financial options pricing with regime switching jump diffusions we present a numerical method to evaluate financial derivatives under regime switching jump diffusion models the prices of european and american options are derived by solving a partial integro differential equation pide and a linear complementarity problem lcp respectively we use the implicit method with three time levels to solve the pide and apply it coupled with the operator splitting method to solve the lcp the proposed method has the advantage not only to avoid any fixed point iteration techniques at each time step but also to evaluate directly the prices of the european and american options at all states of the economy it is proved that the implicit method to solve the pide for the european option problem is stable with the second order accuracy in the discrete 2 norm we perform some numerical simulations to illustrate the analysis of the proposed method under the regime switching jump diffusion models

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

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

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

some convergence results for howard s algorithm this paper deals with convergence results of howard s algorithm for the resolution of min a in mathcal a b a x c a 0 where b a is a matrix c a is a vector and mathcal a is a compact set we show a global superlinear convergence result under a monotonicity assumption on the matrices b a extensions of howard s algorithm for a max min problem of the form max b in mathcal b min a in mathcal a b a b x c a b 0 are also proposed in the particular case of an obstacle problem of the form min ax b x g 0 where a is an n times n matrix satisfying a monotonicity assumption we show the convergence of howard s algorithm in no more than n iterations instead of the usual 2 n bound still in the case of an obstacle problem we establish the equivalence between howard s algorithm and a primal dual active set algorithm m hintermuller k ito and k kunisch siam j optim 13 2002 pp 865 888 the algorithms are illustrated on the discretization of nonlinear pdes arising in the context of mathematical finance american option and merton s portfolio problem of front propagation problems and for the double obstacle problem

a new method for analytical solution of inplane free vibration of rectangular orthotropic plates based on the analysis of infinite systems abstract in this paper a new method for in plane free vibration analysis of rectangular orthotropic plate is presented the cases for completely clamped and completely free plates are considered in detail the boundary value problem is essentially reduced to infinite systems of linear algebraic equations new mathematical theory and associated results are presented for investigating the asymptotic behavior of non trivial solution of quasi regular infinite system by enhancing the theory an effective and accurate algorithm for determining the natural frequencies and modes shapes vibrations is developed the accuracy and computational efficiency of the method are demonstrated by examples

exact solutions for the free in plane vibration of rectangular plates with two opposite edges simply supported two distinct types of simple support boundary conditions are formulated for rectangular plates undergoing in plane free vibration each type of simple support is shown to be analogous to the well known simple support edge conditions encountered in the free transverse vibration analysis of rectangular plates it is shown that exact solutions may be obtained for plate free vibration eigenvalues and mode shapes when two opposite plate edges are given either type of simple support the other two edges being given any combination of classical edge conditions a complete analysis is provided for plate in plane vibration with pairs of clamped or free edge conditions imposed on the non simply supported boundaries and exact eigenvalues are tabulated for a large range of plate aspect ratios this appears to be the first thorough study of these in plane vibration problems with exact solutions

dual control monte carlo method for tight bounds of value function in regime switching utility maximization abstract in this paper we study the dual control approach for the optimal asset allocation problem in a continuous time regime switching market we find the lower and upper bounds of the value function that is a solution to a system of fully coupled nonlinear partial differential equations these bounds can be tightened with additional controls to the dual process we suggest a monte carlo algorithm for computing the tight lower and upper bounds and show the method is effective with a variety of utility functions including power non hara and yaari utilities the latter two utilities are beyond the scope of any current methods available in finding the value function

a finite time horizon optimal stopping problem with regime switching we extend the technique developed in e bayraktar a proof of the smoothness of the finite time horizon american put option for jump diffusion http arxiv org abs math 0703782 2007 to a class of finite time horizonal optimal stopping problems under regime switching models which includes the pricing of american put options the construction involved also leads to a computational procedure for the solutions of such optimal stopping problems

a high order finite element scheme for pricing options under regime switching jump diffusion processes this paper considers the numerical pricing of european american and butterfly options whose asset price dynamics follow the regime switching jump diffusion process in an incomplete market structure and using the no arbitrage pricing principle the option pricing problem under the jump modulated regime switching process is formulated as a set of coupled partial integro differential equations describing different states of a markov chain we develop efficient numerical algorithms to approximate the spatial terms of the option pricing equations using linear and quadratic basis polynomial approximations and solve the resulting initial value problem using exponential time integration various numerical examples are given to demonstrate the superiority of our computational scheme with higher level of accuracy and faster convergence compared to existing methods for pricing options under the regime switching model

accurate evaluation of european and american options under the cgmy process a finite difference method for integro differential equations arising from le vy driven asset processes in finance is discussed the equations are discretized in space by the collocation method and in time by an explicit backward differentiation formula the discretization is shown to be second order accurate for a relevant parameter range determining the degree of the singularity in the le vy measure the singularity is dealt with by means of an integration by parts technique an application of the fast fourier transform gives the overall amount of work o n t n log n rendering the method fast

solving complex pide systems for pricing american option under multi state regime switching jump diffusion model abstract based on exponential time differencing approach an efficient second order method is developed for solving systems of partial integral differential equations the method is implemented to solve american options under multi state regime switching with jumps the method is seen to be strongly stable l stable and avoids any spurious oscillations caused by non smooth initial data the predictor corrector nature of the method makes it highly efficient in solving nonlinear pides in each regime with different volatilities and interest rates penalty method approach is applied to handle the free boundary constraint of american options numerical results are presented to illustrate the performance of the method for american options under merton s jump diffusion models pade approximation of matrix exponential functions and partial fraction splitting technique are applied to construct computationally efficient version of the method efficiency accuracy and reliability of the method are compared with those of the existing methods available in the literature

combined fixed point and policy iteration for hamilton jacobi bellman equations in finance implicit methods for hamilton jacobi bellman hjb partial differential equations give rise to highly nonlinear discretized algebraic equations the classic policy iteration approach may not be efficient in many circumstances in this article we derive sufficient conditions to ensure convergence of a combined fixed point policy iteration scheme for the solution of discretized equations numerical examples are included for a singular stochastic control problem arising in insurance a guaranteed minimum withdrawal benefit where the underlying risky asset follows a jump diffusion and an american option assuming a regime switching process

accurate analytical type solutions for the free in plane vibration of clamped and simply supported rectangular plates highly accurate analytical type solutions are obtained by means of the superposition method for the free in plane vibration frequencies and mode shapes of fully clamped rectangular plates the modes are separated into three distinct families and each family is handled separately excellent agreement is encountered when computed results are compared with results obtained by other methods following the obtaining of solutions for the fully clamped plate it is shown how this work can be exploited to obtain exact levy type solutions for the free in plane vibration modes of the plate with simple support along all edges the pure shear and extensional vibration modes characteristic of the latter plate problem are handled separately again excellent agreement with results obtained by other means is encountered a limited number of mode shapes for the fully clamped plate are presented

travelling waves and static structures in a two dimensional exactly solvable reaction diffusion system exact travelling wave solutions of a fisher type reaction diffusion equation in two spatial dimensions are obtained the waves have nontrivial geometries and possess velocities smaller and larger than that of a plane wave the dispersion relationship and velocity curvature dependence generated by these solutions are characterized static structures with and without circular symmetry are constructed

a posteriori error analysis for a class of integral equations and variational inequalities we consider elliptic and parabolic variational equations and inequalities governed by integro differential operators of order 2s in 0 2 our main motivation is the pricing of european or american options under levy processes in particular pure jump processes or jump diffusion processes with tempered stable processes the problem is discretized using piecewise linear finite elements in space and the implicit euler method in time we construct a residual type a posteriori error estimator which gives a computable upper bound for the actual error in h s norm the estimator is localized in the sense that the residuals are restricted to the discrete non contact region numerical experiments illustrate the accuracy of the space and time estimators and show that they can be used to measure local errors and drive adaptive algorithms

finite element solution of diffusion problems with irregular data diffusion problems occuring in practice often involve irregularities in the initial or boundary data resulting in a local break down of the solution s regularity this may drastically reduce the accuracy of discretization schemes over the whole interval of integration unless certain precautions are taken the diagonal pade schemes of order 2 combined with a standard finite element discretization usually require an unnatural step size restriction in order to achieve even locally optimal accuracy it is shown here that this restriction can be avoided by means of a sample damping procedure which preserves the order of the discretization and in the case 1 does not increase the costs

free in plane vibration analysis of rectangular plates by the method of superposition abstract the superposition method is introduced as a means for obtaining analytical type solutions for free in plane vibration of rectangular plates the governing differential equations and boundary conditions are expressed in dimensionless form the problem of free in plane vibration of the completely free rectangular plate is resolved for illustrative purposes convergence is found to be rapid and excellent agreement between computed results and those obtained by previous authors utilizing the rayleigh ritz energy method is obtained it is pointed out that following procedures analogous to those utilized in resolving lateral plate vibration problems in plane free vibration problems related to point supported plates plates with in plane elastic boundary support etc are now amenable to solution by this method

iterative methods for the solution of a singular control formulation of a gmwb pricing problem discretized singular control problems in finance result in highly nonlinear algebraic equations which must be solved at each timestep we consider a singular stochastic control problem arising in pricing a guaranteed minimum withdrawal benefit gmwb where the underlying asset is assumed to follow a jump diffusion process we use a scaled direct control formulation of the singular control problem and examine the conditions required to ensure that a fast fixed point policy iteration scheme converges our methods take advantage of the special structure of the gmwb problem in order to obtain a rapidly convergent iteration the direct control method has a scaling parameter which must be set by the user we give estimates for bounds on the scaling parameter so that convergence can be expected in the presence of round off error example computations verify that these estimates are of the correct order finally we compare the scaled direct control formulation to a formulation based on a block version of the penalty method huang and forsyth in ima j numer anal 32 320 351 2012 we show that the scaled direct control method has some advantages over the penalty method

optimal switching over multiple regimes this paper studies the optimal switching problem for a general one dimensional diffusion with multiple more than two regimes this is motivated in the real options literature by the investment problem of a firm managing several production modes while facing uncertainties a viscosity solutions approach is employed to carry out a fine analysis on the associated system of variational inequalities leading to sharp qualitative characterizations of the switching regions these characterizations in turn reduce the switching problem into one of finding a finite number of threshold values in a state that would trigger switchings the results of our analysis take several qualitatively different forms depending on model parameters and the issue of when and where it is optimal to switch is addressed the general results are then demonstrated by the three regime case where a quasi explicit solution is obtained and a numerical procedure to find these critical values is devised in terms of the expectation functionals of hitting times for one dimensional diffusions

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

a semi analytic valuation of american options under a two state regime switching economy abstract in this study we develop a semi analytic method to evaluate american options under a two state regime switching economy the two free boundaries corresponding to the states divide the pricing domain into two regions a common continuation region and a transition region non linear partial differential equation pde systems are derived under the black scholes framework for each region the laplace transform method is used to solve the pde systems equations for determining the optimal exercise prices are obtained analytically and solved numerically in the laplace space a numerical inversion technique is then used to obtain the free boundaries and the option prices in the original time space the results of various examples show that our technique is efficient and accurate

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

a penalty method for american options with jump diffusion processes the fair price for an american option where the underlying asset follows a jump diffusion process can be formulated as a partial integral differential linear complementarity problem we develop an implicit discretization method for pricing such american options the jump diffusion correlation integral term is computed using an iterative method coupled with an fft while the american constraint is imposed by using a penalty method we derive sufficient conditions for global convergence of the discrete penalized equations at each timestep finally we present numerical tests which illustrate such convergence