operator theory on the wkb method and bremmer series the well known wkb method and bremmer series in mathematical physics have various modifications and applications basically they are all limited to the finite dimensional case such methods have been extended to the infinite dimensional case in this paper i e operators on a general normed linear space this enables us to apply the methods to solve the problem of transport of radiation through a layer of cloud the derivation is based on input output states and linear operators some elementary properties are exploited along with physical interpretations the convergence of the series is established an example is given to demonstrate the methods and the rate of convergence

a new efficient numerical method for solving american option under regime switching model a system of coupled free boundary problems describing american put option pricing under regime switching is considered in order to build numerical solution firstly a front fixing transformation is applied transformed problem is posed on multidimensional fixed domain and is solved by explicit finite difference method the numerical scheme is conditionally stable and is consistent with the first order in time and second order in space the proposed approach allows the computation not only of the option price but also of the optimal stopping boundary numerical examples demonstrate efficiency and accuracy of the proposed method the results are compared with other known approaches to show its competitiveness

high order compact finite difference scheme for option pricing in stochastic volatility models we derive a new high order compact finite difference scheme for option pricing in stochastic volatility models the scheme is fourth order accurate in space and second order accurate in time under some restrictions theoretical results like unconditional stability in the sense of von neumann are presented where the analysis becomes too involved we validate our findings by a numerical study numerical experiments for the european option pricing problem are presented we observe fourth order convergence for non smooth payoff

valuation of stock loans with regime switching this paper is concerned with stock loan valuation in which the underlying stock price is dictated by geometric brownian motion with regime switching the stock loan pricing is quite different from that for standard american options because the associated variational inequalities may have infinitely many solutions in addition the optimal stopping time equals infinity with positive probability variational inequalities are used to establish values of stock loans and reasonable values of critical parameters such as loan sizes loan rates and service fees in terms of certain algebraic equations numerical examples are included to illustrate the results

methods for pricing american options under regime switching we analyze a number of techniques for pricing american options under a regime switching stochastic process the techniques analyzed include both explicit and implicit discretizations with the focus being on methods which are unconditionally stable in the case of implicit methods we also compare a number of iterative procedures for solving the associated nonlinear algebraic equations numerical tests indicate that a fixed point policy iteration coupled with a direct control formulation is a reliable general purpose method finally we remark that we formulate the american problem as an abstract optimal control problem hence our results are applicable to more general problems as well

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

stability of hypersurfaces with constant mean curvature soit m n compacte orientable et soit x m n r n 1 une immersion a courbure moyenne constante non nulle alors x est stable si et seulement si x m r n 1 est une sphere s n r n 1

far field boundary conditions for black scholes equations in this paper we consider the partial differential equations approach for valuing european style options in order to solve the equations numerically by finite difference or finite element methods it is necessary to introduce an artificial boundary in order to make the computational domain bounded we derive pointwise bounds for the error caused by various boundary conditions imposed on the artificial boundary the error estimates can be used to determine a priori a suitable location for the artificial boundary in terms of a given error tolerance

constructing positive reliable numerical solution for american call options a new front fixing transformation is applied to the black scholes equation for the american call option pricing problem the transformed non linear problem involves homogeneous boundary conditions independent of the free boundary the numerical solution by an explicit finite difference method is positive and monotone stability and consistency of the scheme are studied the explicit proposed method is compared with other competitive implicit ones from the points of view accuracy and computational cost

numerical analysis and simulation of option pricing problems modeling illiquid markets this paper deals with the numerical analysis and simulation of nonlinear black scholes equations modeling illiquid markets where the implementation of a dynamic hedging strategy affects the price process of the underlying asset a monotone difference scheme ensuring nonnegative numerical solutions and avoiding unsuitable oscillations is proposed stability properties and consistency of the scheme are studied and numerical simulations involving changes in the market liquidity parameter are included

option pricing with regime switching by trinomial tree method we present a fast and simple tree model to price simple and exotic options in markov regime switching model mrsm with multi regime we modify the trinomial tree model of boyle 1986 12 by controlling the risk neutral probability measure in different regime states to ensure that the tree model can accommodate the data of all different regimes at the same time preserving its combining tree structure in mrsm the market might not be complete therefore we provide some ideas and discussions on managing the regime switching risk in support of our results

computing american option price under regime switching with rationality parameter american put option pricing under regime switching is modelled by a system of coupled partial differential equations the proposed model combines better the reality of the market by incorporating the regime switching jointly with the emotional behaviour of traders using the rationality parameter approach recently introduced by tagholt gad and lund petersen to cope with possible irrational exercise policy the classical rational exercise is recovered as a limit case of the rational parameter the resulting nonlinear system of pdes is solved by a weighted finite difference method also known as method in order to avoid the need of an iterative method for nonlinear system the term with rationality parameter and the coupling term are treated explicitly next the resulting linear system is solved by thomas algorithm stability conditions for the numerical scheme are studied by using von neumann approach numerical examples illustrate the efficiency and accuracy of the proposed method