Radoslav L. Valkov

Fitted finite volume method for a generalized Black---Scholes equation transformed on finite interval

A generalized Black–Scholes equation is considered on the semi-axis. It is transformed on the interval (0, 1) in order to make the computational domain finite. The new parabolic operator degenerates at the both ends of the interval and we are forced to use the Gärding inequality rather than the classical coercivity. A fitted finite volume element space approximation is constructed. It is proved that the time

A numerical approach for the American call option pricing model

\theta

A Computational Scheme for a Problem in the Zero‐coupon Bond Pricing

-weighted full discretization is uniquely solvable and positivity-preserving. Numerical experiments, performed to illustrate the usefulness of the method, are presented.

Finite-volume difference scheme for the Black-Scholes equation in stochastic volatility models

We present a numerical approach of the free boundary problem for the Black-Scholes equation for pricing the American call option on stocks paying a continuous dividend. A fixed domain transformation of the free boundary problem into a parabolic equation defined on a fixed spatial domain is performed. As a result a nonlinear time-dependent term is involved in the resulting equation. Two iterative numerical algorithms are proposed. Computational experiments, confirming the accuracy of the algorithms are discussed.

Convergence of a finite volume element method for a generalized Black-Scholes equation transformed on finite interval

In this paper we derive a finite volume difference scheme for a degenerate parabolic equation with dynamical boundary conditions of zero‐coupon bond pricing. We show that the system matrix of the discretization scheme is an M‐matrix, so that the discretization is monotone. This provides the non‐negativity of the price whit respect to time if the initial distribution is nonnegative. Then one can prove convergence of the numerical solution with rate of convergence O(h), where h denotes the mesh parameter [2]. Several numerical experiments show higher accuracy with comparison of known difference schemes near the boundary (degeneration).

Petrov-Galerkin analysis for a degenerate parabolic equation in zero-coupon bond pricing

We study numerically the two-dimensional Black-Scholes equation in stochastic volatility models [3]. For these models, starting from the conservative form of the equation, we construct a finite-volume difference scheme using the appropriate boundary conditions. The scheme is first order accurate in the space grid size. We also present some results from numerical experiments that confirm this.

A TWO‐LEVEL SECOND‐ORDER FINITE DIFFERENCE SCHEME FOR THE SINGLE TERM STRUCTURE EQUATION

In this paper we present a convergence analysis of a positivity-preserving fitted finite volume element method (FVEM) for a generalized Black-Scholes equation transformed on finite interval, degenerating on both boundary points. We first formulate the FVEM as a Petrov-Galerkin finite element method using a spatial discretization, previously proposed by the author. The Gärding coercivity of the corresponding discrete bilinear form is established. We obtain stability and error bounds for the solution of the fully-discrete system. Analysis of the impact of the finite domain transformation on the numerical solution of the original problem is given.

Numerical solution of one‐phase Stefan problem for a non‐classical heat equation

A degenerate parabolic equation in the zero-coupon bond pricing (ZCBP) is studied. First, we analyze the time discretization of the equation. Involving weighted Sobolev spaces, we develop a variational analysis to describe qualitative properties of the solution. On each time-level we formulate a Petrov-Galerkin FEM, in which each of the basis functions of the trial space is determined by the finite volume difference scheme in [2, 3]. Using this formulation, we establish the stability of the method with respect to a discrete energy norm and show that the error of the numerical solution in the energy norm is O(h), where h denotes the mesh parameter.

A Positivity-Preserving Splitting Method for 2D Black-Scholes Equations in Stochastic Volatility Models

In the paper [6] the classical single factor term structure equation for models that predict non‐negative interest rates is numerically studied. For these models the authors proposed a second order accurate three‐level finite difference scheme (FDs) using the appropriate boundary conditions at zero. For the same problem we propose a two‐level second‐order accurate FDs. We also propose an effective algorithm for solving the difference schemes, for which also follows the positivity of the numerical solution. The flexibility of our FDs makes it easy to change the drift and diffusion terms in the model. The numerical experiments confirm the second‐order of accuracy of the scheme and the positivity‐convexity property.

Finite volume difference scheme for a degenerate parabolic equation in the zero-coupon bond pricing

We study numerically a free‐boundary value problem for a non‐classical heat equation. The paper considers the so‐called boundary immobilization method. The case of a region that initially has zero thickness is considered. We compare three difference scheme approximations: semi‐implicit, Crank‐ Nicolson and Keller‐box scheme. Numerical experiments are also discussed.