinear convergence rate are obtained numerical results demonstrate that the new algorithm is effective

an exact integration procedure for vector potentials of thin circular loop antennas a direct integration procedure for far zone vector potentials of thin circular loop antennas has been known for many years this method is general in the sense that it leads to simple integrals which have closed form solutions for most commonly assumed loop current distributions however a comparable integration technique has not been available for evaluating the more complicated near zone vector potentials this paper introduces a systematic approach for the exact integration of general near zone vector potentials associated with current carrying circular loop antennas a particular example is considered where this new integration technique is used to find exact solutions to the vector potential and electromagnetic field integrals for loops with a fourier cosine series expansion of the current the observation is made that degenerate forms of these exact representations lead to simplified expressions for the important special cases of a uniform and cosinusoidal current loop two equivalent forms of exact series expansions are derived for the uniform current vector potential and field integrals it is shown that the familiar small loop approximations as well as the classical far field expressions may be obtained as limiting cases of the more general exact series representations for the uniform current loop obtained in this paper convenient asymptotic far field expansions are derived for the loop with a cosinusoidal current distribution finally the far field analysis for the cosinusoidal loop is generalized to loops having an arbitrary current represented by a fourier cosine series

irreducible affine varieties over a free group ii systems in triangular quasi quadratic form and description of residually free groups abstract we shall prove the conjecture of myasnikov and remeslennikov 4 which states that a finitely generated group is fully residually free every finite set of nontrivial elements has nontrivial images under some homomorphism into a free group if and only if it is embeddable in the lyndon s exponential groupfz x which is thez x completion of the free group herez x is the ring of polynomials of one variable with integer coefficients historically lyndon s attempts to solve tarski s famous problem concerning the elementary equivalence of free groups of different ranks led him to introducefz x an free group is a groupgsuch that the class of formulas true ing is the same as the class of formulas true in a nonabelian free group a finitely generated group is free if and only if it is fully residually free 22 our result gives an algebraic description of free groups we shall give an algorithm to represent a solution set of an arbitrary system of equations overfas a union of finite number of irreducible components in the zariski topology onfn the solution set for every system is contained in the solution set of a finite number of systems in triangular form with quadratic words as leading terms the possibility of such a decomposition for a solution set was conjectured by razborov in 20 and also by rips we shall give a description of systems of equations determining irreducible components using methods developed in 13 19 it is possible to find some of these methods in 18 we are thankful to e rips for attracting our attention to these techniques

augmented lagrangian method applied to american option pricing the american option pricing problem is originally formulated as a stochastic optimal stopping time problem it is also equivalent to a variational inequality problem or a complementarity problem involving the black scholes partial differential operator in this paper the corresponding variational inequality problem is discretized by using a fitted finite volume method based on the discretized form an algorithm is developed by applying augmented lagrangian method alm to the valuation of the american option convergence properties of alm are considered by empirical numerical experiments we conclude that alm is more effective than penalty method and lagrangian method and comparable with the projected successive overrelaxation method psor furthermore numerical results show that alm is more robust in terms of computation time under changes in market parameters interest rate and volatility

pricing american bond options using a penalty method we develop a novel numerical method to price american options on a discount bond under the cox ingrosll ross cir model which is governed by a partial differential complementarity problem we first propose a penalty approach to this complementarity problem resulting in a nonlinear partial differential equation pde to numerically solve this nonlinear pde we develop a novel fitted finite volume method for the spatial discretization coupled with a fully implicit time stepping scheme we show that this full discretization scheme is consistent stable and monotone and hence the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed to solve the discretized nonlinear system we design an iterative method and prove that the method is convergent numerical results are presented to demonstrate the accuracy efficiency and robustness of our methods

edge effects in axial arrays of circular rings a boundary value problem is solved for the current induced by any source in particular for an incident plane wave on any number of identical rings that are arranged to form an axisymmetric linear array the net aperture distribution along the finite array is approximated by a combination of three physically identifiable components a term from the related infinite array plus contributions from two independent edge effect waves that originate from each of the two ends however if individually resonant loops are closer than a critical spacing the entire structure exhibits a wild resonance and the simple approximation becomes invalid

entire domain mom analysis of an array of arbitrarily oriented circular loop antennas a general formulation this paper presents a general formulation for the entire domain method of moments analysis of an n element array of arbitrarily oriented coupled circular loop antennas the orientations of the loops with respect to each other are defined through the use of eulerian angles the positions of the loops in space are defined by translation vectors from the center of a loop to another by making use of coordinate transformations and vector algebra a concise set of linear equations for solving the currents on the loops is obtained the formulation is sufficiently general to be used for all the previously reported coupled loop orientations it is also ideal for the study of skewed circular loops which may be oriented in innumerable ways and in turn demand a general formulation for their analysis the use of the general formulation is illustrated by analyzing the coupling currents between collocated skewed loops the formulation is also applied to study the effects on the transfer impedance between a pair of noncollocated skewed loops as one of the loops is rotated about various axes

an upwind finite difference method for a nonlinear black scholes equation governing european option valuation under transaction costs in this paper we develop a numerical method for a nonlinear parabolic partial differential equation arising from pricing european options under transaction costs the method is based on an upwind finite difference scheme for the spatial discretization and a fully implicit time stepping scheme we prove that the system matrix from this scheme is an m matrix and that the approximate solution converges unconditionally to the viscosity solution to the equation by showing that the scheme is consistent monotone and unconditionally stable a newton iterative algorithm is proposed for solving the discretized nonlinear system of which the jacobian matrix is shown to be also an m matrix numerical experiments are performed to demonstrate the accuracy and robustness of the method

the use of grossone in mathematical programming and operations research the concepts of infinity and infinitesimal in mathematics date back to anciens greek and have always attracted great attention very recently a new methodology has been proposed by sergeyev 10 for performing calculations with infinite and infinitesimal quantities by introducing an infinite unit of measure expressed by the numeral grossone an important characteristic of this novel approach is its attention to numerical aspects in this paper we will present some possible applications and use of in operations research and mathematical programming in particular we will show how the use of can be beneficial in anti cycling procedure for the well known simplex method for solving linear programming problems and in defining exact differentiable penalty functions in nonlinear programming

decreasing functions with applications to penalization the theory of increasing positively homogeneous functions defined on the positive orthant is applied to the class of decreasing functions a multiplicative version of the inf convolution operation is studied for decreasing functions modified penalty functions for some constrained optimization problems are introduced that are in general nonlinear with respect to the objective function of the original problem as the perturbation function of a constrained optimization problem is decreasing the theory of decreasing functions is subsequently applied to the study of modified penalty functions the zero duality gap property and the exact penalization

a penalty method for a mixed nonlinear complementarity problem abstract we propose a power penalty method for a mixed nonlinear complementarity problem mncp and show that the solution to the penalty equation converges to that of the mncp exponentially as the penalty parameter approaches infinity provided that the mapping involved in the mncp is both continuous and monotone furthermore a convergence theorem is established when the monotonicity assumption on the mapping is removed to demonstrate the usefulness and the convergence rates of this method we design a non trivial test mncp problem arising in shape preserving bi harmonic interpolation and apply our method to this test problem the numerical results confirm our theoretical findings

numerical solution of linear and nonlinear black scholes option pricing equations this paper deals with the numerical solution of black scholes option pricing partial differential equations by means of semidiscretization technique for the linear case a fourth order discretization with respect to the underlying asset variable allows a highly accurate approximation of the solution for the nonlinear case of interest modeling option pricing with transaction costs semidiscretization technique provides a competitive numerical solution with respect to others recently given in b during m fournier a jungel convergence of a high order compact finite difference scheme for a nonlinear black scholes equation esaim math modelling numer anal modelisation mathematique et analyse numerique 38 2004 359 369 b during black scholes type equations mathematical analysis parameter identification numerical solution dissertation university mainz july 2005

the exact kernel for cylindrical antenna in the numerical calculation for a cylindrical antenna the thin wire kernel is widely used as a convenient approximation to the exact kernel of the electric field equation however this thin wire approximation is valid only if the radius of wire is much less than its length and the wavelength furthermore arbitrarily using this approximation may not obtain the convergent outcomes the expression of the exact kernel for the cylindrical antenna which would be useful in the numerical work is presented

on a new cylindrical harmonic representation for spherical waves an exact series representation is presented for integrals whose integrands are products of cosine and spherical wave functions where the argument of the cosine term can be any integral multiple n of the azimuth angle spl phi this series expansion is shown to have the following form i n e sup jkr0 r sub 0 spl delta sub no jk spl sigma sub m 1 sup spl infin c m n k sup 2 spl rho spl rho 0 m h sub m sup 2 kr sub 0 kr sub 0 sup m it is demonstrated that in the special cases n 0 and n 1 this series representation corresponds to existing expressions for the cylindrical wire kernel and the uniform current circular loop vector potential respectively a new series representation for spherical waves in terms of cylindrical harmonics is then derived using this general series representation finally a closed form far field approximation is developed and is shown to reduce to existing expressions for the cylindrical wire kernel and the uniform current loop vector potential as special cases

the equation xp yq zr and groups that act freely on l trees let g be a finitely generated group that acts freely on a lambda tree where lambda is an ordered abelian group and let x y z be elements in g we show that if x p y q z r with integers p q r geq 4 then x y and z commute

pricing options under jump diffusion processes with fitted finite volume method this paper develops a numerical method for a partial integro differential equation and a partial integro differential complementarity problem arising from european and american options valuations respectively when the underlying assets are driven by a jump diffusion process the method is based on a fitted finite volume scheme for the spatial discretization and the crank nicolson scheme for the time discretization the fully discretized system is solved by an iterative method coupled with an fft for the evaluation of the discretized integral term while the constraint in the american option model is imposed by adding a penalty term to the original partial integro differential complementarity problem we show that the system matrix of the discretized system is an m matrix and propose an algorithm for solving the discretized system numerical experiments are implemented to show the efficiency and robustness of this method

response to comments on exact field expressions for circular loop antennas using spherical functions expansion the author would like to thank the commentators dr zheng and dr li for reviewing and commenting on the work reported in the author disagrees with their conclusion that the author in is criticizing the merits of their published papers the author is merely explaining the differences between the author s work in and the commentators work reported in this should not detract from the contribution they made in as indicated in their comments

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence