Generalized Black-Scholes Equation Research Articles

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

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 Garding 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.

Cubic Spline Method for a Generalized Black-Scholes Equation

We develop a numerical method based on cubic polynomial spline approximations to solve a a generalized Black-Scholes equation. We apply the implicit Euler method for the time discretization and a cubic polynomial spline method for the spatial discretization. We show that the matrix associated with the discrete operator is an M-matrix, which ensures that the scheme is maximum-norm stable. It is proved that the scheme is second-order convergent with respect to the spatial variable. Numerical examples demonstrate the stability, convergence, and robustness of the scheme.

A robust nonuniform B-spline collocation method for solving the generalized Black-Scholes equation

A robust nonuniform B-spline collocation method for solving the generalized Black-Scholes equation

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 G\"{a}rding inequality rather than the classical coercivity. A fitted finite volume element space approximation is constructed. It is proved that the time \theta-weighted full discretization is uniquely solvable and positivity-preserving. Numerical experiments, performed to illustrate the usefulness of the method, are presented.

Finite Difference Methods for Option Pricing under Lévy Processes: Wiener‐Hopf Factorization Approach

In the paper, we consider the problem of pricing options in wide classes of Lévy processes. We propose a general approach to the numerical methods based on a finite difference approximation for the generalized Black-Scholes equation. The goal of the paper is to incorporate the Wiener-Hopf factorization into finite difference methods for pricing options in Lévy models with jumps. The method is applicable for pricing barrier and American options. The pricing problem is reduced to the sequence of linear algebraic systems with a dense Toeplitz matrix; then the Wiener-Hopf factorization method is applied. We give an important probabilistic interpretation based on the infinitely divisible distributions theory to the Laurent operators in the correspondent factorization identity. Notice that our algorithm has the same complexity as the ones which use the explicit-implicit scheme, with a tridiagonal matrix. However, our method is more accurate. We support the advantage of the new method in terms of accuracy and convergence by using numerical experiments.

Multiscale methods for the valuation of American options with stochastic volatility

This paper deals with the efficient valuation of American options. We adopt Heston's approach for a model of stochastic volatility, leading to a generalized Black–Scholes equation called Heston's equation. Together with appropriate boundary conditions, this can be formulated as a parabolic boundary value problem with a free boundary, the optimal exercise price of the option. For its efficient numerical solution, we employ, among other multiscale methods, a monotone multigrid method based on linear finite elements in space and display corresponding numerical experiments.

Spherical Harmonics Applied to Differential and Integro-Differential Equations Arising in Mathematical Finance

This paper is devoted to extend the spherical harmonics technique to the solution of parabolic differential equations and to integro-differential equations. The heat equation and the Black–Scholes equation are solved by using the method of spherical harmonics. We also discuss the Black–Scholes equation in annular domains, and generalized Black–Scholes equations. Finally we solve some integro-differential equation arising in financial models with jumps by using the method of spherical harmonics.

A cubic B-spline collocation method for a numerical solution of the generalized Black–Scholes equation

In this paper, the uniform cubic B-spline collocation method is implemented to find the numerical solution of the generalized Black–Scholes partial differential equation. We use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of a θ -method, θ ∈ [ 1 / 2 , 1 ] ( θ = 1 corresponds to the back-ward Euler method and θ = 1 / 2 corresponds to the Crank–Nicolson method), and a cubic B-spline collocation method on uniform meshes, respectively. The method corresponding to θ = 1 is shown to be unconditionally stable and first order accurate with respect to the time variable and second order accurate with respect to the space variable while the method corresponding to θ = 1 / 2 is shown to be unconditionally stable and second order accurate with respect to both the variables. Finally, the numerical examples demonstrate the stability and accuracy of the method.

A robust and accurate finite difference method for a generalized Black–Scholes equation

In this paper we present a numerical method for a generalized Black–Scholes equation, which is used for option pricing. The method is based on a central difference spatial discretization on a piecewise uniform mesh and an implicit time stepping technique. Our scheme is stable for arbitrary volatility and arbitrary interest rate, and is second-order convergent with respect to the spatial variable. Furthermore, the present paper efficiently treats the singularities of the non-smooth payoff function. Numerical results support the theoretical results.

Transformation Methods for Evaluating Approximations to the Optimal Exercise Boundary for Linear and Nonlinear Black-Scholes Equations

The purpose of this survey chapter is to present a transformation technique that can be used in analysis and numerical computation of the early exercise boundary for an American style of vanilla options that can be modelled by class of generalized Black-Scholes equations. We analyze qualitatively and quantitatively the early exercise boundary for a linear as well as a class of nonlinear Black-Scholes equations with a volatility coefficient which can be a nonlinear function of the second derivative of the option price itself. A motivation for studying the nonlinear Black-Scholes equation with a nonlinear volatility arises from option pricing models taking into account e.g. nontrivial transaction costs, investor's preferences, feedback and illiquid markets effects and risk from a volatile (unprotected) portfolio. We present a method how to transform the free boundary problem for the early exercise boundary position into a solution of a time depending nonlinear nonlocal parabolic equation defined on a fixed domain. We furthermore propose an iterative numerical scheme that can be used in order to find an approximation of the free boundary. In the case of a linear Black-Scholes equation we are able to derive a nonlinear integral equation for the position of the free boundary. We present results of numerical approximation of the early exercise boundary for various types of linear and nonlinear Black-Scholes equations and we discuss dependence of the free boundary on model parameters. Finally, we discuss an application of the transformation method for the pricing equation for American type of Asian options.

Option pricing theory for financial assets with memory

Max Planck was an early proponent of the use of units based on fundamental constants, his particular system included one involving the Planck constant which emerged from his work on black-body radiation. In this review we discuss a number of such systems and how they have a strong interplay with modern metrology and the SI system of units.

Delta hedged option valuation with underlying non-Gaussian returns

The standard Black–Scholes theory of option pricing is extended to cope with underlying return fluctuations described by general probability distributions. A Langevin process and its related Fokker–Planck equation are devised to model the market stochastic dynamics, allowing us to write and formally solve the generalized Black–Scholes equation implied by dynamical hedging. A systematic expansion around a non-perturbative starting point is then implemented, recovering the Matacz's conjectured option pricing expression. We perform an application of our formalism to the real stock market and find clear evidence that while past financial time series can be used to evaluate option prices before the expiry date with reasonable accuracy, the stochastic character of volatility is an essential ingredient that should necessarily be taken into account in analytical option price modeling.

Estimation of local volatilities in a generalized Black–Scholes model

This paper studies a parameter estimation problem for a generalized Black–Scholes equation, which is used for option pricing. In estimating the volatility function from a set of market observations, we use an implicit finite difference scheme. The function space parameter estimation convergence (FSPEC) is proved and numerical simulations were performed.

ANALYTICAL PRICING OF DOUBLE-BARRIER OPTIONS UNDER A DOUBLE-EXPONENTIAL JUMP DIFFUSION PROCESS: APPLICATIONS OF LAPLACE TRANSFORM

We derive explicit formulas for pricing double (single) barrier and touch options with time-dependent rebates assuming that the asset price follows a double-exponential jump diffusion process. We also consider incorporating time-dependent volatility. Assuming risk-neutrality, the value of a barrier option satisfies the generalized Black–Scholes equation with the appropriate boundary conditions. We take the Laplace transform of this equation in time and solve it explicitly. Option price and risk parameters are computed via the numerical inversion of the corresponding solution. Numerical examples reveal that the pricing formulas are easy to implement and they result in accurate prices and risk parameters. Proposed formulas allow fast computing of smile-consistent prices of barrier and touch options.