Enrique Ponsoda

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].

Structure preserving integrators for solving (non-)linear quadratic optimal control problems with applications to describe the flight of a quadrotor

We present structure preserving integrators for solving linear quadratic optimal control problems. The goal is to build methods which can also be used for the integration of nonlinear problems if they are previously linearized. The equations are solved using an iterative method on a fixed mesh with the constraint that at each iteration one can only use results obtained in previous iterations on that fixed mesh. On the other hand, this problem requires the numerical integration of matrix Riccati differential equations whose exact solution is a symmetric positive definite time-dependent matrix which controls the stability of the equation for the state. This property is not preserved, in general, by the numerical methods. We analyze how to build methods for the linear problem taking into account the previous constraints, and we propose second order exponential methods based on the Magnus series expansion which unconditionally preserve positivity for this problem and analyze higher order Magnus integrators. The performance of the algorithms is illustrated with the stabilization of a quadrotor which is an unmanned aerial vehicle.

A matrix approach to the analytic-numerical solution of mixed partial differential systems☆

E. NAVARRO, E. PONSODA AND L. JODAR Departamento de Matem~tica Aplicada, Universidad Polit~cnica de Valencia P.O. Box 22012, Valencia, Spain (Received and accepted March 1994) Abstract--In this paper, analytic-numerical solutions for initial-boundary value problems related to systems of partial differential equations, are proposed. Given an admissible error e > 0 and a finite domain G, a numerical approximation is constructed in terms of the data, so that the error is uniformly upper bounded by e in G. Keywords--Partial differential system, Eigenvalues bound, Error bound, Schur decomposition, Matrix norms, Matrix functions.

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.

Approximate solutions with a priori error bounds for continuous coefficient matrix Riccati equations

In this paper, the exact solution of the nonsymmetric matrix Riccati equation with continuous coefficients is approximated using Fers approximations of the associated underlying linear system. Given an admissible error @e > 0, the order n of Fers truncation is determined so that in the previously guaranteed existence interval, the error of the approximated solution is less than @e. Some qualitative properties of the Fers approximations are given and illustrative examples are included.

Magnus integrators for solving linear-quadratic differential games

We consider Magnus integrators to solve linear-quadratic N-player differential games. These problems require to solve, backward in time, non-autonomous matrix Riccati differential equations which are coupled with the linear differential equations for the dynamic state of the game, to be integrated forward in time. We analyze different Magnus integrators which can provide either analytical or numerical approximations to the equations. They can be considered as time-averaging methods and frequently are used as exponential integrators. We show that they preserve some of the most relevant qualitative properties of the solution for the matrix Riccati differential equations as well as for the remaining equations. The analytical approximations allow us to study the problem in terms of the parameters involved. Some numerical examples are also considered which show that exponential methods are, in general, superior to standard methods.

New efficient numerical methods to describe the heat transfer in a solid medium

The analysis of heat conduction through a solid with heat generation leads to a linear matrix differential equation with separated boundary conditions. We present a symmetric second order exponential integrator for the numerical integration of this problem using the imbedding formulation. An algorithm to implement this explicit method in an efficient way with respect to the computational cost of the scheme is presented. This method can also be used for nonlinear boundary value problems if the quasilinearization technique is considered. Some numerical examples illustrate the performance of this method.

Efficient numerical integration of N th-order non-autonomous linear differential equations

We consider the numerical integration of high-order linear non-homogeneous differential equations, written as first order homogeneous linear equations, and using exponential methods. Integrators like Magnus expansions or commutator-free methods belong to the class of exponential methods showing high accuracy on stiff or oscillatory problems, but the computation of the exponentials or their action on vectors can be computationally costly. The first order differential equations to be solved present a special algebraic structure (associated with the companion matrix) which allows to build new methods (hybrid methods between Magnus and commutator-free methods). The new methods are of similar accuracy as standard exponential methods with a reduced complexity. Additional parameters can be included into the scheme for optimization purposes. We illustrate how these methods can be obtained and present several sixth-order methods which are tested in several numerical experiments.

A second order numerical method for solving advection-diffusion models

The space-time conservation element and solution element (CE-SE) scheme is a method that improves the well-established methods, like finite differences or finite elements: the integral form of the problem exploits the physical properties of conservation of flow, unlike the differential form. Also, this explicit scheme evaluates the variable and its derivative simultaneously in each knot of the partitioned domain. The CE-SE method can be used for solving the advection-diffusion equation. In this paper, a new numerical method for solving the advection-diffusion equation, based in the CE-SE method is developed. This method increases the spatial precision and it is validated with an analytical solution.

A stable numerical method for solving variable coefficient advection-diffusion models

In a recent paper [E. Defez, R. Company, E. Ponsoda, L. Jodar, Aplicacion del metodo CE-SE a la ecuacion de adveccion-difusion con coeficientes variables, Congreso de Metodos Numericos en Ingeniera (SEMNI), Granada, Spain, 2005] a modified space-time conservation element and solution element scheme for solving the advection-diffusion equation with time-dependent coefficients, is proposed. This equation appears in many physical and technological models like gas flow in industrial tubes, conduction of heat in solids or the evaluation of the heating through radiation of microwaves when the properties of the media change with time. This method improves the well-established methods, like finite differences or finite elements: the integral form of the problem exploits the physical properties of conservation of flow, unlike the differential form. Also, this explicit scheme evaluates the variable and its derivative simultaneously in each knot of the partitioned domain. The modification proposed in [E. Defez, R. Company, E. Ponsoda, L. Jodar, Aplicacion del metodo CE-SE a la ecuacion de adveccion-difusion con coeficientes variables, Congreso de Metodos Numericos en Ingeniera (SEMNI), Granada, Spain, 2005] with regard the original method [S.C. Chang, The method of space-time conservation element and solution element. A new approach for solving the Navier-Stokes and Euler equations, J. Comput. Phys. 119 (1995) 295-324] consists of keeping the variable character of the coefficients in the solution element, without considering the linear approximation. In this paper the stability of the proposed method is studied and a CFL condition is obtained.